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The complexity of finitevalued CSPs
 Institute of Informatics, University of Warsaw, Poland
, 2013
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Goldreich’s PRG: Evidence for nearoptimal polynomial stretch
, 2013
"... We explore the connection between pseudorandomness of local functions and integrality gaps for constraint satisfaction problems. Specifically, we study candidate pseudorandom generators f: {0, 1} n → {0, 1} m constructed by applying some fixed predicate P to m randomly chosen sets of input bits. Gol ..."
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We explore the connection between pseudorandomness of local functions and integrality gaps for constraint satisfaction problems. Specifically, we study candidate pseudorandom generators f: {0, 1} n → {0, 1} m constructed by applying some fixed predicate P to m randomly chosen sets of input bits. Goldreich first considered using functions of this form for cryptographic purposes. The security of these functions against LP and SDP hierarchies is related to the integrality gap of random instances of the MaxCSP problem with predicate P: If a random (highly unsatisfiable) instance “looks ” fully satisfiable to an LP or SDP, the LP or SDP cannot distinguish between the output of the PRG and a random string. For a linear number of rounds of the LS+ and SA+ hierarchies, integrality gaps are known for the MaxCSP problem with pairwiseindependent predicate P [BGMT12, TW13]. However, these works typically take m = O(n), whereas for our application to PRGs, we would prefer to take m = n 1+Ω(1) to get PRGs with polynomial stretch. We show integrality gaps for instances with n 1+Ω(1) constraints and further show integrality gaps for instances with twise independent predicates such that m increases with t. In particular, if we consider random instances, we get integrality gap instances with Ω(n t/2+1/6−ɛ) constraints for both the SA+ and LS+ hierarchies after n Ω(1) rounds. If we allow the deletion of a small number of constraints, we obtain an integrality gap instance with Ω(n t/2+1/2−ɛ) constraints. This result is, in a sense, optimal as random planted instances of twise independent CSPs with Õ(n t+1 2) constraints can be solved efficiently. These gap instances can then be used as PRGs with polynomial stretch that are secure against nΩ(1) rounds of SA+ and LS+. 1
Local Distribution and the Symmetry Gap: Approximability of Multiway Partitioning Problems
"... We study the approximability of multiway partitioning problems, examples of which include Multiway Cut, Nodeweighted Multiway Cut, and Hypergraph Multiway Cut. We investigate these problems from the point of view of two possible generalizations: as MinCSPs, and as Submodular Multiway Partition prob ..."
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We study the approximability of multiway partitioning problems, examples of which include Multiway Cut, Nodeweighted Multiway Cut, and Hypergraph Multiway Cut. We investigate these problems from the point of view of two possible generalizations: as MinCSPs, and as Submodular Multiway Partition problems. These two generalizations lead to two natural relaxations that we call respectively the Local Distribution LP, and the Lovász relaxation. The Local Distribution LP is generally stronger than the Lovász relaxation, but applicable only to MinCSP with predicates of constant size. The relaxations coincide in some cases such as Multiway Cut where they are both equivalent to the CKR relaxation. We show that the Lovász relaxation gives a (2 − 2/k)approximation
The complexity of valued constraint satisfaction
 Bulletin of the EATCS
"... We survey recent results on the broad family of problems that can be cast as valued constraint satisfaction problems. We discuss general methods for analysing the complexity of such problems, give examples of tractable cases, and identify general features of the complexity landscape. 1 ..."
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We survey recent results on the broad family of problems that can be cast as valued constraint satisfaction problems. We discuss general methods for analysing the complexity of such problems, give examples of tractable cases, and identify general features of the complexity landscape. 1
Inapproximability Reductions and Integrality Gaps
, 2013
"... In this thesis we prove intractability results for several well studied problems in combinatorial optimization. Closest ..."
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In this thesis we prove intractability results for several well studied problems in combinatorial optimization. Closest
The Approximability of Learning and Constraint Satisfaction Problems
, 2010
"... International Business Machine. The views and conclusions contained in this document are those of the ..."
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International Business Machine. The views and conclusions contained in this document are those of the
Pricing Loss Leaders can be Hard
"... Abstract: Consider the problem of pricing n items under an unlimited supply with m single minded buyers, each of which is interested in at most k of the items. The goal is to price each item with profit margin p1, p2,..., pn so as to maximize the overall profit. There is an O(k)approximation algori ..."
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Abstract: Consider the problem of pricing n items under an unlimited supply with m single minded buyers, each of which is interested in at most k of the items. The goal is to price each item with profit margin p1, p2,..., pn so as to maximize the overall profit. There is an O(k)approximation algorithm by [BB06] when the price on each item must be above its margin cost; i.e., each pi> 0. We investigate the above problem when the seller is allowed to price some of the items below their margin cost. It was shown in [BB06, BBCH08] that by pricing some of the items below cost, the seller could possibly increase the maximum profit by Ω(log n) times. These items sold at low prices to stimulate other profitable sales are usually called “loss leader”. It is unclear what kind of approximation guarantees are achievable when some of the items can be priced below cost. Understanding this question is posed as an open problem in [BB06]. In this paper, we give a strong negative result for the problem of pricing loss leaders. We prove that assuming the Unique Games Conjecture (UGC) [Kho02], there is no constant approximation algorithm for item pricing with prices below cost allowed even when each customer is interested in at most 3 items. Conceptually, our result indicates that although it is possible to make more money by selling some items below their margin cost, it can be computationally intractable to do so.