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Proving Integrality Gaps Without Knowing the Linear Program
 Theory of Computing
, 2002
"... Proving integrality gaps for linear relaxations of NP optimization problems is a difficult task and usually undertaken on a casebycase basis. We initiate a more systematic approach. We prove an integrality gap of 2o(1) for three families of linear relaxations for vertex cover, and our methods see ..."
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Cited by 63 (2 self)
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Proving integrality gaps for linear relaxations of NP optimization problems is a difficult task and usually undertaken on a casebycase basis. We initiate a more systematic approach. We prove an integrality gap of 2o(1) for three families of linear relaxations for vertex cover, and our methods seem relevant to other problems as well.
A linear round lower bound for lovaszschrijver sdp relaxations of vertex cover
 In IEEE Conference on Computational Complexity. IEEE Computer Society
, 2006
"... We study semidefinite programming relaxations of Vertex Cover arising from repeated applications of the LS+ “liftandproject ” method of Lovasz and Schrijver starting from the standard linear programming relaxation. Goemans and Kleinberg prove that after one round of LS+ the integrality gap remains ..."
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Cited by 30 (9 self)
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We study semidefinite programming relaxations of Vertex Cover arising from repeated applications of the LS+ “liftandproject ” method of Lovasz and Schrijver starting from the standard linear programming relaxation. Goemans and Kleinberg prove that after one round of LS+ the integrality gap remains arbitrarily close to 2. Charikar proves an integrality gap of 2 for a stronger relaxation that is, however, incomparable with two rounds of LS+ and is strictly weaker than the relaxation resulting from a constant number of rounds. We prove that the integrality gap remains at least 7/6 − ε after cεn rounds, where n is the number of vertices and cε> 0 is a constant that depends only on ε.
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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Cited by 20 (0 self)
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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.
Priority Algorithms for Graph Optimization Problems
 IN PROCEEDINGS OF THE SECOND WORKSHOP ON APPROXIMATION AND ONLINE ALGORITHMS
, 2005
"... We continue the study of priority or "greedylike" algorithms as initiated in [6] and as extended to graph theoretic problems in [8]. Graph theoretic problems pose some modelling problems that did not exist in the original applications of [6] and [2]. Following [8], we further clarify thes ..."
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Cited by 6 (3 self)
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We continue the study of priority or "greedylike" algorithms as initiated in [6] and as extended to graph theoretic problems in [8]. Graph theoretic problems pose some modelling problems that did not exist in the original applications of [6] and [2]. Following [8], we further clarify these concepts. In the graph theoretic setting there are several natural input formulations for a given problem and we show that priority algorithm bounds in general depend on the input formulation. We study a variety of graph problems in the context of arbitrary and restricted priority models corresponding to known "greedy algorithms".
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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Cited by 2 (2 self)
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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
LovászSchrijver Reformulation
, 2010
"... We discuss the hierarchies of linear and semidefinite programs defined by Lovász and Schrijver [29]. We describe recent progress on these hierarchies in the contexts of algorithm design, computational complexity and proof complexity. ..."
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We discuss the hierarchies of linear and semidefinite programs defined by Lovász and Schrijver [29]. We describe recent progress on these hierarchies in the contexts of algorithm design, computational complexity and proof complexity.