Weighted constraint satisfaction with set variables.

Siu Fai Keung.Thesis (M.Phil.)--Chinese University of Hong Kong, 2006.Includes bibliographical references (leaves 79-83).Abstracts in English and Chinese.Chapter 1 --- Introduction --- p.1Chapter 1.1 --- (Weighted) Constraint Satisfaction --- p.1Chapter 1.2 --- Set Variables --- p.2Chapter 1.3 --- Motivations and Goals --- p.3Chapter 1.4 --- Overview of the Thesis --- p.4Chapter 2 --- Background --- p.6Chapter 2.1 --- Constraint Satisfaction Problems --- p.6Chapter 2.1.1 --- Backtracking Tree Search --- p.8Chapter 2.1.2 --- Consistency Notions --- p.10Chapter 2.2 --- Weighted Constraint Satisfaction Problems --- p.14Chapter 2.2.1 --- Branch and Bound Search --- p.17Chapter 2.2.2 --- Consistency Notions --- p.19Chapter 2.3 --- Classical CSPs with Set Variables --- p.23Chapter 2.3.1 --- Set Variables and Set Domains --- p.24Chapter 2.3.2 --- Set Constraints --- p.24Chapter 2.3.3 --- Searching with Set Variables --- p.26Chapter 2.3.4 --- Set Bounds Consistency --- p.27Chapter 3 --- Weighted Constraint Satisfaction with Set Variables --- p.30Chapter 3.1 --- Set Variables --- p.30Chapter 3.2 --- Set Domains --- p.31Chapter 3.3 --- Set Constraints --- p.31Chapter 3.3.1 --- Zero-arity Constraint --- p.33Chapter 3.3.2 --- Unary Constraints --- p.33Chapter 3.3.3 --- Binary Constraints --- p.36Chapter 3.3.4 --- Ternary Constraints --- p.36Chapter 3.3.5 --- Cardinality Constraints --- p.37Chapter 3.4 --- Characteristics --- p.37Chapter 3.4.1 --- Space Complexity --- p.37Chapter 3.4.2 --- Generalization --- p.38Chapter 4 --- Consistency Notions and Algorithms for Set Variables --- p.41Chapter 4.1 --- Consistency Notions --- p.41Chapter 4.1.1 --- Element Node Consistency --- p.41Chapter 4.1.2 --- Element Arc Consistency --- p.43Chapter 4.1.3 --- Element Hyper-arc Consistency --- p.43Chapter 4.1.4 --- Weighted Cardinality Consistency --- p.45Chapter 4.1.5 --- Weighted Set Bounds Consistency --- p.46Chapter 4.2 --- Consistency Enforcing Algorithms --- p.47Chapter 4.2.1 --- "Enforcing Element, Node Consistency" --- p.48Chapter 4.2.2 --- Enforcing Element Arc Consistency --- p.51Chapter 4.2.3 --- Enforcing Element Hyper-arc Consistency --- p.52Chapter 4.2.4 --- Enforcing Weighted Cardinality Consistency --- p.54Chapter 4.2.5 --- Enforcing Weighted Set Bounds Consistency --- p.56Chapter 5 --- Experiments --- p.59Chapter 5.1 --- Modeling Set Variables Using 0-1 Variables --- p.60Chapter 5.2 --- Softening the Problems --- p.61Chapter 5.3 --- Steiner Triple System --- p.62Chapter 5.4 --- Social Golfer Problem --- p.63Chapter 5.5 --- Discussions --- p.66Chapter 6 --- Related Work --- p.68Chapter 6.1 --- Other Consistency Notions in WCSPs --- p.68Chapter 6.1.1 --- Full Directional Arc Consistency --- p.68Chapter 6.1.2 --- Existential Directional Arc Consistency --- p.69Chapter 6.2 --- Classical CSPs with Set Variables --- p.70Chapter 6.2.1 --- Bounds Reasoning --- p.70Chapter 6.2.2 --- Cardinality Reasoning --- p.70Chapter 7 --- Concluding Remarks --- p.72Chapter 7.1 --- Contributions --- p.72Chapter 7.2 --- Future Work --- p.74List of Symbols --- p.76Bibliography --- p.7

The Complexity of Weighted Boolean #CSP with Mixed Signs

We give a complexity dichotomy for the problem of computing the partition function of a weighted Boolean constraint satisfaction problem. Such a problem is parameterized by a set of rational-valued functions, which generalize constraints. Each function assigns a weight to every assignment to a set of Boolean variables. Our dichotomy extends previous work in which the weight functions were restricted to being non-negative. We represent a weight function as a product of the form (-1)^s g, where the polynomial s determines the sign of the weight and the non-negative function g determines its magnitude. We show that the problem of computing the partition function (the sum of the weights of all possible variable assignments) is in polynomial time if either every weight function can be defined by a "pure affine" magnitude with a quadratic sign polynomial or every function can be defined by a magnitude of "product type" with a linear sign polynomial. In all other cases, computing the partition function is FP^#P-complete.Comment: 24 page

Optimal Polynomial-Time Compression for Boolean Max CSP

In the Boolean maximum constraint satisfaction problem - Max CSP(?) - one is given a collection of weighted applications of constraints from a finite constraint language ?, over a common set of variables, and the goal is to assign Boolean values to the variables so that the total weight of satisfied constraints is maximized. There exists a concise dichotomy theorem providing a criterion on ? for the problem to be polynomial-time solvable and stating that otherwise it becomes NP-hard. We study the NP-hard cases through the lens of kernelization and provide a complete characterization of Max CSP(?) with respect to the optimal compression size. Namely, we prove that Max CSP(?) parameterized by the number of variables n is either polynomial-time solvable, or there exists an integer d ? 2 depending on ?, such that: 1) An instance of Max CSP(?) can be compressed into an equivalent instance with ?(n^d log n) bits in polynomial time, 2) Max CSP(?) does not admit such a compression to ?(n^{d-?}) bits unless NP ? co-NP / poly. Our reductions are based on interpreting constraints as multilinear polynomials combined with the framework of constraint implementations. As another application of our reductions, we reveal tight connections between optimal running times for solving Max CSP(?). More precisely, we show that obtaining a running time of the form ?(2^{(1-?)n}) for particular classes of Max CSPs is as hard as breaching this barrier for Max d-SAT for some d

A Galois Connection for Weighted (Relational) Clones of Infinite Size

A Galois connection between clones and relational clones on a fixed finite domain is one of the cornerstones of the so-called algebraic approach to the computational complexity of non-uniform Constraint Satisfaction Problems (CSPs). Cohen et al. established a Galois connection between finitely-generated weighted clones and finitely-generated weighted relational clones [SICOMP'13], and asked whether this connection holds in general. We answer this question in the affirmative for weighted (relational) clones with real weights and show that the complexity of the corresponding valued CSPs is preserved

Algebraic Properties of Valued Constraint Satisfaction Problem

The paper presents an algebraic framework for optimization problems expressible as Valued Constraint Satisfaction Problems. Our results generalize the algebraic framework for the decision version (CSPs) provided by Bulatov et al. [SICOMP 2005]. We introduce the notions of weighted algebras and varieties and use the Galois connection due to Cohen et al. [SICOMP 2013] to link VCSP languages to weighted algebras. We show that the difficulty of VCSP depends only on the weighted variety generated by the associated weighted algebra. Paralleling the results for CSPs we exhibit a reduction to cores and rigid cores which allows us to focus on idempotent weighted varieties. Further, we propose an analogue of the Algebraic CSP Dichotomy Conjecture; prove the hardness direction and verify that it agrees with known results for VCSPs on two-element sets [Cohen et al. 2006], finite-valued VCSPs [Thapper and Zivny 2013] and conservative VCSPs [Kolmogorov and Zivny 2013].Comment: arXiv admin note: text overlap with arXiv:1207.6692 by other author