In the PERMUTATION CONSTRAINT SATISFACTION PROBLEM (PERMUTATION CSP) we are given a set of variables V and a set of constraints C, in which the constraints are tuples of elements of V. The goal is to find a total orde...
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In the PERMUTATION CONSTRAINT SATISFACTION PROBLEM (PERMUTATION CSP) we are given a set of variables V and a set of constraints C, in which the constraints are tuples of elements of V. The goal is to find a total ordering of the variables, pi : V -> [1,...,vertical bar V vertical bar], which satisfies as many constraints as possible. A constraint (v(1), v(2),..., v(k)) is satisfied by an ordering pi when pi (v(1)) < (pi v(2)) < ... < pi(v(k)). An instance has arity k if all the constraints involve at most k elements. This problem expresses a variety of permutation problems including FEEDBACK ARC SET and BETWEENNESS problems. A naive algorithm, listing all the n! permutations, requires 2(O(n) (log n)) time. Interestingly, PERMUTATION CSP for arity 2 or 3 can be solved by Held-Karptype algorithms in time O*(2(n)), but no algorithm is known for arity at least 4 with running time significantly better than 2(O(n log n)). In this paper we resolve the gap by showing that ARITY 4 PERMUTATION CSP cannot be solved in time 2(O(n log n)) unless the exponential time hypothesis fails. (C) 2012 Elsevier B.V. All rights reserved.
We prove an algorithmic hardness result for finding low-energy states in the so-called continuous random energy model (CREM), introduced by Bovier and Kurkova in 2004 as an extension of Derrida’s generalized random e...
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