An unbounded Sum-of-Squares hierarchy integrality gap for a polynomially solvable problem

Abstract

In this paper we study the complexity of the Min-sum single machine scheduling problem under algorithms from the Sum-of-Squares/Lasserre hierarchy. We prove the first lower bound for this model by showing that the integrality gap is unbounded at level $\Omega (\sqrt{n})$ even for a variant of the problem that is solvable in $O(n \log n)$ time by the Moore–Hodgson algorithm, namely Min-number of late jobs. We consider a natural formulation that incorporates the objective function as a constraint and prove the result by partially diagonalizing the matrix associated with the relaxation and exploiting this characterization. To the best of our knowledge, our result provides the first example where the Sum-of-Squares hierarchy exhibits an unbounded integrality gap for a polynomially solvable problem after non-constant number of levels.

Appendix: Derivation of the SoS hierarchy

In this section we derive the formulation of the SoS hierarchy used in Sect. 2 and give the missing proofs. In our notation we follow the survey by Rothvoß [24] and we use several known derivations [17]. Let $y \in {\mathbb {R}}^{\mathcal {P}_{2t+2}(N)}$ be a vector indexed by the subsets of $\left\{ 1,...,n \right\}$ of size at most $2t+2$, and $M_{t+1}(y)$ the moment matrix of the variables y defined by $[M_{t+1}(y)]_{I,J} = y_{I \cup J}$, for I, J subsets of N such that $|I|,|J| \le t+1$. Similarly, for every constraint $\ell$ define the moment matrix of the constraint $\ell$ as $[M^\ell _t(y)]_{I,J} = \sum _{i = 1}^n A_{\ell i} y_{I \cup J \cup \left\{ i \right\} } - b_\ell y_{I \cup J}$, where $|I|,|J| \le t$.

Definition 7.1

The SoS hierarchy at level t for the set K, denoted by $\text {{SoS}}_t(K)$, is given by the following semidefinite program

$$\begin{aligned} y_\emptyset= & {} 1, \end{aligned}$$

(18)

$$\begin{aligned} M_{t+1}(y)\succeq & {} 0, \end{aligned}$$

(19)

$$\begin{aligned} M_t^\ell (y)\succeq & {} 0 \quad \text { for every constraint }\ell \end{aligned}$$

(20)

Change of variables. A point in the SoS hierarchy is given by a vector $y \in {\mathbb {R}}^{\mathcal {P}_{2t+2}(N)}$, as seen in Definition 7.1. We now change this variable to a vector that is indexed by all the subsets of N in order to obtain a useful decomposition of the moment matrix as a sum of rank-one matrices. Here it is not necessary to distinguish between the moment matrix of the variables and constraints, hence in what follows we denote a generic vector by $w \in {\mathbb {R}}^{\mathcal {P}_{2d}(N)}$, where d is either t or $t+1$.

Definition 7.2

Let $w \in {\mathbb {R}}^{\mathcal {P}_{2d}(N)}$. For every $I \in \mathcal {P}_{2d}(N)$, define a vector $w^N \in {\mathbb {R}}^{\mathcal {P}_{n}(N)}$ such that

$$\begin{aligned} w_I = \sum _{I \subseteq H \subseteq N} w^N_H \end{aligned}$$

To simplify the notation, we note that the moment matrix of the variables is structurally similar to the moment matrix of the constraints: if $z \in {\mathbb {R}}^{\mathcal {P}_{2t}(N)}$ is a vector such that $z_I = \sum _{i = 1}^n A_{\ell i} y_{I\cup \left\{ i \right\} } - b_\ell y_I$ for some $\ell$, then $[M^\ell _{t}(y)]_{I,J} = z_{I \cup J}$. Hence, the following lemma holds for the moment matrix of variables and constraints.

Lemma 7.1

Let $w \in {\mathbb {R}}^{\mathcal {P}_{2d}(N)}$, and $M \in {\mathbb {R}}^{\mathcal {P}_{d}(N)\times \mathcal {P}_{d}(N)}$ such that $M_{I,J} = w_{I \cup J}$. Then

$$\begin{aligned} M = \sum _{H \subseteq N} w^N_H Z_HZ_H^\top \end{aligned}$$

Proof

Since $M_{I,J} = w_{I \cup J}$, we have by the change of variables that

$$\begin{aligned}{}[M]_{I,J} = \sum _{I \cup J \subseteq H \subseteq N} w^N_H = \sum _{H \subseteq N} \chi _{I \cup J}(H) w^N_H \end{aligned}$$

where $\chi _{I \cup J}(H)$ is the 0-1 indicator function such that $\chi _I(H) = 1$ if and only if $I \cup J \subseteq H$. On the other hand, $[Z_HZ_H^\top ]_{I,J} = [Z_H]_I[Z_H]_J = 1$ if $I \cup J \subseteq H$, and 0 otherwise. Therefore $[Z_HZ_H^\top ]_{I,J} = \chi _{I \cup J}(H)$. $\square$

Lemma 7.2

Given $y \in {\mathbb {R}}^{\mathcal {P}_{2t+2}(N)}$, for the vector $z_I = \sum _{i = 1}^n A_{\ell i} y_{I\cup \left\{ i \right\} } - b_\ell y_I$ we have

$$\begin{aligned} z^N_{I} = g_\ell (x_I) y^N_I \end{aligned}$$

(21)

where $g_\ell (x_I) = \sum _{i = 1}^n A_{\ell i}x_i - b_\ell$ is a linear function corresponding to the constraint $\ell$, evaluated at $x_I$ such that $x_i = 1$ if $i \in I$ and $x_i = 0$ otherwise.

Proof

We need to show that this choice of $z^N_I$ yields $z_I = \sum _{I \subseteq H \subseteq N} z^N_H$. Substituting (21) yields

$$\begin{aligned} \sum _{I \subseteq H \subseteq N} z^N_H= & {} \sum _{I \subseteq H \subseteq N} g_\ell (x_H) y^N_H = \sum _{I \subseteq H \subseteq N} \left[ \sum _{i = 1}^n A_{\ell i}x_i - b_\ell \right] _{x = x_H} y^N_H \\= & {} \sum _{I \subseteq H \subseteq N}\left( \sum _{i = 1}^n \left[ A_{\ell i}x_i\right] _{x = x_H}y^N_H - b_\ell y^N_H \right) \\= & {} \sum _{I \subseteq H \subseteq N} \sum _{i = 1}^n \left[ A_{\ell i}x_i\right] _{x = x_H}y^N_H - b_\ell y_I \end{aligned}$$

Here the term $\left[ A_{\ell i}x_i\right] _{x = x_H}y^N_H$ is $A_{\ell i}y^N_H$ if $i \in H$ and 0 otherwise. Taking this into account and changing the order of the sums, the above becomes

$$\begin{aligned} \sum _{i = 1}^n \sum _{I \cup \left\{ i \right\} \subseteq H \subseteq N} A_{\ell i}y^N_H - b_\ell y_I = \sum _{i = 1}^n A_{\ell i} y_{I\cup \left\{ i \right\} } - b_\ell y_I \end{aligned}$$

which proves the claim. $\square$

The above discussion justifies Definition 2.1.