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LiftandProject Integrality Gaps for the Traveling Salesperson Problem
, 2011
"... We study the liftandproject procedures of LovászSchrijver and SheraliAdams applied to the standard linear programming relaxation of the traveling salesperson problem with triangle inequality. For the asymmetric TSP tour problem, Charikar, Goemans, and Karloff (FOCS 2004) proved that the integral ..."
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We study the liftandproject procedures of LovászSchrijver and SheraliAdams applied to the standard linear programming relaxation of the traveling salesperson problem with triangle inequality. For the asymmetric TSP tour problem, Charikar, Goemans, and Karloff (FOCS 2004) proved that the integrality gap of the standard relaxation is at least 2. We prove that after one round of the LovászSchrijver or SheraliAdams procedures, the integrality gap of the asymmetric TSP tour problem is at least 3/2, with a small caveat on which version of the standard relaxation is used. For the symmetric TSP tour problem, the integrality gap of the standard relaxation is known to be at least 4/3, and Cheung (SIOPT 2005) proved that it remains at least 4/3 after o(n) rounds of the LovászSchrijver procedure, where n is the number of nodes. For the symmetric TSP path problem, the integrality gap of the standard relaxation is known to be at least 3/2, and we prove that it remains at least 3/2 after o(n) rounds of the LovászSchrijver procedure, by a simple reduction to Cheung’s result. 1
Improved NPinapproximability for 2variable linear equations (full version)
, 2014
"... An instance of the 2Lin(2) problem is a system of equations of the form “xi + xj = b (mod 2)”. Given such a system in which it’s possible to satisfy all but an fraction of the equations, we show it is NPhard to satisfy all but a C fraction of equations, for any C < 118 = 1.375 (and any 0 < ..."
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An instance of the 2Lin(2) problem is a system of equations of the form “xi + xj = b (mod 2)”. Given such a system in which it’s possible to satisfy all but an fraction of the equations, we show it is NPhard to satisfy all but a C fraction of equations, for any C < 118 = 1.375 (and any 0 < ≤ 18). The previous best result, standing for over 15 years, had 54 in place of 118. Our result provides the best known NPhardness even for the UniqueGames problem, and it also holds for the special case of MaxCut. The precise factor 118 is unlikely to be best possible; we also give a conjecture concerning analysis of Boolean functions which, if true, would yield a larger hardness factor of 32. Our proof is by a modified gadget reduction from a pairwiseindependent predicate. We also show an inherent limitation to this type of gadget reduction. In particular, any such reduction can never establish a hardness factor C greater than 2.54. Previously, no such limitations on gadget reductions was known.
Linear Programming Hierarchies Suffice for Directed Steiner Tree
"... Abstract. We demonstrate that ` rounds of the SheraliAdams hierarchy and 2 ` rounds of the LovászSchrijver hierarchy suffice to reduce the integrality gap of a natural LP relaxation for Directed Steiner Tree in `layered graphs from Ω( k) to O( ` · log k) where k is the number of terminals. Th ..."
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Abstract. We demonstrate that ` rounds of the SheraliAdams hierarchy and 2 ` rounds of the LovászSchrijver hierarchy suffice to reduce the integrality gap of a natural LP relaxation for Directed Steiner Tree in `layered graphs from Ω( k) to O( ` · log k) where k is the number of terminals. This is an improvement over Rothvoss ’ result that 2 ` rounds of the considerably stronger Lasserre SDP hierarchy reduce the integrality gap of a similar formulation to O( ` · log k). We also observe that Directed Steiner Tree instances with 3 layers of edges have only an O(log k) integrality gap in the standard LP relaxation, complementing the known fact that the gap can be as large as Ω( k) in graphs with 4 layers. 1