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 $2d$ and an odd number of variables $n$, we prove that $\frac{n+2d-1}{2}$ 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 $n$ levels to detect the empty integer hull, and conjectured that the SoS/Lasserre rank for the same problem is $n-1$. We disprove this conjecture and derive lower and upper bounds for the rank

Pebble Games and Linear Equations

We give a new, simplified and detailed account of the correspondence between levels of the Sherali-Adams relaxation of graph isomorphism and levels of pebble-game equivalence with counting (higher-dimensional Weisfeiler-Lehman colour refinement). The correspondence between basic colour refinement and fractional isomorphism, due to Ramana, Scheinerman and Ullman, is re-interpreted as the base level of Sherali-Adams and generalised to higher levels in this sense by Atserias and Maneva, who prove that the two resulting hierarchies interleave. In carrying this analysis further, we here give (a) a precise characterisation of the level-k Sherali-Adams relaxation in terms of a modified counting pebble game; (b) a variant of the Sherali-Adams levels that precisely match the k-pebble counting game; (c) a proof that the interleaving between these two hierarchies is strict. We also investigate the variation based on boolean arithmetic instead of real/rational arithmetic and obtain analogous correspondences and separations for plain k-pebble equivalence (without counting). Our results are driven by considerably simplified accounts of the underlying combinatorics and linear algebra

Directed Steiner Tree and the Lasserre Hierarchy

The goal for the Directed Steiner Tree problem is to find a minimum cost tree in a directed graph G=(V,E) that connects all terminals X to a given root r. It is well known that modulo a logarithmic factor it suffices to consider acyclic graphs where the nodes are arranged in L <= log |X| levels. Unfortunately the natural LP formulation has a |X|^(1/2) integrality gap already for 5 levels. We show that for every L, the O(L)-round Lasserre Strengthening of this LP has integrality gap O(L log |X|). This provides a polynomial time |X|^{epsilon}-approximation and a O(log^3 |X|) approximation in O(n^{log |X|) time, matching the best known approximation guarantee obtained by a greedy algorithm of Charikar et al.Comment: 23 pages, 1 figur

On the Integrality Gap of Directed Steiner Tree Problem

In the Directed Steiner Tree problem, we are given a directed graph G = (V,E) with edge costs, a root vertex r ∈ V, and a terminal set X ⊆ V . The goal is to find the cheapest subset of edges that contains an r-t path for every terminal t ∈ X. The only known polylogarithmic approximations for Directed Steiner Tree run in quasi-polynomial time and the best polynomial time approximations only achieve a guarantee of O(|X|^ε) for any constant ε > 0. Furthermore, the integrality gap of a natural LP relaxation can be as bad as Ω(√|X|).  We demonstrate that l rounds of the Sherali-Adams hierarchy suffice to reduce the integrality gap of a natural LP relaxation for Directed Steiner Tree in l-layered graphs from Ω( k) to O(l · log k) where k is the number of terminals. This is an improvement over Rothvoss’ result that 2l rounds of the considerably stronger Lasserre SDP hierarchy reduce the integrality gap of a similar formulation to O(l · log k). We also observe that Directed Steiner Tree instances with 3 layers of edges have only an O(logk) integrality gap bound in the standard LP relaxation, complementing the fact that the gap can be as large as Ω(√k) in graphs with 4 layers. Finally, we consider quasi-bipartite instances of Directed Steiner Tree meaning no edge in E connects two Steiner nodes V − (X ∪ {r}). By a simple reduction from Set Cover, it is still NP-hard to approximate quasi-bipartite instances within a ratio better than O(log|X|). We present a polynomial-time O(log |X|)-approximation for quasi-bipartite instances of Directed Steiner Tree. Our approach also bounds the integrality gap of the natural LP relaxation by the same quantity. A novel feature of our algorithm is that it is based on the primal-dual framework, which typically does not result in good approximations for network design problems in directed graphs

Matchings and Representation Theory

In this thesis we investigate the algebraic properties of matchings via representation theory. We identify three scenarios in different areas of combinatorial mathematics where the algebraic structure of matchings gives keen insight into the combinatorial problem at hand. In particular, we prove tight conditional lower bounds on the computational complexity of counting Hamiltonian cycles, resolve an asymptotic version of a conjecture of Godsil and Meagher in Erdos-Ko-Rado combinatorics, and shed light on the algebraic structure of symmetric semidefinite relaxations of the perfect matching proble

A Comprehensive Analysis of Lift-and-Project Methods for Combinatorial Optimization

In both mathematical research and real-life, we often encounter problems that can be framed as finding the best solution among a collection of discrete choices. Many of these problems, on which an exhaustive search in the solution space is impractical or even infeasible, belong to the area of combinatorial optimization, a lively branch of discrete mathematics that has seen tremendous development over the last half century. It uses tools in areas such as combinatorics, mathematical modelling and graph theory to tackle these problems, and has deep connections with related subjects such as theoretical computer science, operations research, and industrial engineering. While elegant and efficient algorithms have been found for many problems in combinatorial optimization, the area is also filled with difficult problems that are unlikely to be solvable in polynomial time (assuming the widely believed conjecture $\mathcal{P} \neq \mathcal{NP}$). A common approach of tackling these hard problems is to formulate them as integer programs (which themselves are hard to solve), and then approximate their feasible regions using sets that are easier to describe and optimize over. Two of the most prominent mathematical models that are used to obtain these approximations are linear programs (LPs) and semidefinite programs (SDPs). The study of these relaxations started to gain popularity during the 1960's for LPs and mid-1990's for SDPs, and in many cases have led to the invention of strong approximation algorithms for the underlying hard problems. On the other hand, sometimes the analysis of these relaxations can lead to the conclusion that a certain problem cannot be well approximated by a wide class of LPs or SDPs. These negative results can also be valuable, as they might provide insights into what makes the problem difficult, which can guide our future attempts of attacking the problem. One mathematical framework that generates strong LP and SDP relaxations for integer programs is lift-and-project methods. Among many attractive features, an important advantage of this approach is that tighter relaxations can often be obtained without sacrificing polynomial-time solvability. Also, these procedures are able to generate relaxations systematically, without relying on problem-specific observations. Thus, they can be applied to improve any given relaxation. In the past two decades, lift-and-project methods have garnered a lot of research attention. Many operators under this approach have been proposed, most notably those by Sherali and Adams; Lov{\'a}sz and Schrijver; Balas, Ceria and Cornu{\'e}jols; Lasserre; and Bienstock and Zuckerberg. These operators vary greatly both in strength and complexity, and their performances and limitations on many optimization problems have been extensively studied, with the exception of the Bienstock--Zuckerberg operator (and to a lesser degree, the Lasserre operator) in terms of limitations. In this thesis, we aim to provide a comprehensive analysis of the existing lift and project operators, as well as many new variants of these operators that we propose in our work. Our new operators fill the spectrum of lift-and-project operators in a way which makes all of them more transparent, easier to relate to each other, and easier to analyze. We provide new techniques to analyze the worst-case performances as well as relative strengths of these operators in a unified way. In particular, using the new techniques and a recent result of Mathieu and Sinclair, we prove that the polyhedral Bienstock--Zuckerberg operator requires at least $\sqrt{2n}- \frac{3}{2}$ iterations to compute the matching polytope of the $(2n+1)$-clique. We further prove that the operator requires approximately $\frac{n}{2}$ iterations to reach the stable set polytope of the $n$-clique, if we start with the fractional stable set polytope. Moreover, we obtained an example in which the Bienstock--Zuckerberg operator with positive semidefiniteness requires $\Omega(n^{1/4})$ iterations to compute the integer hull of a set contained in $[0,1]^n$. These examples provide the first known instances where the Bienstock--Zuckerberg operators require more than a constant number of iterations to return the integer hull of a given relaxation. In addition to relating the performances of various lift-and-project methods and providing results for specific operators and problems, we provide some general techniques that can be useful in producing and verifying certificates for lift-and-project relaxations. These tools can significantly simply the task of obtaining hardness results for relaxations that have certain desirable properties. Finally, we characterize some sets on which one of the strongest variants of the Sherali--Adams operator with positive semidefinite strengthenings does not perform better than Lov\'{a}sz and Schrijver's weakest polyhedral operator, providing examples where even imposing a very strong positive semidefiniteness constraint does not generate any additional cuts. We then prove that some of the worst-case instances for many known lift-and-project operators are also bad instances for this significantly strengthened version of the Sherali--Adams operator, as well as the Lasserre operator. We also discuss how the techniques we presented in our analysis can be applied to obtain the integrality gaps of convex relaxations