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Convex Relaxations and Integrality Gaps
"... We discuss the effectiveness of linear and semidefinite relaxations in approximating the optimum for combinatorial optimization problems. Various hierarchies of these relaxations, such as the ones defined by Lovász and Schrijver [47], Sherali and Adams [55] and Lasserre [42] generate increasingly st ..."
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We discuss the effectiveness of linear and semidefinite relaxations in approximating the optimum for combinatorial optimization problems. Various hierarchies of these relaxations, such as the ones defined by Lovász and Schrijver [47], Sherali and Adams [55] and Lasserre [42] generate increasingly strong linear and semidefinite programming relaxations starting from a basic one. We survey some positive applications of these hierarchies, where their use yields improved approximation algorithms. We also discuss known lower bounds on the integrality gaps of relaxations arising from these hierarchies, demonstrating limits on the applicability of such hierarchies for certain optimization problems.
Constant Factor Lasserre Integrality Gaps for Graph Partitioning Problems
"... Partitioning the vertices of a graph into two roughly equal parts while minimizing the number of edges crossing the cut is a fundamental problem (called Balanced Separator) that arises in many settings. For this problem, and variants such as the Uniform Sparsest Cut problem where the goal is to mini ..."
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Partitioning the vertices of a graph into two roughly equal parts while minimizing the number of edges crossing the cut is a fundamental problem (called Balanced Separator) that arises in many settings. For this problem, and variants such as the Uniform Sparsest Cut problem where the goal is to minimize the fraction of pairs on opposite sides of the cut that are connected by an edge, there are large gaps between the known approximation algorithms and nonapproximability results. While no constant factor approximation algorithms are known, even APXhardness is not known either. In this work we prove that for balanced separator and uniform sparsest cut, semidefinite programs from the Lasserre hierarchy (which are the most powerful relaxations studied in the literature) have an integrality gap bounded away from 1, even for Ω(n) levels of the hierarchy. This complements recent algorithmic results in [GS11] which used the Lasserre hierarchy to give an approximation scheme for these problems (with runtime depending on the spectrum of the graph). Along the way, we make an observation that simplifies the task of lifting “polynomial constraints ” (such as the global balance constraint in balanced separator) to higher levels of the Lasserre hierarchy. We also obtain a similar result for Max Cut and prove that even linear number of levels of the Lasserre hierarchy have an integrality gap exceeding 18/17 − o(1), though in this case there are known NPhardness results with this gap.
Hypercontractive inequalities via SOS, and the Frankl–Rödl graph
, 2013
"... Our main result is a formulation and proof of the reverse hypercontractive inequality in the sumofsquares (SOS) proof system. As a consequence we show that for any constant 0 < γ ≤ 1/4, the ⌉ certifies the statement “the maximum independent set in SOS/Lasserre SDP hierarchy at degree 4 ⌈ 1 4γ th ..."
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Our main result is a formulation and proof of the reverse hypercontractive inequality in the sumofsquares (SOS) proof system. As a consequence we show that for any constant 0 < γ ≤ 1/4, the ⌉ certifies the statement “the maximum independent set in SOS/Lasserre SDP hierarchy at degree 4 ⌈ 1 4γ the Frankl–Rödl graph FR n γ has fractional size o(1)”. Here FR n γ = (V, E) is the graph with V = {0, 1} n and (x, y) ∈ E whenever ∆(x, y) = (1 − γ)n (an even integer). In particular, we show the degree4 SOS algorithm certifies the chromatic number lower bound “χ(FR n 1/4) = ω(1)”, even though FR n 1/4 is the canonical integrality gap instance for which standard SDP relaxations cannot even certify “χ(FR n 1/4)> 3”. Finally, we also give an SOS proof of (a generalization of) the sharp (2, q)hypercontractive inequality for any even integer q. 1
ON KHOT’S UNIQUE GAMES CONJECTURE
, 2011
"... Abstract. In 2002, Subhash Khot formulated the Unique Games Conjecture, a conjecture about the computational complexity of certain optimization problems. The conjecture has inspired a remarkable body of work, which has clarified the computational complexity of several optimization problems and the e ..."
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Abstract. In 2002, Subhash Khot formulated the Unique Games Conjecture, a conjecture about the computational complexity of certain optimization problems. The conjecture has inspired a remarkable body of work, which has clarified the computational complexity of several optimization problems and the effectiveness of “semidefinite programming ” convex relaxations. In this paper, which assumes no prior knowledge of computational complexity, we describe the context and statement of the conjecture, and we discuss in some detail one specific line of work motivated by it. 1.
Sharpness of KKL on Schreier graphs
"... Recently, the Kahn–Kalai–Linial (KKL) Theorem on influences of functions on {0, 1} n was extended to the setting of functions on Schreier graphs. Specifically, it was shown that for an undirected Schreier graph Sch(G, X, U) with logSobolev constant ρ and generating set U closed under conjugation, ..."
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Recently, the Kahn–Kalai–Linial (KKL) Theorem on influences of functions on {0, 1} n was extended to the setting of functions on Schreier graphs. Specifically, it was shown that for an undirected Schreier graph Sch(G, X, U) with logSobolev constant ρ and generating set U closed under conjugation, if f: X → {0, 1} then E[f] � log(1/M[f]) · ρ · Var[f]. Here E[f] denotes the average of f’s influences, and M[f] denotes their maximum. In this work we investigate the extent to which this result is sharp. We show: – The condition that U is closed under conjugation cannot in general be eliminated. – The logSobolev constant cannot be replaced by the modified logSobolev constant. – The result cannot be improved for the Cayley graph on Sn with transpositions. – The result can be improved for the Cayley graph on Z n m with standard generators. – Talagrand’s strengthened version of KKL also holds in the Schreier graph setting: avg Iu[f] / log(1/Iu[f]) � ρ · Var[f]. u∈U