scholarly journals Generating Hard Satisfiable Formulas by Hiding Solutions Deceptively

2007 ◽  
Vol 28 ◽  
pp. 107-118 ◽  
Author(s):  
H. Jia ◽  
C. Moore ◽  
D. Strain
Keyword(s):  
Polynomial Time ◽  
Constraint Satisfaction ◽  
Search Algorithms ◽  
Constraint Satisfaction Problems ◽  
Experimental Results ◽  
New Method ◽  
Truth Assignment ◽  
Benchmark Instances ◽  
Problem Generator

To test incomplete search algorithms for constraint satisfaction problems such as 3-SAT, we need a source of hard, but satisfiable, benchmark instances. A simple way to do this is to choose a random truth assignment A, and then choose clauses randomly from among those satisfied by A. However, this method tends to produce easy problems, since the majority of literals point toward the "hidden'' assignment A. Last year, Achlioptas, Jia and Moore proposed a problem generator that cancels this effect by hiding both A and its complement. While the resulting formulas appear to be just as hard for DPLL algorithms as random 3-SAT formulas with no hidden assignment, they can be solved by WalkSAT in only polynomial time. Here we propose a new method to cancel the attraction to A, by choosing a clause with t > 0 literals satisfied by A with probability proportional to q^t for some q < 1. By varying q, we can generate formulas whose variables have no bias, i.e., which are equally likely to be true or false; we can even cause the formula to "deceptively'' point away from A. We present theoretical and experimental results suggesting that these formulas are exponentially hard both for DPLL algorithms and for incomplete algorithms such as WalkSAT.

Partial (Neighbourhood) Singleton Arc Consistency for Constraint Satisfaction Problems

2020 ◽  
Vol 174 (3-4) ◽  
pp. 311-344
Author(s):  
Richard J. Wallace
Keyword(s):  
Constraint Satisfaction ◽  
Search Algorithms ◽  
Constraint Satisfaction Problems ◽  
Experimental Results ◽  
Full Version ◽  
Arc Consistency ◽  
Considerable Promise ◽  
Heuristic Strategies ◽  
Backtrack Search

Algorithms based on singleton arc consistency (SAC) show considerable promise for improving backtrack search algorithms for constraint satisfaction problems (CSPs). The drawback is that even the most efficient of them is still comparatively expensive. Even when limited to preprocessing, they give overall improvement only when problems are quite difficult to solve with more typical procedures such as maintained arc consistency (MAC). The present work examines a form of partial SAC and neighbourhood SAC (NSAC) in which a subset of the variables in a CSP are chosen to be made SAC-consistent or neighbourhood-SAC-consistent. Such consistencies, despite their partial character, are still well-characterized in that algorithms have unique fixpoints. Heuristic strategies for choosing an effective subset of variables are described and tested, the best being choice by highest degree and a more complex strategy of choosing by constraint weight after random probing. Experimental results justify the claim that these methods can be nearly as effective as the corresponding full version of the algorithm in terms of values discarded or problems proven unsatisfiable, while significantly reducing the effort required to achieve this.

Modular lazy search for Constraint Satisfaction Problems

2001 ◽  
Vol 11 (5) ◽  
pp. 557-587 ◽  
Author(s):  
THOMAS NORDIN ◽  
ANDREW TOLMACH
Keyword(s):  
Constraint Satisfaction ◽  
Search Algorithms ◽  
General Purpose ◽  
Constant Factor ◽  
Constraint Satisfaction Problems ◽  
Combinatorial Search ◽  
Intermediate Data ◽  
Variable Ordering ◽  
New Algorithms ◽  
Simple Search

We describe a unified, lazy, declarative framework for solving constraint satisfaction problems, an important subclass of combinatorial search problems. These problems are both practically significant and computationally hard. Finding solutions involves combining good general-purpose search algorithms with problem-specific heuristics. Conventional imperative algorithms are usually implemented and presented monolithically, which makes them hard to understand and reuse, even though new algorithms often are combinations of simpler ones. Lazy functional languages, such as Haskell, encourage modular structuring of search algorithms by separating the generation and testing of potential solutions into distinct functions communicating through an explicit, lazy intermediate data structure. But only relatively simple search algorithms have been treated this way in the past. Our framework uses a generic generation and pruning algorithm parameterized by a labeling function that annotates search trees with conflict sets. We show that many advanced imperative search algorithms, including conflict-directed backjumping, backmarking, minimal forward checking, and fail-first dynamic variable ordering, can be obtained by suitable instantiation of the labeling function. More importantly, arbitrary combinations of these algorithms can be built by simply composing their labeling functions. Our modular algorithms are as efficient as the monolithic imperative algorithms in the sense that they make the same number of consistency checks, and most of our algorithms are within a constant factor of their imperative counterparts in runtime and space usage. We believe our framework is especially well-suited for experimenting to find good combinations of algorithms for specific problems.

scholarly journals Circuit Complexity Of Constraint Satisfaction Problems With Few Subpowers

2021 ◽  
Author(s):  
Amir El-Aooiti
Keyword(s):  
Polynomial Time ◽  
Constraint Satisfaction ◽  
Circuit Complexity ◽  
Constraint Satisfaction Problems ◽  
Solution Set ◽  
Alternative Method ◽  
Constraint Language ◽  
Np Complete

Although Constraint Satisfaction Problems (CSPs) are generally known to be NP-complete, placing restrictions on the constraint template can yield tractable subclasses. By studying the operations in the polymorphism of the constraint language, we can construct algorithms which solve our CSP in polynomial time. Previous results for CSPs with Mal’tsev [7] and generalized majority-minority operations [10] were improved to include CSPs with k-edge operations [15]. We present an alternative method to solve k-edge CSPs by utilizing Boolean trees placing the problem in the class NC2 . We do this by arranging the logical formulas describing the CSP into a Boolean tree where each leaf represents a constraint in the CSP. We take the conjunction of the constraint formulas yielding partial solutions at every step until we are left with a solution set at the root of the tree which satisfies all of the constraints.

scholarly journals Compiling CSPs: A Complexity Map of (Non-Deterministic) Multivalued Decision Diagrams

2014 ◽  
Vol 23 (04) ◽  
pp. 1460015 ◽  
Author(s):  
Jérôme Amilhastre ◽  
Hélène Fargier ◽  
Alexandre Niveau ◽  
Cédric Pralet
Keyword(s):  
Polynomial Time ◽  
Constraint Satisfaction ◽  
Binary Decision Diagrams ◽  
Constraint Satisfaction Problems ◽  
Decision Diagrams ◽  
Np Hard ◽  
Ordered Binary Decision Diagrams ◽  
Binary Decision

Constraint Satisfaction Problems (CSPs) offer a powerful framework for representing a great variety of problems. The difficulty is that most of the requests associated with CSPs are NP-hard. When these requests have to be addressed online, Multivalued Decision Diagrams (MDDs) have been proposed as a way to compile CSPs. In the present paper, we draw a compilation map of MDDs, in the spirit of the NNF compilation map, analyzing MDDs according to their succinctness and to their tractable transformations and queries. Deterministic ordered MDDs are a generalization of ordered binary decision diagrams to non-Boolean domains: unsurprisingly, they have similar capabilities. More interestingly, our study puts forward the interest of non-deterministic ordered MDDs: when restricted to Boolean domains, they capture OBDDs and DNFs as proper subsets and have performances close to those of DNNFs. The comparison to classical, deterministic MDDs shows that relaxing the determinism requirement leads to an increase in succinctness and allows more transformations to be satisfied in polynomial time (typically, the disjunctive ones). Experiments on random problems confirm the gain in succinctness.

Efficient Singleton Consistency by Combining Forward Checking and Bound Consistency

2014 ◽  
Vol 23 (04) ◽  
pp. 1460017
Author(s):  
Jinsong Guo ◽  
Hongbo Li ◽  
Zhanshan Li ◽  
Yonggang Zhang ◽  
Xianghua Jia
Keyword(s):  
Constraint Satisfaction ◽  
Search Algorithm ◽  
Computational Cost ◽  
Search Space ◽  
Search Algorithms ◽  
Constraint Satisfaction Problems ◽  
Local Consistency ◽  
Arc Consistency ◽  
Strong Bound

Maintaining local consistencies can improve the efficiencies of the search algorithms solving constraint satisfaction problems (CSPs). Comparing with arc consistency which is the most widely used local consistency, stronger local consistencies can make the search space smaller while they require higher computational cost. In this paper, we make an attempt on the compromise between the pruning ability and the computational cost. A new local consistency called singleton strong bound consistency (SSBC) and its light version, light SSBC, are proposed. The search algorithm maintaining light SSBC can outperform MAC on a considerable number of problems.

SUCCESSIVE SEARCH METHOD FOR SOLVING VALUED CONSTRAINT SATISFACTION AND OPTIMIZATION PROBLEMS

2002 ◽  
Vol 11 (03) ◽  
pp. 425-436 ◽  
Author(s):  
MOHAMED TOUNSI ◽  
PHILIPPE DAVID
Keyword(s):  
Lower Bound ◽  
Constraint Satisfaction ◽  
Optimization Problems ◽  
Main Idea ◽  
Earth Observation ◽  
Constraint Satisfaction Problems ◽  
Search Method ◽  
New Method ◽  
Earth Observation Satellite ◽  
Russian Doll Search

In this paper we introduce a new method based on Russian Doll Search (RDS) for solving optimization problems expressed as Valued Constraint Satisfaction Problems (VCSPs). The RDS method solves problems of size n (where n is the number of variables) by replacing one search by n successive searches on nested subproblems using the results of each search to produce a better lower bound. The main idea of our method is to introduce the variables through the successive searches not one by one but by sets of k variables. We present two variants of our method: the first one where the number k is fixed, noted kfRDS; the second one, kvRDS, where k can be variable. Finally, we show that our method improves RDS on daily management of an earth observation satellite.

scholarly journals Using Pivot Consistency to Decompose and Solve Functional CSPs

1995 ◽  
Vol 2 ◽  
pp. 447-474 ◽  
Author(s):  
P. David
Keyword(s):  
Structural Properties ◽  
Constraint Satisfaction ◽  
Constraint Satisfaction Problems ◽  
New Method ◽  
Functional Constraints ◽  
Local Consistency ◽  
Semantic Properties

Many studies have been carried out in order to increase thesearch efficiency of constraint satisfaction problems; among them,some make use of structural properties of the constraintnetwork; others take into account semantic properties of theconstraints, generally assuming that all the constraints possessthe given property. In this paper, we propose a new decompositionmethod benefiting from both semantic properties of functional constraints (not bijective constraints) and structuralproperties of the network; furthermore, not all the constraints needto be functional. We show that under some conditions, the existenceof solutions can be guaranteed. We first characterize a particularsubset of the variables, which we name a root set. We thenintroduce pivot consistency, a new local consistency which is aweak form of path consistency and can be achieved in O(n^2d^2)complexity (instead of O(n^3d^3) for path consistency), and wepresent associated properties; in particular, we show that anyconsistent instantiation of the root set can be linearly extended to a solution, which leads to the presentation of the aforementioned new method for solving by decomposing functional CSPs.

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