Abstract
The theta body of a graph, introduced by Grötschel, Lovász, and Schrijver (in 1986), is a tractable relaxation of the independentset polytope derived from the Lovász theta number. In this paper, we recursively extend the theta body, and hence the theta number, to hypergraphs. We obtain fundamental properties of this extension and relate it to the highdimensional Hoffman bound of Filmus, Golubev, and Lifshitz. We discuss two applications: trianglefree graphs and Mantel’s theorem, and bounds on the density of triangleavoiding sets in the Hamming cube.
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1 Introduction
The theta number of a graph, introduced by Lovász [18] to determine the Shannon capacity of the pentagon, is one of the founding results of semidefinite programming and has inspired numerous developments in combinatorics (see Grötschel, Lovász, and Schrijver [12, Chapter 9] and Schrijver [23, Chapter 67]), coding theory (see Schrijver [24]), and discrete geometry (see Oliveira and Vallentin [5]). It is a graph parameter that provides at the same time an upper bound for the independence number of a graph and a lower bound for the chromatic number of its complement, a result known as Lovász’s sandwich theorem. The theta number also has weighted variants, and both Lovász’s original parameter and its variants can be computed in polynomial time. To this day, the only known polynomialtime algorithms to compute a maximumweight independent set or a minimumweight coloring in a perfect graph compute the weighted theta number as a subroutine.
The sandwich theorem has a geometrical counterpart, the theta body. The theta body of a graph \(G = (V, E)\) was introduced by Grötschel, Lovász, and Schrijver [13]; it is the convex body \({{\,\textrm{TH}\,}}(G) \subseteq \mathbb {R}^V\) given by the feasible region of the optimization program defining the theta number. It contains the independentset polytope of G and is contained in the polytope defined by the clique inequalities of G. One can optimize linear functions over the theta body in polynomial time, that is, the weak optimization problem over \({{\,\textrm{TH}\,}}(G)\) can be solved in polynomial time. The theta body provides a characterization of perfect graphs: \({{\,\textrm{TH}\,}}(G)\) is a polytope, and in this case is exactly the independentset polytope, if and only if G is a perfect graph.
In this paper we extend the definition of the theta body from graphs to hypergraphs, derive fundamental properties of this extension, and discuss applications.
1.1 Independence in Hypergraphs
Let \(H = (V, E)\) be an runiform hypergraph for some integer \(r \ge 1\), so V is a finite set and \(E \subseteq \left( {\begin{array}{c}V\\ r\end{array}}\right) \), where \(\left( {\begin{array}{c}V\\ r\end{array}}\right) \) denotes the set of relement subsets of V. For \(r=2\) this gives the usual notion of a graph, while the case \(r=1\) is somewhat degenerate but convenient for inductive arguments. The complement of H is the runiform hypergraph \({\overline{H}}\) with vertex set V in which an rsubset e of V is an edge if and only if e is not an edge of H.
A set \(I \subseteq V\) is independent in H if no edge of H is contained in I. Given a weight function \(w \in \mathbb {R}^V\), the weighted independence number of H is
where \(w(I) = \sum _{x \in I} w(x)\). When \(w = \textbf{1}\) is the constantone function, \(\alpha (H, w)\) is the independence number of H, denoted simply by \(\alpha (H)\). Computing the independence number of a graph is a known NPhard problem [16] and computing its hypergraph counterpart is also NPhard.
The independentset polytope of H is the convex hull of characteristic functions of independent sets of H, namely
where \(\chi _S \in \mathbb {R}^V\) is the characteristic function of \(S \subseteq V\). The weighted independence number \(\alpha (H, w)\) can be computed by maximizing \(w^\textsf{T}f\) over \(f \in {{\,\textrm{IND}\,}}(H)\), and so optimizing over \({{\,\textrm{IND}\,}}(H)\) is an NPhard problem.
A clique of H is a set \(C \subseteq V\) such that every rsubset of C is an edge. Note that cliques of H are independent sets of \({\overline{H}}\) and vice versa. Note also that any set with fewer than r elements is both a clique and an independent set (the same happens with graphs: single vertices are both cliques and independent sets).
If C is a clique of H and if \(f \in {{\,\textrm{IND}\,}}(H)\), then \(f(C) \le r  1\). These valid inequalities for \({{\,\textrm{IND}\,}}(H)\) are called clique inequalities; they give a relaxation of the independentset polytope, namely the polytope
Clearly, \({{\,\textrm{IND}\,}}(H) \subseteq {{\,\textrm{QIND}\,}}(H) \subseteq [0, 1]^V\). The integer vectors in \({{\,\textrm{QIND}\,}}(H)\) are precisely the characteristic functions of independent sets, and so the integer hull of \({{\,\textrm{QIND}\,}}(H)\) is \({{\,\textrm{IND}\,}}(H)\).
Since cliques of H are independent sets of \({\overline{H}}\), the separation problem over \({{\,\textrm{QIND}\,}}(H)\) consists of finding a maximumweight independent set of \({\overline{H}}\), and it is therefore NPhard. As a consequence, optimizing over \({{\,\textrm{QIND}\,}}(H)\) is NPhard as well.
1.2 The Theta Body of Graphs and Hypergraphs
Grötschel, Lovász, and Schrijver [13] defined the theta body of a graph G: a convex relaxation of \({{\,\textrm{IND}\,}}(G)\) stronger than \({{\,\textrm{QIND}\,}}(G)\) over which it is possible to optimize a linear function in polynomial time.
For a symmetric matrix A, write
where \(a = {{\,\textrm{diag}\,}}A\) is the diagonal of A. The theta body of a graph \(G = (V, E)\) is
(This specific formulation was given by Lovász and Schrijver [17].) Here and elsewhere, positive semidefinite matrices are always symmetric.
The theta body is a closed and convex set satisfying
for every graph G; since \({{\,\textrm{QIND}\,}}(G)\) is bounded, the theta body is compact. Moreover, optimizing over the theta body is the same as solving a semidefinite program, and in this case this can be done to any desired precision in polynomial time using either the ellipsoid method [12, Chapter 9] or the interiorpoint method [4].
The Lovász theta number of G for a weight function \(w \in \mathbb {R}^V\) is obtained by optimizing over the theta body, namely
for \(w = \textbf{1}\) we recover the theta number as originally defined by Lovász [18], which we denote simply by \(\vartheta (G)\). Immediately we get
Our aim is to extend the definition of the theta body, and therefore of the theta number, to runiform hypergraphs for \(r \ge 3\). We do so recursively, and the base of our recursion is \(r = 1\). By taking this as the base, we can give uniform proofs without relying on what is known about the theta body of a graph. So we will always take \(r = 1\) as the base, unless this choice would lead us into trouble.
Let \(H = (V, E)\) be an runiform hypergraph for \(r \ge 2\). Given \(x \in V\), the link of x in H is the \((r1)\)uniform hypergraph \(H_x\) with vertex set
in which an \((r1)\)subset e of \(V_x\) is an edge if and only if \(e \cup \{x\}\) is an edge of H.
Given a matrix \(A \in \mathbb {R}^{V \times V}\) and \(x \in V\), let \(A_x \in \mathbb {R}^V\) denote the row of A indexed by x, that is, \(A_x(y) = A(x, y)\). If \(f:V \rightarrow \mathbb {R}\) is a function and \(U \subseteq V\) is a set, denote by f[U] the restriction of f to U.
We are now ready to give our main definition.
Definition 1.1
Let \(H = (V, E)\) be an runiform hypergraph. For \(r = 1\), the theta body of H is \({{\,\textrm{TH}\,}}(H) = {{\,\textrm{IND}\,}}(H)\). For \(r\ge 2\), the theta body of H is
where, if a link \(H_x\) is empty, no constraint is imposed on the row \(F_x\).
Since the links of an runiform hypergraph are \((r1)\)uniform hypergraphs, we have a recursive definition. When \(r = 2\), we have \({{\,\textrm{TH}\,}}(H_x) = \{0\}\) for every nonempty link, and so we recover the usual definition (2) of the theta body of a graph.
The theta number can now be extended to hypergraphs: given a weight function \(w \in \mathbb {R}^V\), the theta number of H for w is
For unit weights, we write \(\vartheta (H)\) instead of \(\vartheta (H, \textbf{1})\).
In Sect. 2, we will see how \({{\,\textrm{TH}\,}}(H)\) defined above is in many ways analogous to the theta body of a graph defined in (2). In particular, we will see in Theorem 2.1 that
and therefore \(\alpha (H, w) \le \vartheta (H, w)\) for every weight function w. Moreover, as shown in Theorem 2.5, it is possible to optimize linear functions over \({{\,\textrm{TH}\,}}(H)\) in polynomial time.
1.3 The Weighted Fractional Chromatic Number
Let \(H = (V, E)\) be an runiform hypergraph for some \(r \ge 2\). The chromatic number of H, denoted by \(\chi (H)\), is the minimum number of colors needed to color the vertices of H in such a way that no edge is monochromatic. In other words, \(\chi (H)\) is the minimum number of disjoint independent sets needed to partition the vertex set of H.
Given \(w \in \mathbb {R}_+^V\), the weighted fractional chromatic number of H is
When \(w = \textbf{1}\) is the constantone function, \(\chi ^*(H, w)\) is the fractional chromatic number, denoted simply by \(\chi ^*(H)\). Note also that k is not specified, so we may consider any number of independent sets. In this way, if \(w = \textbf{1}\) and the \(\lambda _i\) are required to be integers, then we get the chromatic number, so \(\chi ^*(H) \le \chi (H)\).
For the chromatic or weighted fractional chromatic number, the case \(r = 1\) is degenerate: if the hypergraph has an edge, then there is no coloring, hence the restriction to \(r \ge 2\).
For a graph \(G = (V, E)\) and a weight function \(w \in \mathbb {R}_+^V\), it is known [12, Chapter 9] that \(\vartheta (G, w) \le \chi ^*({\overline{G}}, w)\). (The same inequality for the chromatic number and \({w = \textbf{1}}\) was proved by Lovász [18].) Corollary 2.3 generalizes this inequality to the setting of hypergraphs: if \(H = (V, E)\) is an runiform hypergraph and \(w \in \mathbb {R}_+^V\) is a weight function, then \(\vartheta (H, w) \le (r1)\chi ^*({\overline{H}}, w)\).
1.4 The Hoffman Bound
The Lovász theta number is also related to a wellknown spectral upper bound for the independence number of regular graphs, originally due to Hoffman. If G is a dregular graph on n vertices and if \(\lambda \) is the smallest eigenvalue of its adjacency matrix, then
this upper bound for the independence number is known as the Hoffman bound.
The Hoffman bound connects spectral graph theory with extremal combinatorics, and as such has found many applications in combinatorics and theoretical computer science. Recently, it has been extended to the highdimensional setting of edgeweighted hypergraphs by Filmus, Golubev, and Lifshitz [10], who also derived interesting applications in extremal set theory.
Lovász [18, Theorem 9] showed that the theta number \(\vartheta (G)\) is always at least as good as the Hoffman bound, that is, \(\alpha (G) \le \vartheta (G) \le \lambda n/(d  \lambda )\) for every dregular graph G. In Sect. 3 we will extend Lovász’s result to hypergraphs, showing that the hypergraph theta number \(\vartheta (H)\) is also at least as good as the highdimensional Hoffman bound.
1.5 The Antiblocker of the Theta Body
A convex set \(K \subseteq \mathbb {R}^n\) is of antiblocking type if \(\emptyset \ne K \subseteq \mathbb {R}_+^n\) and if \(x \in K\) and \(0 \le y \le x\) implies that \(y \in K\). The antiblocker of K is
Note that the antiblocker of a convex set of antiblocking type is also a convex set of antiblocking type. If K is also assumed to be closed, then \(A(A(K)) = K\) (see Grötschel, Lovász, and Schrijver [12, p. 11]).
If G is a graph, then the antiblocker of \({{\,\textrm{TH}\,}}(G)\) is \({{\,\textrm{TH}\,}}({\overline{G}})\) (see Grötschel, Lovász, and Schrijver [12, Chapter 9]). This fact is essential for proving that a graph is perfect if and only if its theta body is a polytope.
The same, however, does not hold for hypergraphs in general. In Sect. 4 we will describe the antiblocker of \({{\,\textrm{TH}\,}}(H)\) explicitly, and this will lead to another relaxation of \({{\,\textrm{IND}\,}}(H)\) and corresponding bounds for the weighted independence number and the weighted fractional chromatic number.
1.6 Symmetry and Applications
When a hypergraph is highly symmetric, it is possible to greatly simplify the optimization problem giving the theta number, as we explore in Sect. 5.
By exploiting symmetry we are able to explicitly compute the theta number in the following two illustrative cases. In Sect. 6 we consider a family of hypergraphs related to Mantel’s theorem in extremal graph theory. In this toy example, we compute the theta number of these hypergraphs, showing that it gives a tight bound for the independence number leading to a proof of Mantel’s theorem.
In Sect. 7 we consider 3uniform hypergraphs over the Hamming cube whose edges are all triangles with a given side length in Hamming distance. We give a closed formula for the theta number, and we show numerical results supporting our conjecture (see Conjecture 7.3) that the density of such triangleavoiding sets in the Hamming cube decays exponentially fast with the dimension.
1.7 Notation
For an integer \(n\ge 1\) we write \([n] = \{1, \ldots , n\}\). For a set V and \(S \subseteq V\) we denote by \(\chi _S:V \rightarrow \mathbb {R}\) the characteristic function of S, which is defined by \(\chi _S(x) = 1\) if \(x \in S\) and \(\chi _S(x) = 0\) otherwise. If \(f:V \rightarrow \mathbb {R}\) is a function and \(S \subseteq V\), then \(f(S) = \sum _{x \in S} f(x)\). The collection of all rsubsets of V is denoted by \(\left( {\begin{array}{c}V\\ r\end{array}}\right) \).
If \(H = (V, E)\) is an runiform hypergraph, we denote by \({\overline{H}}\) the complement of H, which is the hypergraph with vertex set V and edge set \(\left( {\begin{array}{c}V\\ r\end{array}}\right) \setminus E\).
We denote by \({{\,\textrm{diag}\,}}A\) the vector giving the diagonal of a square matrix A. The trace inner product between symmetric matrices A, \(B \in \mathbb {R}^{n \times n}\) is \(\langle A, B \rangle = {{\,\textrm{tr}\,}}AB = \sum _{i,j=1}^n A_{ij} B_{ij}\). Positive semidefinite matrices are always symmetric. For a symmetric matrix A with diagonal a, we write
2 Properties of the Theta Body
Given an runiform hypergraph \(H = (V, E)\) for \(r \ge 2\), it is useful to consider the lifted version of the theta body as given in Definition 1.1, namely
Note that \({{\,\textrm{TH}\,}}(H)\) is the projection of \({{\,\textrm{LTH}\,}}(H)\) onto the subspace of diagonal matrices, being therefore a projected spectrahedron.
Theorem 2.1
If H is an runiform hypergraph, then \({{\,\textrm{TH}\,}}(H)\) is compact, convex, and satisfies
Proof
The proof proceeds by induction on r. The base case is \(r = 1\), for which the statement is easily seen to hold.
Assume \(r \ge 2\). By the induction hypothesis, the statement of the theorem holds for the theta body of every link. This implies that \({{\,\textrm{LTH}\,}}(H)\) is convex, and hence \({{\,\textrm{TH}\,}}(H)\) is convex.
Let us show that \({{\,\textrm{LTH}\,}}(H)\) is compact and, since \({{\,\textrm{TH}\,}}(H)\) is a projection of \({{\,\textrm{LTH}\,}}(H)\), it will follow that \({{\,\textrm{TH}\,}}(H)\) is compact.
Let \((F_k)_{k \ge 1}\) be a sequence of points in \({{\,\textrm{LTH}\,}}(H)\) that converges to F. Immediately we have that \(R(F)\) is positive semidefinite. Now fix \(x \in V\) and let \(a^\textsf{T}f \le \beta \) be any valid inequality for \({{\,\textrm{TH}\,}}(H_x)\). Then
and we see that \(F_x[V_x] \in F(x, x) {{\,\textrm{TH}\,}}(H_x)\), proving that \({{\,\textrm{LTH}\,}}(H)\) is closed.
To see that \({{\,\textrm{LTH}\,}}(H)\) is bounded, note that for every \(x \in V\) the \(2\times 2\) submatrix
of \(R(F)\) is positive semidefinite (where \(f = {{\,\textrm{diag}\,}}F\)), hence \(f(x)  f(x)^2 \ge 0\) and so \(F(x, x) = f(x) \le 1\) for all \(x \in V\). This implies that \({{\,\textrm{tr}\,}}F \le V\) and, since F is positive semidefinite, the Frobenius norm^{Footnote 1} of F is at most V. This finishes the proof that \({{\,\textrm{LTH}\,}}(H)\) is compact.
It remains to show that (4) holds. For the first inclusion, let \(I \subseteq V\) be an independent set. For every \(x \in V\), if \(x \in I\), then \(I \cap V_x\) is an independent set of the link \(H_x\), so by the induction hypothesis \(\chi _I[V_x] \in {{\,\textrm{TH}\,}}(H_x)\). It follows that \(\chi _I \chi _I^\textsf{T}\in {{\,\textrm{LTH}\,}}(H)\), and so \({{\,\textrm{IND}\,}}(H) \subseteq {{\,\textrm{TH}\,}}(H)\).
For the second inclusion in (4), note first that \({{\,\textrm{TH}\,}}(H) \subseteq [0, 1]^V\). Let \(C \subseteq V\) be a clique and let \(F \in {{\,\textrm{LTH}\,}}(H)\); write \(f = {{\,\textrm{diag}\,}}F\). If \(C \le r  1\), then since \(f \le \textbf{1}\) we have \(f(C) \le C \le r  1\) and we are done.
So assume \(C \ge r\). Since \(R(F)\) is positive semidefinite we have
Since \(C \ge r\), every relement subset of C is an edge of H, and so for every \(x \in C\) we know that \(C \setminus \{x\} \subseteq V_x\) is a clique of the link \(H_x\). By the induction hypothesis we know that \({{\,\textrm{TH}\,}}(H_x) \subseteq {{\,\textrm{QIND}\,}}(H_x)\), hence
Together with (5) we get \(0 \le (r1)^2  (r1)f(C)\), whence \(f(C) \le r1\) as wished. \(\square \)
As a corollary we get that \({{\,\textrm{TH}\,}}(H)\) is a formulation of \({{\,\textrm{IND}\,}}(H)\), that is, the integer hull of the theta body is the independentset polytope.
Corollary 2.2
If H is an runiform hypergraph and if \(f \in {{\,\textrm{TH}\,}}(H)\) is an integral vector, then f is the characteristic function of an independent set of H.
Proof
As \({{\,\textrm{TH}\,}}(H) \subseteq {{\,\textrm{QIND}\,}}(H) \subseteq [0, 1]^V\), we know that f is a 0–1 vector that satisfies all clique inequalities, and the conclusion follows. \(\square \)
Since \({{\,\textrm{IND}\,}}(H) \subseteq {{\,\textrm{TH}\,}}(H)\), it follows immediately from the definition (3) that \(\alpha (H, w) \le \vartheta (H, w)\) for every \(w \in \mathbb {R}^V\). What about a lower bound for the chromatic number?
For a graph \(G = (V, E)\) and \(w \in \mathbb {R}_+^V\) we also have \(\vartheta (G, w) \le \chi ^*({\overline{G}}, w)\). (Recall the definition of \(\chi ^*\) from Sect. 1.3.) A simple example shows that the same cannot be true for hypergraphs in general. Indeed, fix \(r \ge 3\) and let H be the complete runiform hypergraph on r vertices (that is, H has exactly one edge containing all of its vertices). The complement of H is the empty hypergraph. Then \(\vartheta (H) = r1\), whereas \(\chi ^*({\overline{H}}) = 1\), and the inequality fails to hold. It can, however, be extended, and this simple example shows that the extension is tight.
Corollary 2.3
If H is an runiform hypergraph and \(w \in \mathbb {R}_+^V\), then \(\alpha (H, w) \le \vartheta (H, w)\). If moreover \(r \ge 2\), then \(\vartheta (H, w) \le (r1) \chi ^*({\overline{H}}, w)\).
Proof
The first statement follows immediately from \({{\,\textrm{IND}\,}}(H) \subseteq {{\,\textrm{TH}\,}}(H)\).
The second statement follows from \({{\,\textrm{TH}\,}}(H) \subseteq {{\,\textrm{QIND}\,}}(H)\). Indeed, let \(\lambda _1\), ..., \(\lambda _k\) be nonnegative numbers and \(C_1\), ..., \(C_k\) be independent sets of \({\overline{H}}\) such that \(\lambda _1 \chi _{C_1} + \cdots + \lambda _k \chi _{C_k} = w\). If \(f \in {{\,\textrm{TH}\,}}(H)\), then f satisfies all clique inequalities, so since each \(C_i\) is a clique of H we have \(\chi _{C_i}^\textsf{T}f = f(C_i) \le r1\). Hence
and we are done. \(\square \)
Just like the theta body of a graph, the theta body of a hypergraph can be shown to be of antiblocking type (see Sect. 1.5 for background), and this leads to an inequality description of the theta body in terms of the theta number.
Theorem 2.4
If \(H = (V, E)\) is an runiform hypergraph, then \({{\,\textrm{TH}\,}}(H)\) is of antiblocking type and \({{\,\textrm{TH}\,}}(H) = \{\, f \in \mathbb {R}_+^V: w^\textsf{T}f \le \vartheta (H, w)\) for all \(w \in \mathbb {R}_+^V\, \}\).
Proof
We proceed by induction on r. The statement is immediate for the base case \(r = 1\), so assume \(r \ge 2\). We claim: if \(w \in \mathbb {R}^V\) and \(w_+(x) = \max \{0, w(x)\}\) for all \(x \in V\), then \(\vartheta (H, w) = \vartheta (H, w_+)\).
Since \({{\,\textrm{TH}\,}}(H) \subseteq \mathbb {R}_+^V\), it is clear that \(\vartheta (H, w) \le \vartheta (H, w_+)\) for every \(w \in \mathbb {R}^V\); let us now prove the reverse inequality. Let \(F \in {{\,\textrm{LTH}\,}}(H)\) be a matrix such that \(w_+^\textsf{T}f = \vartheta (H, w_+)\), where \(f = {{\,\textrm{diag}\,}}F\). Let \(S = \{\, x \in V: w(x) \ge 0\,\}\) and denote by \({\bar{F}}\) the Hadamard (entrywise) product of F and \(\chi _S \chi _S^\textsf{T}\); write \({\bar{f}} = {{\,\textrm{diag}\,}}{\bar{F}}\).
Note that \(R({\bar{F}})\) is the Hadamard product of \(R(F)\) and \((1; \chi _S)(1; \chi _S)^\textsf{T}\), hence \(R({\bar{F}})\) is positive semidefinite. For every \(x \in V\) we have \(0 \le {\bar{F}}_x[V_x] \le F_x[V_x]\), and so the induction hypothesis implies that \({\bar{F}}_x[V_x] \in {\bar{F}}(x, x) {{\,\textrm{TH}\,}}(H_x)\). Hence \({\bar{F}} \in {{\,\textrm{LTH}\,}}(H)\), and \(\vartheta (H, w) \ge w^\textsf{T}{\bar{f}} = w_+^\textsf{T}f = \vartheta (H, w_+)\), proving the claim.
The inequality description follows immediately. Every \(f \in {{\,\textrm{TH}\,}}(H)\) is nonnegative and satisfies \(w^\textsf{T}f \le \vartheta (H, w)\) for all \(w \in \mathbb {R}_+^V\). For the reverse inclusion note that, since \({{\,\textrm{TH}\,}}(H)\) is closed and convex,
So let \(f\ge 0\) be such that \(w^\textsf{T}f \le \vartheta (H, w)\) for all \(w \in \mathbb {R}_+^V\). For \(w \in \mathbb {R}^V\), let \(w_+\) be defined as above, so \(w_+ \ge 0\). Then, for every \(w \in \mathbb {R}^V\) we have
and we see that \(f \in {{\,\textrm{TH}\,}}(H)\).
To finish, let us show that the theta body is of antiblocking type. If \(f \in {{\,\textrm{TH}\,}}(H)\) and \(0\le g \le f\), then for every \(w \in \mathbb {R}_+^V\) we have \(w^\textsf{T}g \le w^\textsf{T}f \le \vartheta (H, w)\), and so \(g \in {{\,\textrm{TH}\,}}(H)\). \(\square \)
Finally, for every fixed \(r \ge 1\) it is possible to optimize over \({{\,\textrm{TH}\,}}(H)\) in polynomial time. More precisely, in the language of Grötschel, Lovász, and Schrijver [12, Chapter 4], we have:
Theorem 2.5
If \(r \ge 1\) is fixed, then the weak optimization problem over \({{\,\textrm{TH}\,}}(H)\) can be solved in polynomial time for every runiform hypergraph H.
Proof
The result is trivial for \(r = 1\). For graphs, that is, \(r = 2\), the statement was proven by Grötschel, Lovász, and Schrijver [12, Theorem 9.3.30], and here it is easier to take \(r = 2\) as our base case, as will become clear soon. So we assume that \(r \ge 3\) and that the weak optimization problem can be solved in polynomial time for \((r1)\)uniform hypergraphs; we want to show how to solve the weak optimization problem in polynomial time for runiform hypergraphs.
Let \(H = (V, E)\) be an runiform hypergraph. If we show that we can solve the weak optimization problem over the convex set \({{\,\textrm{LTH}\,}}(H)\), then we are done. It suffices [12, Chapter 4] to show that \({{\,\textrm{LTH}\,}}(H)\) has the required inscribed and circumscribed balls of appropriate size, and that the weak membership problem for \({{\,\textrm{LTH}\,}}(H)\) can be solved in polynomial time.
It can be easily checked that all entries of a matrix in \({{\,\textrm{LTH}\,}}(H)\) are bounded in absolute value by 1, and so the origincentered ball of radius V circumscribes the theta body. To find an inscribed ball, note that the fulldimensional convex set
is a subset of \({{\,\textrm{LTH}\,}}(H)\), so it contains a ball which is also contained in \({{\,\textrm{LTH}\,}}(H)\). (This assertion fails when H is a graph, which is why we take \(r = 2\) as the base to simplify the proof.)
Now, given a symmetric matrix \(F \in \mathbb {R}^{V \times V}\), to test whether \(F \in {{\,\textrm{LTH}\,}}(H)\) we first test whether \(R(F)\) is positive semidefinite using (for instance) Cholesky decomposition. By induction, the weak optimization problem for each link can be solved in polynomial time, hence so can the weak membership problem for each link. We then finish by calling the weak membership oracle for each link. \(\square \)
3 Relationship to the Hoffman Bound
Let G be a dregular graph on n vertices and let \(\lambda \) be the smallest eigenvalue of its adjacency matrix. Hoffman showed that
the righthand side above became know as the Hoffman bound. Hoffman never published this particular result, though he did publish a similar lower bound [15] for the chromatic number which also came to be known as the Hoffman bound; see Haemers [14] for a historical account. Lovász [18] showed that the theta number is always at least as good as the Hoffman bound and that, when the graph is edge transitive, both bounds coincide. The Hoffman bound has also been extended to certain infinite graphs, and its relation to extensions of the theta number has been studied [1].
Filmus, Golubev, and Lifshitz [10] extended the Hoffman bound to edgeweighted hypergraphs and described several applications to extremal combinatorics. Our goal in this section is to show that our extension of the theta number to hypergraphs is always at least as good as the extended Hoffman bound. We begin with the extension of Filmus, Golubev, and Lifshitz.
To simplify the presentation and to be consistent with the setup used so far, we restrict ourselves to weighted hypergraphs without loops. A weighted r uniform hypergraph is a pair \(X = (V, \mu )\) where V is a finite set, called the vertex set of the hypergraph, and \(\mu \) is a probability measure on \(\left( {\begin{array}{c}V\\ r\end{array}}\right) \). The underlying hypergraph of X is the runiform hypergraph on V whose edge set is the support of \(\mu \).
Let \(X = (V, \mu )\) be a weighted runiform hypergraph and let \(H = (V, E)\) be its underlying hypergraph. For \(i = 1\), ..., \(r1\), the measure \(\mu \) induces a probability measure \(\mu ^{(i)}\) on \(\left( {\begin{array}{c}V\\ i\end{array}}\right) \) by the following experiment: we first choose an edge e of X according to \(\mu \) and then we choose an isubset of e uniformly at random. Concretely, for \(\sigma \in \left( {\begin{array}{c}V\\ i\end{array}}\right) \) we have
Note that \(\mu ^{(1)}\) can be seen as a weight function on V. We define the independence number of X as \(\alpha (X) = \alpha (H, \mu ^{(1)})\).
Let \(X^{(i)} \subseteq \left( {\begin{array}{c}V\\ i\end{array}}\right) \) be the support of \(\mu ^{(i)}\). We may assume, without loss of generality, that \(X^{(1)} = V\), since vertices not in the support of \(\mu ^{(1)}\) are isolated and do not contribute to the independence number.
The link of \(\sigma \in X^{(i)}\) is the weighted \((ri)\)uniform hypergraph \(X_\sigma = (V, \mu _\sigma )\), where \(\mu _\sigma \) is the probability measure on \(\left( {\begin{array}{c}V\\ ri\end{array}}\right) \) defined by the following experiment: sample a random edge \(e \in \left( {\begin{array}{c}V\\ r\end{array}}\right) \) according to \(\mu \) conditioned on \(\sigma \subseteq e\) and output \(e \setminus \sigma \). We also say that \(X_\sigma \) is an i link of X. Concretely, for \(e \in \left( {\begin{array}{c}V {\setminus } \sigma \\ ri\end{array}}\right) \) we have
For a vertex \(x \in V\) we write \(X_x\) instead of \(X_{\{x\}}\) for the link of x. Note that the underlying hypergraph of \(X_x\), minus its isolated vertices, is exactly \(H_x\), the link of x in the underlying hypergraph of X, which we have used so far.
Equip \(\mathbb {R}^V\) with the inner product
for f, \(g \in \mathbb {R}^V\). Since V is the support of \(\mu ^{(1)}\), this inner product is nondegenerate. The normalized adjacency operator of X is the operator \(T_X\) on \(\mathbb {R}^V\) such that
for all \(f \in \mathbb {R}^V\). Here, \(\mu _x^{(1)} = (\mu _x)^{(1)}\) is the measure on V induced by the measure \(\mu _x\) defining the link of x. Combine (6) and (7) to get
Now use (6) and (8) to see that
for every \(x \in V\) and \(y \in V_x\). Hence
for f, \(g \in \mathbb {R}^V\). It follows at once that \(T_X\) is selfadjoint and thus has real eigenvalues.
Note that \(T_X\textbf{1}= \textbf{1}\), hence the constant one vector is an eigenvector of \(T_X\) with associated eigenvalue 1. Moreover, the largest eigenvalue of \(T_X\) is 1. Indeed, recall that if \(A \in \mathbb {R}^{n \times n}\) is a matrix and if \(\lambda \) is an eigenvalue of A, then \(\lambda  \le \Vert A\Vert _\infty = \max _{i \in [n]} \sum _{j=1}^n A_{ij}\). Since \(\Vert T_X\Vert _\infty = 1\) by construction, it follows that 1 is the largest eigenvalue of \(T_X\).
Let \(\lambda (X)\) be the smallest eigenvalue of \(T_X\), which is negative since \({{\,\textrm{tr}\,}}T_X = 0\) as is clear from (10). For \(i = 1\), ..., \(r2\), let \(\lambda _i(X)\) be the minimum possible eigenvalue of the normalized adjacency operator of any ilink of X, that is,
and set \(\lambda _0(X) = \lambda (X)\).
With this notation, the Hoffman bound of X introduced by Filmus, Golubev, and Lifshitz [10] is
Say \(G = (V, E)\) is a dregular graph and introduce on its edges the uniform probability measure, obtaining a weighted 2uniform hypergraph \(X = (V, \mu )\). In this case, \(\mu ^{(1)}\) is the uniform probability measure on V and the normalized adjacency operator \(T_X\) is simply the adjacency matrix of G divided by d. If \(\lambda \) is the smallest eigenvalue of the adjacency matrix of G, then \(\lambda (X) = \lambda / d\) and the highdimensional Hoffman bound reads
which is, up to normalization, the Hoffman bound for \(\alpha (G)\).
Filmus, Golubev, and Lifshitz showed that \(\alpha (X) \le {{\,\textrm{Hoff}\,}}(X)\) and that this bound does not change when one takes tensor powers of the hypergraph, a fact that has implications for some problems in extremal combinatorics. The next theorem relates the hypergraph theta number to the highdimensional Hoffman bound.
Theorem 3.1
If \(X = (V, \mu )\) is a weighted runiform hypergraph for some \(r \ge 2\) and if H is its underlying hypergraph, then \(\alpha (X) \le \vartheta (H, \mu ^{(1)}) \le {{\,\textrm{Hoff}\,}}(X)\).
A few remarks before the proof. The theta number is a bound for the weighted independence number, where the weights are placed on the vertices. The Hoffman bound on the other hand is defined for an edgeweighted hypergraph, and since edge weights naturally induce vertex weights, it is possible to compare it to the theta number. However, not every weight function on vertices can be derived from a weight function on edges, so in this sense the theta number applies in more general circumstances.
Moreover, even when a vertexweight function \(w:V \rightarrow \mathbb {R}_+\) can be derived from an edgeweight function, it is not clear how to efficiently find an edgeweight function that gives w and for which the Hoffman bound gives a good upper bound for \(\alpha (H, w)\). A natural idea is to compute the optimal Hoffman bound, that is, to find the edge weights inducing w for which the corresponding Hoffman bound is the smallest possible. This was proposed by Filmus, Golubev, and Lifshitz [10, Sect. 4.3], but for \(r \ge 3\) the resulting optimization problem has a nonconvex objective function, and it is not clear how to solve it efficiently. In contrast, one can always efficiently compute the theta number of a hypergraph (see Theorem 2.5), and Theorem 3.1 says that the bound so obtained will always be at least as good as the optimal Hoffman bound.
Finally, an important property of the extension of the Hoffman bound is that it is invariant under the tensor power operation, while it is unclear whether the hypergraph theta number behaves nicely under natural hypergraph products.
Proof of Theorem 3.1
By definition we have \(\alpha (X) = \alpha (H, \mu ^{(1)})\) which, by Corollary 2.3, is at most \(\vartheta (H, \mu ^{(1)})\).
The proof of the inequality \(\vartheta (H, \mu ^{(1)}) \le {{\,\textrm{Hoff}\,}}(X)\) proceeds by induction on r. The base is \(r = 2\), in which case the statement was shown by Lovász [18, Theorem 9]. (Note that the Hoffman bound is not defined for \(r = 1\), which is why we take \(r = 2\) as the base.)
So assume \(r \ge 3\). Let \(f \in {{\,\textrm{TH}\,}}(H)\) be such that \((\mu ^{(1)})^\textsf{T}f = \vartheta (H, \mu ^{(1)})\) and let \(F \in {{\,\textrm{LTH}\,}}(H)\) be a matrix such that \(f = {{\,\textrm{diag}\,}}F\). Since \(R(F)\) is positive semidefinite, by taking the Schur complement we see that \(F  f f^\textsf{T}\) is also positive semidefinite, and so
To finish the proof it then suffices to show that
Since \(F \in {{\,\textrm{LTH}\,}}(H)\), we have \(F_x[V_x] \in F(x, x) {{\,\textrm{TH}\,}}(H_x)\) for every \(x \in V\), and so
By induction, \(\vartheta (H_x, \mu _x^{(1)}) \le {{\,\textrm{Hoff}\,}}(X_x)\), hence
where \(M = \max _{x \in V} {{\,\textrm{Hoff}\,}}(X_x)\). Use (9) on the lefthand side above to get
which already looks much closer to (11).
We work henceforth on the space \(\mathbb {R}^V\) equipped with the nondegenerate inner product \((\cdot ,\cdot )\) defined above. Since F is positive semidefinite, let \(g_1\), ..., \(g_n\) be an orthonormal basis of eigenvectors of F, with associated nonnegative eigenvalues \(\lambda _1\), ..., \(\lambda _n\). We then have \(F(x, y) = \sum _{i=1}^n \lambda _i g_i(x) g_i(y)\) and
and, using (10),
Let \(\textbf{1}= v_1\), \(v_2\), ..., \(v_n\) be an orthonormal basis of eigenvectors of \(T_X\) with associated eigenvalues \(1 = \alpha _1 \ge \alpha _2 \ge \cdots \ge \alpha _n = \lambda (X)\). For every i we have
whence \(\sum _{j=2}^n (g_i, v_j)^2 = 1  (g_i, \textbf{1})^2\). It follows that
Combine this with (15) and (12) to get
By (13) this implies that
Since \(\alpha _n < 0\) and hence \(1  \alpha _n > 0\), using (14) we finally get
We are now essentially done. Indeed, \(\lambda _i(X_x) \ge \lambda _{i+1}(X)\) for all \(x \in V\) and \(i = 0\), ..., \(r2\), hence
The lefthand side above is precisely \({{\,\textrm{Hoff}\,}}(X_x)\) and the righthand side is equal to \(\lambda _0(X) + (1  \lambda _0(X)) {{\,\textrm{Hoff}\,}}(X)\). We conclude that
Since \(\alpha _n = \lambda _0(X)\) by definition, this combined with (16) gives (11), as wished. \(\square \)
We mentioned above that the Hoffman bound coincides with the theta number when the graph is edge transitive. More generally, if a weighted hypergraph and all its lowerorder links are vertex transitive, then the Hoffman bound coincides with the theta number. The proof of this assertion is an adaptation of the proof of Theorem 3.1: use the results of Sect. 5 to take an invariant matrix \(F \in {{\,\textrm{LTH}\,}}(H)\) and check that since the hypergraph and all its lowerorder links are vertex transitive, every inequality in the proof is tight.
4 The Antiblocker
Recall the definition of antiblocker from Sect. 1.5. If G is a graph, then the antiblocker of \({{\,\textrm{TH}\,}}(G)\) is \({{\,\textrm{TH}\,}}({\overline{G}})\) (see Grötschel, Lovász, and Schrijver [12, Chapter 9]). The same does not hold for hypergraphs in general. Consider, for instance, the hypergraph \(H = ([r], \{[r]\})\) and notice that \(f = \chi _{[r1]} \in {{\,\textrm{IND}\,}}(H)\) and \(g = \chi _{[r]} \in {{\,\textrm{IND}\,}}({\overline{H}})\). So \(f \in {{\,\textrm{TH}\,}}(H)\) and \(g \in {{\,\textrm{TH}\,}}({\overline{H}})\), but \(f^\textsf{T}g = r1\). Hence, for \(r \ge 3\), \({{\,\textrm{TH}\,}}({\overline{H}})\) is not the antiblocker of \({{\,\textrm{TH}\,}}(H)\).
It seems from this simple example that we are off by a factor of \(r1\), so is \(A({{\,\textrm{TH}\,}}(H)) = (r1)^{1} {{\,\textrm{TH}\,}}({\overline{H}})\)? The answer is again no, and the smallest example is the hypergraph on \(\{1, \ldots , 5\}\) with edges \(\{1, 2, 5\}\), \(\{1, 3, 4\}\), and \(\{2, 3, 4\}\).
To describe the antiblocker of the theta body, we start by defining an alternative theta number inspired by the dual of the theta number for graphs. For a number \(\lambda \) and a symmetric matrix A with diagonal a, write
Given an runiform hypergraph H for \(r \ge 2\) and a weight function \(w \in \mathbb {R}_+^V\), denote by \(\vartheta ^\circ (H, w)\) both the semidefinite program below and its optimal value:
where \({\overline{V}}_x\) is the vertex set of the link of x in \({\overline{H}}\).
Now define
This is a nonempty, closed, and convex set of antiblocking type contained in \([0, 1]^V\). Indeed, to see that \({{\,\mathrm{TH^\circ }\,}}(H) \subseteq [0, 1]^V\), fix \(x \in V\) and let \(w = \chi _{\{x\}}\). Then \(\lambda = 1\) and \(Z = \chi _{\{x\}} \chi _{\{x\}}^\textsf{T}\) is a feasible solution of \(\vartheta ^\circ (H, w)\), whence \(\vartheta ^\circ (H, w) \le 1\), implying that \(g(x) = w^\textsf{T}g \le 1\) for every \(g \in {{\,\mathrm{TH^\circ }\,}}(H)\), as we wanted.
Theorem 4.1
If \(H = (V, E)\) is an runiform hypergraph for \(r \ge 2\), then:

(1)
\(\vartheta ^\circ (H, w) = \max \{\, w^\textsf{T}g: g \in {{\,\mathrm{TH^\circ }\,}}(H)\,\}\) for every \(w \in \mathbb {R}_+^V\);

(2)
\(\vartheta (H, l) \vartheta ^\circ ({\overline{H}}, w) \ge l^\textsf{T}w\) for every l, \(w \in \mathbb {R}_+^V\);

(3)
\(A({{\,\textrm{TH}\,}}(H)) = {{\,\mathrm{TH^\circ }\,}}({\overline{H}})\).
If G is a graph, then \(A({{\,\textrm{TH}\,}}(G)) = {{\,\textrm{TH}\,}}({\overline{G}})\), and hence \({{\,\textrm{TH}\,}}(G) = {{\,\mathrm{TH^\circ }\,}}(G)\). For runiform hypergraphs with \(r \ge 3\), this is no longer always the case.
Proof
The proof of (1) will require the following facts:

(i)
if \(w \in \mathbb {R}_+^V\) and \(\alpha \ge 0\), then \(\vartheta ^\circ (H, \alpha w) = \alpha \vartheta ^\circ (H, w)\);

(ii)
if w, \(w' \in \mathbb {R}_+^V\), then \(\vartheta ^\circ (H, w + w') \le \vartheta ^\circ (H, w) + \vartheta ^\circ (H, w')\);

(iii)
if w, \(w' \in \mathbb {R}_+^V\) and \(w' \le w\), then \(\vartheta ^\circ (H, w') \le \vartheta ^\circ (H, w)\).
To show (i), note that if \((\lambda , Z)\) is a feasible solution of \(\vartheta ^\circ (H, w)\), then \((\alpha \lambda , \alpha Z)\) is a feasible solution of \(\vartheta ^\circ (H, \alpha w)\). For (ii), simply take feasible solutions for w and \(w'\) and note that their sum is a feasible solution for \(w + w'\). For (iii), we show that if \(w' \le w\) differs from w in a single entry \(x \in V\), then the inequality holds; by applying this result repeatedly, we then get (iii).
Indeed, fix \(x \in V\) and let \((\lambda , Z)\) be an optimal solution of \(\vartheta ^\circ (H, w)\). If \({\bar{Z}}\) is the Hadamard product of Z and \((\textbf{1} \chi _{\{x\}})(\textbf{1} \chi _{\{x\}})^\textsf{T}\), then \((\lambda , {\bar{Z}})\) is a feasible solution of \(\vartheta ^\circ (H, {\bar{w}})\), where \({\bar{w}}(x) = 0\) and \({\bar{w}}(y) = w(y)\) for \(y \ne x\). By taking convex combinations of \((\lambda , {\bar{Z}})\) and \((\lambda , Z)\), we then see that \(\vartheta ^\circ (H, w') \le \lambda = \vartheta ^\circ (H, w)\) for every \(w'\) such that \(0 \le w'(x) \le w(x)\) and \(w'(y) = w(y)\) for \(y \ne x\).
Back to (1), suppose \(\max \{\, w^\textsf{T}g: g \in {{\,\mathrm{TH^\circ }\,}}(H)\,\} < \vartheta ^\circ (H, w)\). Since \({{\,\mathrm{TH^\circ }\,}}(H)\) is a compact set, Theorem 8.1 from Appendix 1 gives us a function \(y:\mathbb {R}_+^V \rightarrow \mathbb {R}_+\), of finite support, such that
and together with (i), (ii), and (iii) we get a contradiction.
To see (2), fix l, \(w \in \mathbb {R}_+^V\) and let \((\lambda , Z)\) be an optimal solution of \(\vartheta ^\circ ({\overline{H}}, w)\). If \(w = 0\), then the result is immediate, so assume \(w \ne 0\) and therefore \(\vartheta ^\circ ({\overline{H}}, w) > 0\). Then \(\lambda ^{1} Z \in {{\,\textrm{LTH}\,}}(H)\) so \(\lambda ^{1} w \in {{\,\textrm{TH}\,}}(H)\) and
proving (2).
To finish, we prove that if \(f \in {{\,\textrm{TH}\,}}(H)\) and \(g \in {{\,\mathrm{TH^\circ }\,}}({\overline{H}})\), then \(f^\textsf{T}g \le 1\), as (3) then follows by using Lehman’s lengthwidth inequality^{Footnote 2} together with (1) and (2). So take \(f \in {{\,\textrm{TH}\,}}(H)\) and \(g \in {{\,\mathrm{TH^\circ }\,}}({\overline{H}})\). Let \(A \in {{\,\textrm{LTH}\,}}(H)\) be such that \(\vartheta (H, g) = g^\textsf{T}a\), where \(a = {{\,\textrm{diag}\,}}A\). Note that \(\lambda = 1\) and \(Z = A\) is a feasible solution of \(\vartheta ^\circ ({\overline{H}}, a)\), so \(\vartheta ^\circ ({\overline{H}}, a) \le 1\). Hence
and we are done. \(\square \)
The antiblocker offers another relaxation of the independentset polytope: we have the following analogue of Theorem 2.1 and Corollary 2.3.
Theorem 4.2
If H is an runiform hypergraph for \(r \ge 2\), then
and \((r1)^{1} \alpha (H, w) \le \vartheta ^\circ (H, w) \le \chi ^*({\overline{H}}, w)\) for every \(w \in \mathbb {R}_+^V\).
Proof
The antiblocker of \({{\,\textrm{IND}\,}}(H)\) is \((r1)^{1} {{\,\textrm{QIND}\,}}({\overline{H}})\) (see Theorem 9.4 in Schrijver [25]). Since also \(A(\alpha K) = \alpha ^{1}A(K)\) for every convex set of antiblocking type K and \(\alpha > 0\), we get (18) directly from Theorems 2.1 and 4.1.
It follows that \((r1)^{1}\alpha (H, w) \le \vartheta ^\circ (H, w)\). The proof of \(\vartheta ^\circ (H, w) \le \chi ^*({\overline{H}}, w)\) is a straightforward modification of the proof of Corollary 2.3. \(\square \)
5 Exploiting Symmetry
When a hypergraph is highly symmetric, the optimization problem over the theta body or its lifted counterpart can be significantly simplified. We enter the realm of invariant semidefinite programs, a topic which has been thoroughly explored in the last decade [2]. In this section, we discuss the aspects of the general theory that are most relevant to our applications.
Let V be a finite set and let \(\Gamma \) be a finite group that acts on V. The action of \(\Gamma \) extends naturally to a function \(f \in \mathbb {R}^V\): given \(\sigma \in \Gamma \) we define
Similarly, the action extends to matrices \(A \in \mathbb {R}^{V \times V}\) by setting
for every \(\sigma \in \Gamma \). We say that \(f\in \mathbb {R}^V\) is \(\Gamma \) invariant if \(\sigma f = f\) for all \(\sigma \in \Gamma \). We define \(\Gamma \)invariant matrices likewise.
An automorphism \(\sigma \) of a hypergraph \(H = (V, E)\) is a permutation of V that preserves edges: if \(e \in E\), then \(\sigma e \in E\). The set of all automorphisms forms a group under function composition, called the automorphism group of H and denoted by \({{\,\textrm{Aut}\,}}(H)\).
Let \(H = (V, E)\) be an runiform hypergraph for \(r \ge 2\). Consider first the optimization problem over \({{\,\textrm{LTH}\,}}(H)\): given \(w \in \mathbb {R}_+^V\), we want to find
If \(\Gamma \subseteq {{\,\textrm{Aut}\,}}(H)\) is a group and w is \(\Gamma \)invariant, then when solving the optimization problem above we may restrict ourselves to \(\Gamma \)invariant matrices F.
Indeed, for \(x \in V\), let \(H_x = (V_x, E_x)\) be the link of x. Since \(\Gamma \subseteq {{\,\textrm{Aut}\,}}(H)\), for every \(x \in V\) and every \(\sigma \in \Gamma \) we have that \(V_{\sigma x} = \sigma V_x\) and \(E_{\sigma x} = \sigma E_x\), hence \(H_{\sigma x} = \sigma H_x\). It follows that \({{\,\textrm{TH}\,}}(H_{\sigma x}) = \sigma {{\,\textrm{TH}\,}}(H_x)\), where the action of \(\sigma \) maps the function \(f:V_x \rightarrow \mathbb {R}\) to the function \(\sigma f:V_{\sigma x} \rightarrow \mathbb {R}\) by \((\sigma f)(\sigma y) = f(y)\) for \(y \in V_x\).
This implies that, if \(F \in {{\,\textrm{LTH}\,}}(H)\), then \(\sigma F \in {{\,\textrm{LTH}\,}}(H)\) for every \(\sigma \in \Gamma \). Since w is invariant, the objective values of F and \(\sigma F\) coincide for every \(\sigma \in \Gamma \). Use the convexity of \({{\,\textrm{LTH}\,}}(H)\) to conclude that, if \(F \in {{\,\textrm{LTH}\,}}(H)\), then
also belongs to \({{\,\textrm{LTH}\,}}(H)\). Now \({\bar{F}}\) is \(\Gamma \)invariant and has the same objective value as F, hence when solving (19) we can restrict ourselves to \(\Gamma \)invariant matrices.
If \(\Gamma \) is a large group, this restriction allows us to simplify (19) considerably using standard techniques [2]. The case when \(\Gamma \) acts transitively on V is of particular interest to us.
Theorem 5.1
If \(H = (V, E)\) is an runiform hypergraph for \(r \ge 2\) and if \(\Gamma \subseteq {{\,\textrm{Aut}\,}}(H)\) acts transitively on V, then the optimal value of (19) for \(w = \textbf{1}\) is equal to the optimal value of the problem
where \(x_0 \in V\) is any fixed vertex and J is the allones matrix.
Proof
Note that \(w = \textbf{1}\) is \(\Gamma \)invariant, so when solving (19) we can restrict ourselves to \(\Gamma \)invariant matrices. Let F be a \(\Gamma \)invariant feasible solution of (19) and set \(A = V (\textbf{1}^\textsf{T}f)^{1} F\), where \(f = {{\,\textrm{diag}\,}}F\).
Since \(\Gamma \) acts transitively, all diagonal entries of F are equal, hence A is a feasible solution of (20). Now \(R(F)\) is positive semidefinite, and hence the Schur complement \(F  f f^\textsf{T}\) is also positive semidefinite. So
and we see that the optimal value of (20) is at least that of (19).
For the reverse inequality, let A be a feasible solution of (20). Since the action of \(\Gamma \) is transitive, we immediately get that \(A(x, x) = 1\) for all \(x \in V\); it is a little more involved, though mechanical, to verify that \(A_x[V_x] \in {{\,\textrm{TH}\,}}(H_x)\) for all \(x \in V\).
So set \(F = V^{2} \langle J, A\rangle A\) and \(f = {{\,\textrm{diag}\,}}F\); note that \(f = V^{2} \langle J, A\rangle \textbf{1}\). Since \(\textbf{1}^\textsf{T}f = V^{1} \langle J, A\rangle \), if we show that F is a feasible solution of (19), then we are done, and to show that F is feasible for (19) it suffices to show that \(R(F)\) is positive semidefinite.
This in turn can be achieved by showing that the Schur complement \(F  f f^\textsf{T}\) is positive semidefinite. Indeed, note that since A is \(\Gamma \)invariant, the constant vector \(\textbf{1}\) is an eigenvector of A with eigenvalue \(V^{1} \langle J, A\rangle \). Hence \(\textbf{1}\) is an eigenvector of both F and \(f f^\textsf{T}\) with the same eigenvalue; since all other eigenvalues of \(f f^\textsf{T}\) are zero and F is positive semidefinite, we are done. \(\square \)
Symmetry also simplifies testing whether a given vector is in the theta body.
Theorem 5.2
Let \(H = (V, E)\) be an runiform hypergraph with \(r \ge 2\) and let \(\Gamma \subseteq {{\,\textrm{Aut}\,}}(H)\) be a group. A \(\Gamma \)invariant vector \(f \in \mathbb {R}^V\) is in \({{\,\textrm{TH}\,}}(H)\) if and only if \(f \ge 0\) and \(w^\textsf{T}f \le \vartheta (H, w)\) for every \(\Gamma \)invariant \(w \in \mathbb {R}_+^V\).
Proof
Necessity being trivial from Theorem 2.4, let us prove sufficiency. If \(w\in \mathbb {R}_+^V\) is any weight function, then since f is \(\Gamma \)invariant we have that
where \({\bar{w}} = \Gamma ^{1}\sum _{\sigma \in \Gamma } \sigma ^{1} w\). Note that \({\bar{w}}\) is \(\Gamma \)invariant.
We claim that \(\vartheta (H, {\bar{w}}) \le \vartheta (H, w)\). Indeed, since \({\bar{w}}\) is \(\Gamma \)invariant, let \(g \in {{\,\textrm{TH}\,}}(H)\) be a \(\Gamma \)invariant vector such that \({\bar{w}}^\textsf{T}g = \vartheta (H, {\bar{w}})\). Then \((\sigma w)^\textsf{T}g = w^\textsf{T}(\sigma ^{1} g) = w^\textsf{T}g\) for all \(\sigma \in \Gamma \), and so \(w^\textsf{T}g = {\bar{w}}^\textsf{T}g\), hence \(\vartheta (H, w) \ge {\bar{w}}^\textsf{T}g\), proving the claim.
Now use to claim to get \(w^\textsf{T}f = {\bar{w}}^\textsf{T}f \le \vartheta (H, {\bar{w}}) \le \vartheta (H, w)\), and with Theorem 2.4 we are done. \(\square \)
6 TriangleEncoding Hypergraphs and Mantel’s Theorem
In a 1910 issue of the journal Wiskundige Opgaven, published by the Royal Dutch Mathematical Society, Mantel [19] asked what perhaps turned out to be the first question of extremal graph theory; in modern terminology: how many edges can a trianglefree graph on n vertices have? The complete bipartite graph on n vertices with parts of size \(\lfloor n/2\rfloor \) and \(\lceil n/2\rceil \) is trianglefree and has \(\lfloor n^2/4\rfloor \) edges. Can we do better?
The answer came in the same issue, supplied by Mantel and several others: a trianglefree graph on n vertices has at most \(\lfloor n^2/4\rfloor \) edges. Mantel’s theorem was later generalized by Turán to \(K_r\)free graphs for \(r \ge 4\).
Given an integer \(n \ge 3\), we want to find the largest trianglefree graph on \([n] = \{1, \ldots , n\}\). So we construct a 3uniform hypergraph \(H_n = (V_n, E_n)\) as follows:

the vertices of \(H_n\) are the edges of the complete graph \(K_n\) on [n];

three vertices of \(H_n\), corresponding to three edges of \(K_n\), form an edge of \(H_n\) if they form a triangle in \(K_n\).
Independent sets of \(H_n\) thus correspond to trianglefree subgraphs of \(K_n\), and the independentset polytope of \(H_n\) coincides with the Turán polytope studied by Raymond [21]. In order to illustrate our methods we will compute the theta number \(\vartheta (H_n)\), which provides an upper bound of \(n^2/4\) for the independence number of \(H_n\). This bound, rounded down, coincides with the lower bound \(\lfloor n^2/4\rfloor \) given by the complete bipartite graph, showing that the theta number is essentially tight for this infinite family of hypergraphs. Incidentally, this gives another proof of Mantel’s theorem, though not a particularly short one.
The symmetric group \(\mathcal {S}_n\) on n elements acts on [n], and therefore on \(V_n\), and this action preserves edges of \(H_n\), hence \(\mathcal {S}_n\) is a subgroup of \({{\,\textrm{Aut}\,}}(H_n)\). The action of \(\mathcal {S}_n\) is also transitive, so we set \(x_0 = \{1, 2\}\) and use Theorem 5.1 to get
The link of \(x_0 = \{1,2\}\) is the graph with vertex set
and edge set
that is, it is a matching with \(2(n2)\) vertices and \(n2\) edges (see Figure). 1
The row \(A_{x_0}\) of an \(\mathcal {S}_n\)invariant matrix \(A \in \mathbb {R}^{V_n \times V_n}\) is invariant under the stabilizer of \(x_0\), and so \(A_{x_0}[V_{x_0}]\) is a constant function since the stabilizer acts transitively on \((V_n)_{x_0}\). Theorem 5.2 then implies that \(A_{x_0}[(V_n)_{x_0}] \in {{\,\textrm{TH}\,}}((H_n)_{x_0})\) if and only if
since \(n2 \le \alpha ((H_n)_{x_0}) \le \vartheta ((H_n)_{x_0}) \le \chi (\overline{(H_n)_{x_0}}) \le n2\).
We simplify this problem further by computing a basis of the space of \(\mathcal {S}_n\)invariant symmetric matrices in \(\mathbb {R}^{V_n \times V_n}\). The action of \(\mathcal {S}_n\) on \(V_n\) extends naturally to an action on \(V_n \times V_n\). There are three orbits of \(V_n \times V_n\) under this action, namely \(R_k = \{\, (x, y): x \cap y = 2k\, \}\) for \(k = 0\), 1, and 2. So a basis of the invariant subspace is given by the matrices \(A_k\) such that
Note that \(A_0\) is the identity matrix.
A feasible solution of (21) is then of the form
for some real numbers \(\alpha \) and \(\beta \). We see moreover that \(A(x_0, \{1,3\}) = \alpha \), and so (22) becomes \(0 \le \alpha \le 1/2\). The objective function is
For the positive semidefiniteness constraint on A, we observe that \(\{A_0, A_1, A_2\}\) is the Johnson scheme \(\mathcal {J}(n, 2)\) (see Godsil and Meagher [11, Chapter 6]). The algebra spanned by the scheme (its BoseMesner algebra) is commutative, unital, and closed under transposition; its matrices then share a common basis of eigenvectors, say \(v_1\), \(v_2\), and \(v_3\), and can therefore be simultaneously diagonalized. The eigenvalues of \(v_1\), \(v_2\), and \(v_3\) for each matrix are (see Theorem 6.5.2, ibid.):
Putting it all together, our original problem can be rewritten as
This is a linear program on two variables. Using the dual, or finding all vertices of the primal feasible region, it is easy to verify that one optimal solution is
for all \(n \ge 4\). This gives us an optimal value of \(n^2/4\), which rounded down coincides with the lower bound coming from complete bipartite graphs.
7 TriangleAvoiding Sets in the Hamming Cube
For an integer \(n \ge 1\), consider the Hamming cube \(\mathbb {H}^n = \{0,1\}^n\) equipped with the Hamming distance, which for x, \(y \in \mathbb {H}^n\) is denoted by d(x, y) and equals the number of bits in which x and y differ. A classical problem in coding theory is to determine the parameter A(n, d), which is the maximum size of a subset I of \(\mathbb {H}^n\) such that \(d(x, y) \ge d\) for all distinct x, \(y \in I\).
If we let G(n, d) be the graph with vertex set \(\mathbb {H}^n\) in which x, \(y \in \mathbb {H}^n\) are adjacent if \(d(x, y) < d\), then \(A(n, d) = \alpha (G(n, d))\). A simple variant of the Lovász theta number of G(n, d), obtained by requiring that F in (2) be nonnegative as well, then provides an upper bound for A(n, d), which is easy to compute given the abundant symmetry of G(n, d). This bound, known as the linear programming bound, was originally described by Delsarte [6]; its relation to the theta number was later discovered by McEliece, Rodemich, and Rumsey [20] and Schrijver [22].
We now consider a hypergraph analogue of this problem. Let \(s \ge 1\) be an integer. Three distinct points \(x_1\), \(x_2\), \(x_3 \in \mathbb {H}^n\) form an s triangle if \(d(x_i, x_j) = s\) for all \(i \ne j\). It is easy to show that there is an striangle in \(\mathbb {H}^n\) if and only if s is even and \(0 < s \le \lfloor 2n/3\rfloor \).
We want to find the largest size of a set of points in \(\mathbb {H}^n\) that avoids striangles. More precisely, given integers n, \(s \ge 1\), we consider the hypergraph H(n, s) whose vertex set is \(\mathbb {H}^n\) and whose edges are all striangles and we want to find its independence number. The theta number \(\vartheta (H(n, s))\) defined in (3) gives us an upper bound.
To compute \(\vartheta (H(n, s))\), start by noting that \({{\,\textrm{Iso}\,}}(\mathbb {H}^n)\), the group of isometries of \(\mathbb {H}^n\), is a subgroup of the automorphism group of \(H = H(n, s)\) and, since it acts transitively on \(\mathbb {H}^n\), we can use Theorem 5.1 to simplify our problem. To do so we choose \(x_0 = 0\).
The vertex set of the link \(H_0\) of 0 is \(\mathbb {H}^n_s\), the set of all words of weight s, the weight of a word being the number of 1 s in it; two words are adjacent in \(H_0\) if they are at distance s. The isometry group \({{\,\textrm{Iso}\,}}(\mathbb {H}^n_s)\) of \(\mathbb {H}^n_s\) is a subgroup of the automorphism group of \(H_0\).
If \(A:\mathbb {H}^n \times \mathbb {H}^n \rightarrow \mathbb {R}\) is an \({{\,\textrm{Iso}\,}}(\mathbb {H}^n)\)invariant symmetric matrix, then A(x, y) depends only on d(x, y), and so \(a = A_0[V_0]\) is a constant function. We write A(t) for the value of A(x, y) when \(d(x, y) = t\).
By Theorem 5.2, we have \(a \in {{\,\textrm{TH}\,}}(H_0)\) if and only if \(a \ge 0\) and \(w^\textsf{T}a \le \vartheta (H_0, w)\) for every \({{\,\textrm{Iso}\,}}(\mathbb {H}^n_s)\)invariant \(w \in \mathbb {R}_+^{V_0}\). Since \({{\,\textrm{Iso}\,}}(\mathbb {H}^n_s)\) acts transitively on \(\mathbb {H}^n_s\), every such invariant w is constant, and we conclude that \(A_0[V_0] \in {{\,\textrm{TH}\,}}(H_0)\) if and only if \(0 \le \mathbb {H}^n_s A(s) \le \vartheta (H_0)\).
The problem can be further simplified. A matrix \(A:\mathbb {H}^n \times \mathbb {H}^n \rightarrow \mathbb {R}\) is \({{\,\textrm{Iso}\,}}(\mathbb {H}^n)\)invariant and positive semidefinite if and only if there are numbers \(a_0\), ..., \(a_n \ge 0\) such that
where \(K^n_k\) is the Krawtchouk polynomial of degree k, normalized so \(K^n_k(0) = 1\). This polynomial can be defined on integers \(t \in \{0, \ldots ,n\}\) by the formula
If \(E_k(x, y) = K^n_k(d(x, y))\), then we have the orthogonality relations \(\langle E_k, E_l\rangle = 0\) for \(k \ne l\); see Dunkl [8].
With this characterization, and noting that \(E_0 = J\) is the allones matrix, we have
Rewriting (20), we see that \(\vartheta (H(n,s))\) is the optimal value of the problem
Here, we have omitted the constraint \(0 \le \mathbb {H}_s^n A(s)\), since it is automatically satisfied by the optimal solution.
Problem (25) has only two constraints, and so its optimal solution admits a simple expression. With \(M_K^n(s) = \min \{\, K^n_k(s): {k = 0, \dots ,~n}\,\}\) for \(s \ge 0\) we have:
Theorem 7.1
If \(n \ge 1\) is an integer and \(0 < s \le \lfloor 2n/3\rfloor \) is an even integer, then
Proof
Write \(H = H(n, s)\). By our choice of s, there are striangles in \(\mathbb {H}^n\), so \(H_0\) is a nonempty graph. Hence \(\vartheta (H_0) \le \chi (\overline{H_0}) < \mathbb {H}^n_s\), and so a feasible solution of (25) has to use some variable \(a_k\) for \(k > 0\).
To solve our problem we want to maximize \(a_0\) keeping the convex combination
below \(\mathbb {H}^n_s^{1} \vartheta (H_0)\). We cannot achieve this by using only \(a_0\), so the best way to do it is to let \(k^*\) be such that \(K^n_{k^*}(s) = M_K^n(s)\) and use only the variables \(a_0\) and \(a_{k^*}\). This leads us to the system
whose solution yields exactly (26). \(\square \)
To compute \(\vartheta (H_0)\) we again use symmetry. Let \(A:\mathbb {H}^n_s \times \mathbb {H}^n_s \rightarrow \mathbb {R}\) be a matrix. If A is \({{\,\textrm{Iso}\,}}(\mathbb {H}^n_s)\)invariant, then A(x, y) depends only on d(x, y), and so we write A(t) for the value of A(x, y) when \(d(x, y) = t\). The matrix A is \({{\,\textrm{Iso}\,}}(\mathbb {H}^n_s)\)invariant and positive semidefinite if and only if there are numbers \(a_0\), ..., \(a_s \ge 0\) such that
(note that Hamming distances in \(\mathbb {H}^n_s\) are always even), where \(Q^{n,s}_k\) is the Hahn polynomial of degree k, normalized so \(Q_k^{n,s}(0) = 1\). For an integer \(0 \le t \le s\), these polynomials are given by the formula
If \(E_k(x, y) = Q^{n,s}_k(d(x, y) / 2)\), then \(\langle E_k, E_l\rangle = 0\) whenever \(k \ne l\) (see Delsarte [7], in particular Theorem 5, and Dunkl [9]).
With this characterization, \(\langle J, A\rangle = \mathbb {H}^n_s^2 a_0\) since \(E_0 = J\). Rewriting (20), we see that \(\vartheta (H_0)\) is the optimal value of the problem
Writing \(M^n_Q(s) = \min \{\, Q^{n,s}_k(s/2): {k=0, \dots ,~s}\,\}\), we have the analogue of Theorem 7.1.
Theorem 7.2
If \(n \ge 1\) is an integer and \(0 < s \le \lfloor 2n/3\rfloor \) is an even integer, then
Proof
Adapt the proof of Theorem 7.1. \(\square \)
The upshot is that \(\vartheta (H(n,s))\) may be expressed entirely in terms of the parameters
Very similar expressions can be derived for the theta number in the more general setting of q ary cubes \(\{0, \dots , q1\}^n\) for any integer \(q \ge 2\); in this case we must use Krawtchouk polynomials with weight \((q1)/q\) (see Dunkl [8]) and qary Hahn polynomials [7].
The theta number for hypergraphs can also be extended to some wellbehaved infinite hypergraphs, and can be used in particular to provide upper bounds for the density of simplexavoiding sets on the sphere and in Euclidean space [3]. For triangleavoiding sets on the sphere \(S^{n1}\), for instance, the bound obtained is like the one in Theorems 7.1 and 7.2, with both the Krawtchouk and Hahn polynomials replaced by Gegenbauer (ultraspherical) polynomials \(P^n_k\) (resp. \(P^{n1}_k\)), which are the orthogonal polynomials on the interval \([1, 1]\) for the weight function \((1x^2)^{(n3)/2}\). In this setting, the link of a vertex \(x \in S^{n1}\) is a scaled copy of \(S^{n2}\).
This bound can be analyzed asymptotically, yielding an upper bound for the density of simplexavoiding sets that decays exponentially fast in the dimension of the underlying space. The key point in the analysis is to show exponential decay of the parameter \(M_P^n(t) = \min \{\, P^n_k(t): k \ge 0 \,\}\) for \(t \in (0,1)\). This is done in two steps. First, one uses results on the asymptotic behavior of the roots of Gegenbauer polynomials to show that \(\min \{\, P^n_k(t): k \ge 0 \,\}\) is attained at \(k = \Omega (n)\). Then, one shows that \(P^n_k(t)\) tends to 0 exponentially fast if \(k = \Omega (n)\) by exploiting a particular integral representation for the Gegenbauer polynomials [3, Lemma 4.2].
The same can be attempted for the Hamming cube: how does the density of a subset of \(\mathbb {H}^n\) that avoids striangles behave as n goes to infinity? For a fixed s, the answer is simple, since \(\mathbb {H}^n_s\) is exponentially smaller than \(\mathbb {H}^n\). We should therefore consider a regime where s and n both tend to infinity; for instance, we could take \(s = s(n, c)\) to be the even integer closest to n/c for some \(c > 1\). Numerical evidence (see Fig. 7) supports the following conjecture.
Conjecture 7.3
With s(n, c) defined as above, \(\vartheta (H(n, s(n, c))) / 2^n\) decays exponentially fast with n for every fixed \(c > 2\), whereas \(\vartheta (H(n, s(n, 2))) / 2^n\) decays linearly fast with n.
We leave open the question of whether this conjecture, for \(c > 2\), can be proven using Theorems 7.1 and 7.2. Following the strategy of CastroSilva, Oliveira, Slot, and Vallentin [3], it is possible to show that the minima in (28) and (29) are attained at \(k = \Omega (n)\), using results on the roots of Krawtchouk and Hahn polynomials. For \(c = 2\), it appears that the minimum in (28) is always attained at \(k = 2\) when n is a multiple of 4, implying in this case that \(M_n(n/2) = K_2^n(n/2) = 1/(n1)\). The remaining obstacle to finishing the analysis of the asymptotic behavior of \(M_K^n(s)\) and \(M_Q^n(s)\) is the lack of a suitable integral representation for the Krawtchouk and Hahn polynomials, as was available for the Gegenbauer polynomials.
Notes
The Frobenius norm of a symmetric matrix \(A \in \mathbb {R}^{n \times n}\) is the Euclidean norm of A considered as an \(n^2\)dimensional vector. If A has eigenvalues \(\lambda _1\), ..., \(\lambda _n\), then the square of the Frobenius norm is \(\lambda _1^2 + \cdots + \lambda _n^2\), hence the Frobenius norm is at most \({{\,\textrm{tr}\,}}A = \lambda _1 + \cdots + \lambda _n\) when all eigenvalues are nonnegative, that is, when A is positive semidefinite.
See Theorem 9.5 in Schrijver [25], where the result is stated for polyhedra, but the same proof works for convex sets as well.
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This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie SkłodowskaCurie agreement No 764759. The third author was at the Centrum Wiskunde & Informatica, Amsterdam, when this research was carried out. The fourth author is partially supported by the SFB/TRR 191 “Symplectic Structures in Geometry, Algebra and Dynamics” and by the project “Spectral bounds in extremal discrete geometry” (project number 414898050), both funded by the DFG.
Appendix A. Duality for Optimization Over Compact Convex Sets
Appendix A. Duality for Optimization Over Compact Convex Sets
In the proof of Theorem 4.1 we used a kind of linear programming dual of the optimization problem over \({{\,\mathrm{TH^\circ }\,}}(H)\) to get a contradiction, but \({{\,\mathrm{TH^\circ }\,}}(H)\) is not necessarily a polyhedron. Since it is convex and compact, however, a kind of strong duality holds. The following theorem should be known, but we could not find a suitable reference.
Theorem 8.1
let I be a set and for every \(i \in I\) let \(a_i \in \mathbb {R}^n\) and \(\beta _i \in \mathbb {R}\); write \(S = \{\, x \in \mathbb {R}_+^n: a_i^\textsf{T}x \le \beta _i\text { for }i \in I\,\}\). If S is nonempty and compact, then for every \(c \in \mathbb {R}^n\) we have that \(\max \{\, c^\textsf{T}x: x \in S\,\}\) is the optimal value of
Proof
It is easy to show that \(\max \le \inf \): if \(x \in S\) and if y is a feasible solution of (30), then
For the reverse inequality, start by observing that we may assume that I is countable. Indeed, center on each rational point in \(\mathbb {R}^{n+1}\) balls of radii 1/k for each integer \(k \ge 1\). Inside every such ball choose a point \((a_i, \beta _i)\), for \(i \in I\), if such a point exists. This gives a countable subset of I defining the same set S.
So say \(I = \{1, 2, \ldots \}\) and for an integer \(k \ge 1\) write
We claim that there is \(k_0\) such that \(S_{k_0}\) is compact. If not, then for every \(k \ge 1\) there is a nonzero \(z_k \in \mathbb {R}_+^n\) such that \(a_i^\textsf{T}z_k \le 0\) for all \(1 \le i \le k\). If we normalize these points so \(\Vert z_k\Vert = 1\) for every k, then the sequence \((z_k)\) has a converging subsequence, and we may assume that the sequence itself converges, say to a point z with \(\Vert z\Vert = 1\). Note that \(z \ge 0\). Moreover, for every \(i \ge 1\) we have
and since S is nonempty it follows that S is unbounded, a contradiction.
So for every \(k \ge k_0\) let \(x_k^*\) be an optimal solution of \(\max \{\, c^\textsf{T}x: x \in S_k\,\}\). Since \(S_{k_0}\) is bounded and \(x^*_k \in S_{k_0}\) for every \(k \ge k_0\), the sequence \((x^*_k)\) has a converging subsequence; assume the sequence itself converges to \(x^*\). Then \(x^* \ge 0\) and for \(i \ge 1\) we have
so \(x^* \in S\). Moreover, since \(c^\textsf{T}x_k^* \ge \max \{\, c^\textsf{T}x: x \in S\,\}\) for every k, we conclude that \(x^*\) is an optimal solution of \(\max \{\, c^\textsf{T}x: x \in S\,\}\).
The strong duality theorem of linear programming gives us, for every \(k \ge k_0\), a function \(y_k:I \rightarrow \mathbb {R}_+\), supported on [k], such that
Each \(y_k\) is a feasible solution of (30), and it follows that the optimal value of (30) is \(\le c^\textsf{T}x^*\). \(\square \)
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CastroSilva, D., de Oliveira Filho, F.M., Slot, L. et al. A Recursive Theta Body for Hypergraphs. Combinatorica 43, 909–938 (2023). https://doi.org/10.1007/s00493023000409
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DOI: https://doi.org/10.1007/s00493023000409
Keywords
 Hypergraph independence number
 Hypergraph chromatic number
 Lovász theta number
 Theta body
 Hoffman bound
 Semidefinite programming