Local consistency as a reduction between constraint satisfaction problems

We study the use of local consistency methods as reductions between constraint satisfaction problems (CSPs), and promise version thereof, with the aim to classify these reductions in a similar way as the algebraic approach classifies gadget reductions between CSPs. This research is motivated by the requirement of more expressive reductions in the scope of promise CSPs. While gadget reductions are enough to provide all necessary hardness in the scope of (finite domain) non-promise CSP, in promise CSPs a wider class of reductions needs to be used.We provide a general framework of reductions, which we call consistency reductions, that covers most (if not all) reductions recently used for proving NP-hardness of promise CSPs. We prove some basic properties of these reductions, and provide the first steps towards understanding the power of consistency reductions by characterizing a fragment associated to arc-consistency in terms of polymorphisms of the template. In addition to showing hardness, consistency reductions can also be used to provide feasible algorithms by reducing to a fixed tractable (promise) CSP, for example, to solving systems of affine equations. In this direction, among other results, we describe the well-known Sherali-Adams hierarchy for CSP in terms of a consistency reduction to linear programming

Variable and value elimination in binary constraint satisfaction via forbidden patterns

Variable or value elimination in a constraint satisfaction problem (CSP) can be used in preprocessing or during search to reduce search space size. A variable elimination rule (value elimination rule) allows the polynomial-time identification of certain variables (domain elements) whose elimination, without the introduction of extra compensatory constraints, does not affect the satisfiability of an instance. We show that there are essentially just four variable elimination rules and three value elimination rules defined by forbidding generic sub-instances, known as irreducible existential patterns, in arc-consistent CSP instances. One of the variable elimination rules is the already-known Broken Triangle Property, whereas the other three are novel. The three value elimination rules can all be seen as strict generalisations of neighbourhood substitution.Comment: A full version of an IJCAI'13 paper to appear in Journal of Computer and System Sciences (JCSS

Report on BCTCS 2016: The 32nd British Colloquium for Theoretical Computer Science 22–24 March 2016, Queen’s University Belfast

Report on BCTCS 2016: The 32nd British Colloquium for Theoretical Computer Science 22–24 March 2016, Queen’s University Belfas

ON CLASSICAL AND QUANTUM CONSTRAINT SATISFACTION PROBLEMS IN THE TRIAL AND ERROR MODEL

Ph.DDOCTOR OF PHILOSOPH

Statistical Physics Approaches to Unique Games

We show how two techniques from statistical physics can be adapted to solve a variant of the notorious Unique Games problem, potentially opening new avenues towards the Unique Games Conjecture. The variant, which we call Count Unique Games, is a promise problem in which the "yes" case guarantees a certain number of highly satisfiable assignments to the Unique Games instance. In the standard Unique Games problem, the "yes" case only guarantees at least one such assignment. We exhibit efficient algorithms for Count Unique Games based on approximating a suitable partition function for the Unique Games instance via (i) a zero-free region and polynomial interpolation, and (ii) the cluster expansion. We also show that a modest improvement to the parameters for which we give results would be strong negative evidence for the truth of the Unique Games Conjecture

Variable Elimination in Binary CSP via Forbidden Patterns

International audienceA variable elimination rule allows the polynomialtime identification of certain variables whose elimination does not affect the satisfiability of an instance. Variable elimination in the constraint satisfaction problem (CSP) can be used in preprocessing or during search to reduce search space size. We show that there are essentially just four variable elimination rules defined by forbidding generic sub-instances, known as irreducible patterns, in arc-consistent CSP instances. One of these rules is the Broken Triangle Property, whereas the other three are novel

Optimal Inapproximability Results for MAX-CUT and Other 2-Variable CSPs?

In this paper we show a reduction from the Unique Games problem to the problem of approximating MAX-CUT to within a factor of αGW + ∈, for all ∈ \u3e 0; here αGW ≈ .878567 denotes the approximation ratio achieved by the Goemans-Williamson algorithm [26]. This implies that if the Unique Games Conjecture of Khot [37] holds then the Goemans-Williamson approximation algorithm is optimal. Our result indicates that the geometric nature of the Goemans-Williamson algorithm might be intrinsic to the MAX-CUT problem. Our reduction relies on a theorem we call Majority Is Stablest. This was introduced as a conjecture in the original version of this paper, and was subsequently confirmed in [45]. A stronger version of this conjecture called Plurality Is Stablest is still open, although [45] contains a proof of an asymptotic version of it. Our techniques extend to several other two-variable constraint satisfaction problems. In particular, subject to the Unique Games Conjecture, we show tight or nearly tight hardness results for MAX-2SAT, MAX-q-CUT, and MAX-2LIN(q). For MAX-2SAT we show approximation hardness up to a factor of roughly .943. This nearly matches the .940 approximation algorithm of Lewin, Livnat, and Zwick [41]. Furthermore, we show that our .943... factor is actually tight for a slightly restricted version of MAX-2SAT. For MAX-q-CUT we show a hardness factor which asymptotically (for large q) matches the approximation factor achieved by Frieze and Jerrum [25], namely 1 − 1/q + 2(ln q)/q2 . For MAX-2LIN(q) we show hardness of distinguishing between instances which are (1−∈)-satisfiable and those which are not even, roughly, (q−∈/2)-satisfiable. These parameters almost match those achieved by the recent algorithm of Charikar, Makarychev, and Makarychev [10]. The hardness result holds even for instances in which all equations are of the form xi − xj = c. At a more qualitative level, this result also implies that 1 − ∈ vs. ∈ hardness for MAX-2LIN(q) is equivalent to the Unique Games Conjecture