# On Sub-determinants and the Diameter of Polyhedra

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## Abstract

We derive a new upper bound on the diameter of a polyhedron \(P = \{x {\in } {\mathbb {R}}^n :Ax\le b\}\), where \(A \in {\mathbb {Z}}^{m\times n}\). The bound is polynomial in \(n\) and the largest absolute value of a sub-determinant of \(A\), denoted by \(\Delta \). More precisely, we show that the diameter of \(P\) is bounded by \(O(\Delta ^2 n^4\log n\Delta )\). If \(P\) is bounded, then we show that the diameter of \(P\) is at most \(O(\Delta ^2 n^{3.5}\log n\Delta )\). For the special case in which \(A\) is a totally unimodular matrix, the bounds are \(O(n^4\log n)\) and \(O(n^{3.5}\log n)\) respectively. This improves over the previous best bound of \(O(m^{16}n^3(\log mn)^3)\) due to Dyer and Frieze (Math Program 64:1–16, 1994).

## Keywords

Diameter of polyhedra Polyhedral graph Total unimodularity## 1 Introduction

One of the fundamental open problems in optimization and discrete geometry is the question whether the diameter of a polyhedron can be bounded by a polynomial in the dimension and the number of its defining inequalities.The problem is readily explained: A *polyhedron* is a set of the form \(P = \{x \in {\mathbb {R}}^n :Ax\le b\}\), where \(A \in {\mathbb {R}}^{m\times n}\) is a matrix and \(b \in {\mathbb {R}}^m\) is an \(m\)-dimensional vector. A *vertex* of \(P\) is a point \(x^* \in P\) such that there exist \(n\) linearly independent rows of \(A\) whose corresponding inequalities of \(Ax\le b\) are satisfied by \(x^*\) with equality. Throughout this paper, we assume that the polyhedron \(P\) is *pointed*, i.e. it has vertices, which is equivalent to saying that the matrix \(A\) has full column-rank. Two different vertices \(x^*\) and \(y^*\) are *neighbors* if they are the endpoints of an *edge* of the polyhedron, i.e. there exist \(n-1\) linearly independent rows of \(A\) whose corresponding inequalities of \(Ax\le b\) are satisfied with equality both by \(x^*\) and \(y^*\). In this way, we obtain the undirected *polyhedral graph* with edges being pairs of neighboring vertices of \(P\). This graph is connected. The *diameter* of \(P\) is the smallest natural number that bounds the length of a shortest path between any pair of vertices in this graph. The question is now as follows:

Can the diameter of a polyhedron \(P = \{x \in {\mathbb {R}}^n :Ax\le b\}\) be bounded by a polynomial in \(m\) and \(n\)?

The belief in a positive answer to this question is called the *polynomial Hirsch conjecture*. Despite a lot of research effort during the last 50 years, the gap between lower and upper bounds on the diameter remains huge. While, when the dimension \(n\) is fixed, the diameter can be bounded by a linear function of \(m\) [2, 16], for the general case the best upper bound, due to Kalai and Kleitman [13], is \(m^{1 + \log n}\). The best lower bound is of the form \((1+\varepsilon ) \cdot m\) for some \(\varepsilon >0\) in fixed and sufficiently large dimension \(n\). This is due to a celebrated result of Santos [21] who disproved the, until then longstanding, original *Hirsch conjecture* for polytopes. The Hirsch conjecture stated that the diameter of a bounded polyhedron^{1} is at most \(m-n\). Interestingly, this huge gap (polynomial versus quasi-polynomial) is also not closed in a very simple combinatorial abstraction of polyhedral graphs [8]. However, it was shown by Vershynin [22] that every polyhedron can be perturbed by a small random amount so that the expected diameter of the resulting polyhedron is bounded by a polynomial in \(n\) and \(\ln m\). See Kim and Santos [14] for a recent survey.

In light of the importance and apparent difficulty of the open question above, many researchers have shown that it can be answered in an affirmative way in some special cases. Naddef [19] proved that the Hirsch conjecture holds true for \(0/1\)-polytopes. Orlin [20] provided a quadratic upper bound for flow-polytopes. Brightwell et al. [5] showed that the diameter of the transportation polytope is linear in \(m\) and \(n\), and a similar result holds for the dual of a transportation polytope [1] and the axial \(3\)-way transportation polytope [7].

The results on flow polytopes and classical transportation polytopes concern polyhedra defined by *totally unimodular matrices*, i.e., integer matrices whose sub-determinants are \(0,\pm 1\). For such polyhedra Dyer and Frieze [6] had previously shown that the diameter is bounded by a polynomial in \(n\) and \(m\). Their bound is \(O(m^{16}n^3(\log mn)^3)\). Their result is also algorithmic: they show that there exists a randomized simplex-algorithm that solves linear programs defined by totally unimodular matrices in polynomial time.

Our main result is a generalization and considerable improvement of the diameter bound of Dyer and Frieze. We show that the diameter of a polyhedron \(P = \{x \in {\mathbb {R}}^n :Ax\le b\}\), with \(A \in {\mathbb {Z}}^{m\times n}\) is bounded by \(O(\Delta ^2 n^4\log n\Delta )\). Here, \(\Delta \) denotes the largest absolute value of a *sub-determinant* of \(A\). If \(P\) is bounded, i.e., a *polytope*, then we can show that the diameter of \(P\) is at most \(O(\Delta ^2 n^{3.5}\log n\Delta )\). To compare our bound with the one of Dyer and Frieze one has to set \(\Delta \) above to one and obtains \(O(n^4\log n)\) and \(O(n^{3.5}\log n)\) respectively. Notice that our bound is independent of \(m\), i.e., the number of rows of \(A\).

### 1.1 The Proof Method

Let \(u\) and \(v\) be two vertices of \(P\). We estimate the maximum number of iterations of two breadth-first-search explorations of the polyhedral graph, one initiated at \(u\), the other initiated at \(v\), until a common vertex is discovered. The diameter of \(P\) is at most twice this number of iterations. The main idea in the analysis is to reason about the normal cones of vertices of \(P\) and to exploit a certain volume expansion property.

We can assume that \(P = \{x \in {\mathbb {R}}^n :Ax\le b\}\) is *non-degenerate*, i.e., each vertex has exactly \(n\) tight inequalities. This can be achieved by slightly perturbing the right-hand side vector \(b\): in this way the diameter can only grow. Notice that the polyhedron is then also full-dimensional. We denote the polyhedral graph of \(P\) by \(G_P = (V,E)\). Let \(v \in V\) now be a vertex of \(P\). The *normal cone* \(C_v\) of \(v\) is the set of all vectors \(c \in {\mathbb {R}}^n\) such that \(v\) is an optimal solution of the linear program \(\max \{c^Tx:x \in {\mathbb {R}}^n, \, Ax\le b\}\). The normal cone \(C_v\) of a vertex of \(v\) is a full-dimensional simplicial polyhedral cone. Two vertices \(v\) and \(v'\) are adjacent if and only if \(C_v\) and \(C_{v'}\) share a facet. No two distinct normal cones share an interior point. Furthermore, if \(P\) is a polytope, then the union of the normal cones of vertices of \(P\) is the complete space \({\mathbb {R}}^n\).

*volume*of a set \(U\subseteq V\) of vertices as the volume of the union of the normal cones of \(U\) intersected with the

*unit ball*\(B_n = \{ x \in {\mathbb {R}}^n :\Vert x\Vert _2\le 1\}\), i.e.,

Together with the integrality of \(A\), the bound \(\Delta \) on the subdeterminants guarantees that the angle between one facet of a normal cone \(C_v\) and the opposite ray is not too small. We combine this fact, which we formalize in Lemma 3, with an isoperimetric inequality to show that the volume of \(\mathcal {N}(I)\) is large relative to the volume of \(I\).

### **Lemma 1**

We provide the proof of this lemma in the next section. Our diameter bound for polytopes is an easy consequence:

### **Theorem 2**

Let \(P=\{x \in {\mathbb {R}}^n :Ax\le b\}\) be a polytope where all subdeterminants of \(A\in {\mathbb {Z}}^{m\times n}\) are bounded by \(\Delta \) in absolute value. The diameter of \(P\) is bounded by \(O(\Delta ^2 n^{3.5}\log n\Delta )\).

### *Proof*

We estimate the maximum number of iterations of breadth-first-search until the total volume of the discovered vertices exceeds \((1/2) \cdot {\mathrm {vol}}(B_n)\). This is an upper bound on the aforementioned maximum number of iterations of two breadth-first-search explorations until a common vertex is discovered.

### *Remark*

The result of Dyer and Frieze [6] is also based on analyzing expansion properties via isoperimetric inequalities. It is our choice of normal cones as the natural geometric representation, and the fact that we only ask for volume expansion instead of expansion of the graph itself, that allows us to get a better bound. Expansion properties of the graphs of general classes of polytopes have also been studied elsewhere in the literature, e.g. [11, 12].

### 1.2 Organization of the Paper

The next section is devoted to a proof of the volume-expansion property, i.e., Lemma 1. The main tool that is used here is a classical *isoperimetric inequality* that states that among measurable subsets of a sphere with fixed volume, spherical caps have the smallest circumference. Section 3 deals with unbounded polyhedra. Compared to the case of polytopes, the problem that arises here is the fact that the union of the normal cones is not the complete space \({\mathbb {R}}^n\). To tackle this case, we rely on an isoperimetric inequality of Lovász and Simonovits [17]. Finally, we discuss how our bound can be further generalized. In fact, not all sub-determinants of \(A\) need to be at most \(\Delta \) but merely the entries of \(A\) and the \((n-1)\)-dimensional sub-determinants have to be bounded by \(\Delta \), which yields a slightly stronger result.

## 2 Volume Expansion

*cone*is a subset of \({\mathbb {R}}^n\) that is closed under the multiplication with non-negative scalars. The intersection of a cone with the unit ball \(B_n\) is called a

*spherical cone*. Recall that \(C_v\) denotes the normal cone of the vertex \(v\) of \(P\). We denote the spherical cone \(C_v \cap B_n\) by \(S_v\) and, for a subset \(U\subseteq V\), the spherical cone \(\bigcup _{v \in U} S_v\) by \(S_U\). Our goal is to show that the following inequality holds for each \(I\subseteq V\) with \({\mathrm {vol}}(S_I) \le \frac{1}{2} {\mathrm {vol}}(B_n)\):

*dockable surface*\(D(S)\), see Fig. 1.

The *base* of \(S\) is the intersection of \(S\) with the unit sphere. We denote the area of the base by \(B(S)\). By *area* we mean the \((n-1)\)-dimensional measure of some surface. Furthermore, \(L(S)\) denotes the length of the relative boundary of the base of \(S\). We use the term *length* to denote the measure of an \((n-2)\)-dimensional volume, see Fig. 1.

### 2.1 Area to Volume Ratio of a Spherical Simplicial Cone

We will first derive inequality (4).

### **Lemma 3**

### *Proof*

### 2.2 An Isoperimetric Inequality for Spherical Cones

We now derive the lower bound (5) on the area to volume ratio for a general spherical cone. To do that, we assume that the spherical cone has the least favorable shape for the area to volume ratio and derive the inequality for cones of this shape. Here one uses classical isoperimetric inequalities. The basic isoperimetric inequality states that the measurable subset of \({\mathbb {R}}^n\) with a prescribed volume and minimal area is the ball of this volume. In this paper, we need Lévy’s isoperimetric inequality, see e.g. [10, Theorem 2.1], which can be seen as an analogous result for spheres: it states that a measurable subset of the sphere of prescribed area and minimal boundary is a spherical cap.

*cone of revolution*if there exist a vector \(v\) and an angle \(0<\theta \le \pi /2\) such that \(S\) is the set of vectors in the unit ball that form an angle of at most \(\theta \) with \(v\):

### **Lemma 4**

The spherical cone of given volume with minimum lateral surface is a cone of revolution.

### *Proof*

By the first equation of (3), every spherical cone of volume \(V\) intersects the unit sphere in a surface of area \(nV\). Furthermore, by the second equation of (3), the length of the boundary of this surface is proportional to the area of the lateral surface of the cone. Then the problem of finding the spherical cone of volume \(V\) with the minimum lateral surface can be rephrased as follows: Find a surface of area \(nV\) on the unit sphere having the boundary of minimum length. By Lévy’s isoperimetric inequality for spheres, the optimal shape for such a surface is a spherical cap. As observed above, this corresponds to a cone of revolution.\(\square \)

### **Lemma 5**

### *Proof*

Let \(H\) be the hyperplane containing the boundary of the base of \(S\). Then \(H\) divides \(S\) into two parts: a truncated cone \(K\) and the convex hull of a spherical cap. The radius \(r\) of the base of \(K\) is bounded by one.

Finally we are now ready to consider the case of an arbitrary spherical cone.

### **Lemma 6**

### *Proof*

This was the final step in the proof of Lemma 1 and thus we have also proved Theorem 2, our main result on polytopes. The next section is devoted to unbounded polyhedra.

## 3 The Case of an Unbounded Polyhedron

If the polyhedron \(P\) is unbounded, then the union of the normal cones of all vertices of \(P\) forms a proper subset \(K'\) of \(\mathbb R^n\): namely, \(K'\) is the set of objective functions \(c\) for which the linear program \(\max \{c^Tx:x\in P\}\) has finite optimum. Similarly, the set \(K'\cap B_n\) is a proper subset of \(B_n\). Then, given the union of the spherical cones that have already been discovered by the breadth-first-search (we denote this set by \(S\)), we should redefine the dockable surface of \(S\) as that part of the lateral surface of \(S\) that is shared by some neighboring cones. In other words, we should exclude the part lying on the boundary of \(K'\cap B_n\). However, this implies that Lemma 6 cannot be immediately applied.

### **Theorem 7**

We now illustrate how the above result can be used in our context. Let \(K = K'\cap B_n\) and observe that \(K\) is a convex and compact set. Let \(S\subseteq K\) be the union of the spherical cones that have already been discovered by the breadth-first-search. We define the dockable surface of \(S\) as that part of the lateral surface of \(S\) that is disjoint from the boundary of \(K\). We denote by \(D'(S)\) the area of the dockable surface of \(S\). We can prove the following analogue of Lemma 6:

### **Lemma 8**

If \({\mathrm {vol}}(S) \le \frac{1}{2} {\mathrm {vol}}(K)\), then \(D'(S)\ge {\mathrm {vol}}(S)\).

### *Proof*

Following the same approach as that used for the case of a polytope, one can show the following result for polyhedra.

### **Theorem 9**

Let \(P=\{x \in {\mathbb {R}}^n :Ax\le b\}\) be a polyhedron, where all sub-determinants of \(A\in {\mathbb {Z}}^{m\times n}\) are bounded by \(\Delta \) in absolute value. Then the diameter of \(P\) is bounded by \(O(\Delta ^2 n^4\log n\Delta )\). In particular, if \(A\) is totally unimodular, then the diameter of \(P\) is bounded by \(O(n^4\log n)\).

## 4 Remarks

### 4.1 Which Sub-Determinants Enter the Bound?

### **Theorem 10**

Let \(P=\{x \in {\mathbb {R}}^n :Ax\le b\}\) be a polyhedron, where the entries of \(A\) (respectively the sub-determinants of \(A\) of size \(n-1\)) are bounded in absolute value by \(\Delta _1\) (respectively \(\Delta _{n-1}\)). Then the diameter of \(P\) is bounded by \(O(\Delta _1\Delta _{n-1} n^4\log n\Delta _1)\). Moreover, if \(P\) is a polytope, its diameter is bounded by \(O(\Delta _1\Delta _{n-1} n^{3.5}\log n\Delta _1)\).

### 4.2 A More General Geometric Setting

Since our result was first announced in [3], Brunsch and Röglin [4] have found an algorithm to compute a short path between two given vertices of a non-degenerate polyhedron \(P = \{ x \in {\mathbb {R}}^n :Ax \le b\}\) that runs in expected polynomial time in \(n,m\) and \(1/\delta \), where \(\delta \) is a lower bound on the sine of the angle of a row of \(A\) to the subspace of \(n-1\) other rows of \(A\). The expected length of the path is \(O(m n^2 / \delta ^2)\). If \(A \in {\mathbb {Z}}^{m\times n}\), then \(\delta \ge 1/(\Delta _1 \Delta _{n-1} n)\), where \(\Delta _1\) and \(\Delta _{n-1}\) are, as before, bounds on the absolute values of \(1\times 1\) and \((n-1)\times (n-1)\) sub-determinants.

Our proof technique applies in this setting as well. We have volume expansion since the normal cones cannot be too flat. The parameter \(\delta \) is a measure for this flatness. In this setting, Lemma 1 reads as follows.

### **Lemma 11**

### **Theorem 12**

Let \(P=\{x \in {\mathbb {R}}^n :Ax\le b\}\) be a polytope where the sine of the angle of any row of \(A\) to the subspace generated by \(n-1\) other rows of \(A\) is at least \(\delta \). The diameter of \(P\) is bounded by \(O(n^{2.5}/\delta \cdot \ln (n/\delta ))\).

## Footnotes

## Notes

### Acknowledgments

This work was carried out while all authors were at EPFL (École Polytechnique Fédérale de Lausanne), Switzerland. The authors acknowledge support from the DFG Focus Program 1307 within the project “Algorithm Engineering for Real-time Scheduling and Routing” and from the Swiss National Science Foundation within the project “Set-partitioning integer programs and integrality gaps”.

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