Complexity of n-Queens Completion (Extended Abstract)

The n-Queens problem is to place n chess queens on an n by n chessboard so that no two queens are on the same row, column or diagonal. The n-Queens Completion problem is a variant, dating to 1850, in which some queens are already placed and the solver is asked to place the rest, if possible. We show that n-Queens Completion is both NP-Complete and #P-Complete. A corollary is that any non-attacking arrangement of queens can be included as a part of a solution to a larger n-Queens problem. We introduce generators of random instances for n-Queens Completion and the closely related Blocked n-Queens and Excluded Diagonals Problem. We describe three solvers for these problems, and empirically analyse the hardness of randomly generated instances. For Blocked n-Queens and the Excluded Diagonals Problem, we show the existence of a phase transition associated with hard instances as has been seen in other NP-Complete problems, but a natural generator for n-Queens Completion did not generate consistently hard instances. The significance of this work is that the n-Queens problem has been very widely used as a benchmark in Artificial Intelligence, but conclusions on it are often disputable because of the simple complexity of the decision problem. Our results give alternative benchmarks which are hard theoretically and empirically, but for which solving techniques designed for n-Queens need minimal or no change.

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Isomorphisms of NP complete problems on random instances

[1993] Proceedings of the Eigth Annual Structure in Complexity Theory Conference ◽

10.1109/sct.1993.336540 ◽

2002 ◽

Cited By ~ 4

Author(s):

J. Belanger ◽

J. Wang

Keyword(s):

Complete Problems ◽

Random Instances ◽

Np Complete

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Classical Generalized Probabilistic Satisfiability

Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence ◽

10.24963/ijcai.2017/126 ◽

2017 ◽

Cited By ~ 1

Author(s):

Carlos Caleiro ◽

Filipe Casal ◽

Andreia Mordido

Keyword(s):

Phase Transition ◽

Linear Inequalities ◽

Satisfiability Problem ◽

Probabilistic Logic ◽

Phase Transition Behavior ◽

Free Theory ◽

Polynomial Reduction ◽

Complete Problems ◽

Probabilistic Satisfiability ◽

Np Complete

We analyze a classical generalized probabilistic satisfiability problem (GGenPSAT) which consists in deciding the satisfiability of Boolean combinations of linear inequalities involving probabilities of classical propositional formulas. GGenPSAT coincides precisely with the satisfiability problem of the probabilistic logic of Fagin et al. and was proved to be NP-complete. Here, we present a polynomial reduction of GGenPSAT to SMT over the quantifier-free theory of linear integer and real arithmetic. Capitalizing on this translation, we implement and test a solver for the GGenPSAT problem. As previously observed for many other NP-complete problems, we are able to detect a phase transition behavior for GGenPSAT.

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When Subgraph Isomorphism is Really Hard, and Why This Matters for Graph Databases

Journal of Artificial Intelligence Research ◽

10.1613/jair.5768 ◽

2018 ◽

Vol 61 ◽

pp. 723-759 ◽

Cited By ~ 10

Author(s):

Ciaran McCreesh ◽

Patrick Prosser ◽

Christine Solnon ◽

James Trimble

Keyword(s):

Phase Transition ◽

Isomorphism Problem ◽

Subgraph Isomorphism ◽

Graph Databases ◽

Complete Problems ◽

Real World Problem ◽

Np Complete ◽

Problem Instances ◽

Pattern Graph ◽

Indexing Technique

The subgraph isomorphism problem involves deciding whether a copy of a pattern graph occurs inside a larger target graph. The non-induced version allows extra edges in the target, whilst the induced version does not. Although both variants are NP-complete, algorithms inspired by constraint programming can operate comfortably on many real-world problem instances with thousands of vertices. However, they cannot handle arbitrary instances of this size. We show how to generate "really hard" random instances for subgraph isomorphism problems, which are computationally challenging with a couple of hundred vertices in the target, and only twenty pattern vertices. For the non-induced version of the problem, these instances lie on a satisfiable / unsatisfiable phase transition, whose location we can predict; for the induced variant, much richer behaviour is observed, and constrainedness gives a better measure of difficulty than does proximity to a phase transition. These results have practical consequences: we explain why the widely researched "filter / verify" indexing technique used in graph databases is founded upon a misunderstanding of the empirical hardness of NP-complete problems, and cannot be beneficial when paired with any reasonable subgraph isomorphism algorithm.

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NUMBER OF SOLUTIONS OF SATISFIABILITY INSTANCES—APPLICATIONS TO KNOWLEDGE BASES

International Journal of Pattern Recognition and Artificial Intelligence ◽

10.1142/s0218001489000061 ◽

1989 ◽

Vol 03 (01) ◽

pp. 53-65 ◽

Cited By ~ 2

Author(s):

J.C. SIMON ◽

O. DUBOIS

Keyword(s):

Artificial Intelligence ◽

Propositional Logic ◽

Statistical Approach ◽

Knowledge Bases ◽

Zero Order ◽

Number Of Solutions ◽

Exponential Process ◽

Complete Problems ◽

Np Complete ◽

Logical Rules

In propositional logic (zero order) a system of logical rules may be put under the form of a conjunction of disjunction, i.e. a “satisfiability” or SAT-problem. SAT is central to NP-complete problems. Any result obtained on SAT would have consequences for a lot of problems important in artificial intelligence. We deal with the question of the number N of solutions of SAT. Firstly, any system of SAT clauses may be transformed in a system of independent clauses by an exponential process; N may be computed exactly. Secondly, by a statistical approach, results are obtained showing that for a given SAT-instance, it should be possible to find an estimate of N with a margin of confidence in polynomial time. Thirdly, we demonstrate the usefulness of these ideas on large knowledge bases.

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POLYNOMIAL OBSERVABLES IN THE GRAPH PARTITIONING PROBLEM

International Journal of Modern Physics C ◽

10.1142/s0129183101001456 ◽

2001 ◽

Vol 12 (01) ◽

pp. 13-18

Author(s):

M. A. MARCHISIO

Keyword(s):

Phase Transition ◽

Graph Partitioning ◽

Graph Partitioning Problem ◽

Complete Problems ◽

Partitioning Problem ◽

Np Complete

Although NP-Complete problems are the most difficult decisional problems, it is possible to discover in them polynomial (or easy) observables. We study the Graph Partitioning Problem showing that it is possible to recognize in it two correlated polynomial observables. The particular behavior of one of them with respect to the connectivity of the graph suggests the presence of a phase transition in partitionability.

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On the Classification of NP Complete Problems and Their Duality Feature

International Journal of Computer Science and Information Technology ◽

10.5121/ijcsit.2018.10106 ◽

2018 ◽

Vol 10 (1) ◽

pp. 67-78 ◽

Cited By ~ 1

Author(s):

Wenhong Tian ◽

Wenxia Guo ◽

Majun He

Keyword(s):

Complete Problems ◽

Np Complete

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Reconstructing a Graph from its Neighborhood Lists

Combinatorics Probability Computing ◽

10.1017/s0963548300000535 ◽

1993 ◽

Vol 2 (2) ◽

pp. 103-113 ◽

Cited By ~ 10

Author(s):

Martin Aigner ◽

Eberhard Triesch

Keyword(s):

Decision Problem ◽

Graph Isomorphism ◽

Bipartite Graphs ◽

Labeled Graph ◽

Np Complete

Associate to a finite labeled graph G(V, E) its multiset of neighborhoods (G) = {N(υ): υ ∈ V}. We discuss the question of when a list is realizable by a graph, and to what extent G is determined by (G). The main results are: the decision problem is NP-complete; for bipartite graphs the decision problem is polynomially equivalent to Graph Isomorphism; forests G are determined up to isomorphism by (G); and if G is connected bipartite and (H) = (G), then H is completely described.

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