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

2018 ◽  
Vol 61 ◽  
pp. 723-759 ◽  
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.

scholarly journals A Convex Relaxation Bound for Subgraph Isomorphism

2012 ◽  
Vol 2012 ◽  
pp. 1-18
Author(s):  
Christian Schellewald
Keyword(s):  
Lower Bound ◽  
Structural Information ◽  
Convex Relaxation ◽  
Maximum Clique ◽  
Combinatorial Problem ◽  
Isomorphism Problem ◽  
Maximum Clique Problem ◽  
Subgraph Isomorphism ◽  
Semidefinite Programming Relaxation ◽  
Problem Instances

In this work a convex relaxation of a subgraph isomorphism problem is proposed, which leads to a new lower bound that can provide a proof that a subgraph isomorphism between two graphs can not be found. The bound is based on a semidefinite programming relaxation of a combinatorial optimisation formulation for subgraph isomorphism and is explained in detail. We consider subgraph isomorphism problem instances of simple graphs which means that only the structural information of the two graphs is exploited and other information that might be available (e.g., node positions) is ignored. The bound is based on the fact that a subgraph isomorphism always leads to zero as lowest possible optimal objective value in the combinatorial problem formulation. Therefore, for problem instances with a lower bound that is larger than zero this represents a proof that a subgraph isomorphism can not exist. But note that conversely, a negative lower bound does not imply that a subgraph isomorphism must be present and only indicates that a subgraph isomorphism can not be excluded. In addition, the relation of our approach and the reformulation of the largest common subgraph problem into a maximum clique problem is discussed.

A Backtracking Algorithmic Toolbox for Solving the Subgraph Isomorphism Problem

2021 ◽  
pp. 208-246
Author(s):  
Jurij Mihelič ◽  
Uroš Čibej ◽  
Luka Fürst
Keyword(s):  
Independent Set ◽  
Isomorphism Problem ◽  
Lessons Learned ◽  
Subgraph Isomorphism ◽  
Graph Pattern Matching ◽  
Practical Applications ◽  
Graph Pattern ◽  
Complete Problems ◽  
Constraint Satisfaction Programming ◽  
Core Problem

The subgraph isomorphism problem asks whether a given graph is a subgraph of another graph. It is one of the most general NP-complete problems since many other problems (e.g., Hamiltonian cycle, clique, independent set, etc.) have a natural reduction to subgraph isomorphism. Furthermore, there is a variety of practical applications where graph pattern matching is the core problem. Developing efficient algorithms and solvers for this problem thus enables good solutions to a variety of different practical problems. In this chapter, the authors present and experimentally explore various algorithmic refinements and code optimizations for improving the performance of subgraph isomorphism solvers. In particular, they focus on algorithms that are based on the backtracking approach and constraint satisfaction programming. They gather experiences from many state-of-the-art algorithms as well as from their engagement in this field. Lessons learned from engineering such a solver can be utilized in many other fields where backtracking is a prominent approach for solving a particular problem.

scholarly journals Complexity of n-Queens Completion (Extended Abstract)

2018 ◽  
Author(s):  
Ian P. Gent ◽  
Christopher Jefferson ◽  
Peter Nightingale
Keyword(s):  
Artificial Intelligence ◽  
Phase Transition ◽  
Decision Problem ◽  
Completion Problem ◽  
Complete Problems ◽  
Random Instances ◽  
Np Complete

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.

Rapid Close-to-Optimum Optimization by Genetic Algorithms

1997 ◽  
Vol 08 (05) ◽  
pp. 1103-1117
Author(s):  
R. Hackl ◽  
I. Morgenstern
Keyword(s):  
Optimal Solution ◽  
Global Optimum ◽  
Traveling Salesman ◽  
Real Word ◽  
Practical Application ◽  
Standard Problem ◽  
Complete Problems ◽  
Np Complete ◽  
Problem Instances ◽  
The Traveling Salesman Problem

In this article we consider the optimization of np-complete problems with a genetic algorithm. For "real word" problems we regard it to be sufficient to get close to the optimal solution without any guarantee of ever hitting it. Our algorithm was tested on two problem classes: the traveling salesman problem and the product ordering problem; the first is a standard problem, the latter a problem we were confronted with in a practical application. For all investigated problem instances we found very good solutions (<0.2% above optimum) in each run and even the global optimum in some runs on a Pentium/100 MHz-PC. For one instance of the TSP problem we could verify that the time spent to find the optimum follows a logarithmic normal distribution.

scholarly journals An Optimization of Closed Frequent Subgraph Mining Algorithm

2017 ◽  
Vol 17 (1) ◽  
pp. 3-15
Author(s):  
J. Demetrovics ◽  
H. M. Quang ◽  
N. V. Anh ◽  
V. D. Thi
Keyword(s):  
Graph Mining ◽  
Isomorphism Problem ◽  
Subgraph Isomorphism ◽  
Frequent Subgraph Mining ◽  
Complete Problem ◽  
Area Of Interest ◽  
Subgraph Mining ◽  
Mining Algorithm ◽  
Frequent Subgraph ◽  
Np Complete

Abstract Graph mining isamajor area of interest within the field of data mining in recent years. Akey aspect of graph mining is frequent subgraph mining. Central to the entire discipline of frequent subgraph mining is the concept of subgraph isomorphism. One major issue in early subgraph isomorphism research concerns computational complexity. Normally, the subgraph isomorphism problem is NP-complete. Previous studies of frequent subgraph mining have not solved NP-complete problem in the subgraph isomorphism. In this paper, we proposeanew algorithm which can deal with this problem. The proposed algorithm can solve the subgraph isomorphism in polynomial time in some settings. Moreover, the new algorithm is proved theoretically more effective than previous studies in closed frequent subgraph mining.

scholarly journals Classical Generalized Probabilistic Satisfiability

2017 ◽  
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.

scholarly journals POLYNOMIAL OBSERVABLES IN THE GRAPH PARTITIONING PROBLEM

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.

scholarly journals Complexity of n-Queens Completion

2017 ◽  
Vol 59 ◽  
pp. 815-848 ◽  
Author(s):  
Ian P. Gent ◽  
Christopher Jefferson ◽  
Peter Nightingale
Keyword(s):  
Artificial Intelligence ◽  
Phase Transition ◽  
Decision Problem ◽  
Completion Problem ◽  
Complete Problems ◽  
Random Instances ◽  
Np Complete

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.

scholarly journals Diamond Subgraphs in the Reduction Graph of a One-Rule String Rewriting System

2021 ◽  
Vol 178 (3) ◽  
pp. 173-185
Author(s):  
Arthur Adinayev ◽  
Itamar Stein
Keyword(s):  
Isomorphism Problem ◽  
Complete Characterization ◽  
Hasse Diagram ◽  
Subgraph Isomorphism ◽  
Rewriting Systems ◽  
Rewriting System ◽  
String Rewriting ◽  
Reduction Graph

In this paper, we study a certain case of a subgraph isomorphism problem. We consider the Hasse diagram of the lattice Mk (the unique lattice with k + 2 elements and one anti-chain of length k) and find the maximal k for which it is isomorphic to a subgraph of the reduction graph of a given one-rule string rewriting system. We obtain a complete characterization for this problem and show that there is a dichotomy. There are one-rule string rewriting systems for which the maximal such k is 2 and there are cases where there is no maximum. No other intermediate option is possible.

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