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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
The Complexity of Valued Constraint Satisfaction
"... 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
The Complexity of FiniteValued CSPs (Extended Abstract)
, 2013
"... Let Γ be a set of rationalvalued functions on a fixed finite domain; such a set is called a finitevalued constraint language. The valued constraint satisfaction problem, VCSP(Γ), is the problem of minimising a function given as a sum of functions from Γ. We establish a dichotomy theorem with resp ..."
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Let Γ be a set of rationalvalued functions on a fixed finite domain; such a set is called a finitevalued constraint language. The valued constraint satisfaction problem, VCSP(Γ), is the problem of minimising a function given as a sum of functions from Γ. We establish a dichotomy theorem with respect to exact solvability for all finitevalued languages defined on domains of arbitrary finite size. We show that every core language Γ either admits a binary idempotent and symmetric fractional polymorphism in which case the basic linear programming relaxation solves any instance of VCSP(Γ) exactly, or Γ satisfies a simple hardness condition that allows for a polynomialtime reduction from MaxCut to VCSP(Γ). In other words, there is a single algorithm for all tractable cases and a single reason