Samuli Leppänen

On the Hardest Problem Formulations for the 0/1 Lasserre Hierarchy

The Lasserre/Sum-of-Squares (SoS) hierarchy is a systematic procedure for constructing a sequence of increasingly tight semidefinite relaxations. It is known that the hierarchy converges to the 0/1 polytope in n levels and captures the convex relaxations used in the best available approximation algorithms for a wide variety of optimization problems. In this paper we characterize the set of 0/1 integer linear problems and unconstrained 0/1 polynomial optimization problems that can still have an integrality gap at level n-1. These problems are the hardest for the Lasserre hierarchy in this sense.

Improved Approximation for the Maximum Duo-Preservation String Mapping Problem

In this paper we present improved approximation results for the max duo-preservation string mapping problem (MPSM) introduced in [Chen et al., Theoretical Computer Science, 2014] that is complementary to the well-studied min common string partition problem (MCSP). When each letter occurs at most k times in each string the problem is denoted by k-MPSM. First, we prove that k-MPSM is APX-Hard even when k = 2. Then, we improve on the previous results by devising two distinct algorithms: the first ensures approximation ratio 8/5 for k = 2 and ratio 3 for k = 3, while the second guarantees approximation ratio 4 for any bigger value of k. Finally, we address the approximation of constrained maximum induced subgraph (CMIS, a generalization of MPSM, also introduced in [Chen et al., Theoretical Computer Science, 2014]), and improve the best known 9-approximation for 3-CMIS to a 6-approximation, by using a configuration LP to get a better linear relaxation. We also prove that such a linear program has an integrality gap of k, which suggests that no constant approximation (i.e. independent of k) can be achieved through rounding techniques.

Tight Sum-Of-Squares Lower Bounds for Binary Polynomial Optimization Problems

We give two results concerning the power of the Sum-of-Squares(SoS)/Lasserre hierarchy. For binary polynomial optimization problems of degree

A Lasserre Lower Bound for the Min-Sum Single Machine Scheduling Problem

2d

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

and an odd number of variables

Sum-of-squares rank upper bounds for matching problems

n

Sum-of-Squares Rank Upper Bounds for Matching Problems

, we prove that

Sum-of-Squares Hierarchy Lower Bounds for Symmetric Formulations

\frac{n+2d-1}{2}

Sum-of-squares hierarchy lower bounds for symmetric formulations.

levels of the SoS/Lasserre hierarchy are necessary to provide the exact optimal value. This matches the recent upper bound result by Sakaue, Takeda, Kim and Ito. Additionally, we study a conjecture by Laurent, who considered the linear representation of a set with no integral points. She showed that the Sherali-Adams hierarchy requires

On the Lasserre/Sum-of-Squares Hierarchy with Knapsack Covering Inequalities.

n