scholarly journals Cardinality Encodings for Graph Optimization Problems

2017 ◽  
Author(s):  
Alexey Ignatiev ◽  
Antonio Morgado ◽  
Joao Marques-Silva
Keyword(s):  
Optimization Problems ◽  
State Of The Art ◽  
Maximum Clique ◽  
Maximum Clique Problem ◽  
Concrete Case ◽  
Large Graphs ◽  
Sparse Graphs ◽  
Maximum Satisfiability ◽  
Graph Optimization ◽  
Sparse Networks

Different optimization problems defined on graphs find application in complex network analysis. Existing propositional encodings render impractical the use of propositional satisfiability (SAT) and maximum satisfiability (MaxSAT) solvers for solving a variety of these problems on large graphs. This paper has two main contributions. First, the paper identifies sources of inefficiency in existing encodings for different optimization problems in graphs. Second, for the concrete case of the maximum clique problem, the paper develops a novel encoding which is shown to be far more compact than existing encodings for large sparse graphs. More importantly, the experimental results show that the proposed encoding enables existing SAT solvers to compute a maximum clique for large sparse networks, often more efficiently than the state of the art.

scholarly journals Abstract Cores in Implicit Hitting Set MaxSat Solving (Extended Abstract)

2021 ◽  
Author(s):  
Jeremias Berg ◽  
Fahiem Bacchus ◽  
Alex Poole
Keyword(s):  
Linear Programming ◽  
Optimization Problems ◽  
State Of The Art ◽  
Empirical Evaluation ◽  
Boolean Satisfiability ◽  
Compact Representation ◽  
Maximum Satisfiability ◽  
Hitting Set ◽  
Combinatorial Optimization Problems ◽  
Core Extraction

Maximum satisfiability (MaxSat) solving is an active area of research motivated by numerous successful applications to solving NP-hard combinatorial optimization problems. One of the most successful approaches for solving MaxSat instances from real world domains are the so called implicit hitting set (IHS) solvers. IHS solvers decouple MaxSat solving into separate core-extraction (i.e. reasoning) and optimization steps which are tackled by a Boolean satisfiability (SAT) and an integer linear programming (IP) solver, respectively. While the approach shows state-of-the-art performance on many industrial instances, it is known that there exists instances on which IHS solvers need to extract an exponential number of cores before terminating. Motivated by the simplest of these problematic instances, we propose abstract cores, a compact representation for a potentially exponential number of regular cores. We demonstrate how to incorporate abstract core reasoning into the IHS algorithm and report on an empirical evaluation demonstrating, that including abstract cores into a state-of-the-art IHS solver improves its performance enough to surpass the best performing solvers of the 2019 MaxSat Evaluation.

scholarly journals Reduced Cost Fixing for Maximum Satisfiability

2018 ◽  
Author(s):  
Fahiem Bacchus ◽  
Antti Hyttinen ◽  
Matti Järvisalo ◽  
Paul Saikko
Keyword(s):  
Integer Programming ◽  
Real World ◽  
Recent Trend ◽  
Optimization Problems ◽  
State Of The Art ◽  
Boolean Satisfiability ◽  
Maximum Satisfiability ◽  
Hitting Set ◽  
Sat Solvers ◽  
Speed Up

Maximum satisfiability (MaxSAT) offers a competitive approach to solving NP-hard real-world optimization problems. While state-of-the-art MaxSAT solvers rely heavily on Boolean satisfiability (SAT) solvers, a recent trend, brought on by MaxSAT solvers implementing the so-called implicit hitting set (IHS) approach, is to integrate techniques from the realm of integer programming (IP) into the solving process. This allows for making use of additional IP solving techniques to further speed up MaxSAT solving. In this line of work, we investigate the integration of the technique of reduced cost fixing from the IP realm into IHS solvers, and empirically show that reduced cost fixing considerable speeds up a state-of-the-art MaxSAT solver implementing the IHS approach.

scholarly journals Some Zero-One Linear Programming Reformulations for the Maximum Clique Problem

2021 ◽  
Vol 27_NS1 (1) ◽  
pp. 32-47
Author(s):  
Ákos Beke ◽  
Sándor Szabó ◽  
Bogdán Zavalnij
Keyword(s):  
Linear Programming ◽  
Optimization Problems ◽  
Maximum Clique ◽  
Independent Set ◽  
Maximum Clique Problem ◽  
Linear Programs ◽  
Combinatorial Optimization Problems ◽  
Edge Covering ◽  
Covering Set ◽  
Clique Problem

Many combinatorial optimization problems can be expressed in terms of zero-one linear programs. For the maximum clique problem the so-called edge reformulation is applied most commonly. Two less frequently used LP equivalents are the independent set and edge covering set reformulations. The number of the constraints (as a function of the number of vertices of the ground graph) is asymptotically quadratic in the edge and the edge covering set LP reformulations and it is exponential in the independent set reformulation, respectively. F. D. Croce and R. Tadei proposed an approach in which the number of the constraints is equal to the number of the vertices. In this paper we are looking for possible tighter variants of these linear programs.

scholarly journals Fast Algorithms for the Maximum Clique Problem on Massive Sparse Graphs

2013 ◽  
pp. 156-169 ◽  
Author(s):  
Bharath Pattabiraman ◽  
Md. Mostofa Ali Patwary ◽  
Assefaw H. Gebremedhin ◽  
Wei-keng Liao ◽  
Alok Choudhary
Keyword(s):  
Maximum Clique ◽  
Fast Algorithms ◽  
Maximum Clique Problem ◽  
Sparse Graphs ◽  
Clique Problem

scholarly journals Finding Most Compatible Phylogenetic Trees over Multi-State Characters

2020 ◽  
Vol 34 (02) ◽  
pp. 1544-1551
Author(s):  
Tuukka Korhonen ◽  
Matti J„ärvisalo
Keyword(s):  
Phylogenetic Trees ◽  
Optimization Problems ◽  
Hybrid Approach ◽  
Optimization Techniques ◽  
Evolutionary Tree ◽  
Character State ◽  
Perfect Phylogeny ◽  
Maximum Satisfiability ◽  
Graph Optimization ◽  
Boolean Optimization

The reconstruction of the evolutionary tree of a set of species based on qualitative attributes is a central problem in phylogenetics. In the NP-hard perfect phylogeny problem the input is a set of taxa (species) and characters (attributes) on them, and the task is to find an evolutionary tree that describes the evolution of the taxa so that each character state evolves only once. However, in practical situations a perfect phylogeny rarely exists, motivating the maximum compatibility problem of finding the largest subset of characters admitting a perfect phylogeny. Various declarative approaches, based on applying integer programming (IP), answer set programming (ASP) and pseudo-Boolean optimization (PBO) solvers, have been proposed for maximum compatibility. In this work we develop a new hybrid approach to solving maximum compatibility for multi-state characters, making use of both declarative optimization techniques (specifically maximum satisfiability, MaxSAT) and an adaptation of the Bouchitt'e-Todinca approach to triangulation-based graph optimization problems. Empirically our approach outperforms in scalability the earlier proposed approaches w.r.t. various parameters underlying the problem.

scholarly journals A Semi-exact Algorithm for Quickly Computing A Maximum Weight Clique in Large Sparse Graphs

2021 ◽  
Vol 72 ◽  
pp. 39-67
Author(s):  
Shaowei Cai ◽  
Jinkun Lin ◽  
Yiyuan Wang ◽  
Darren Strash
Keyword(s):  
Large Scale ◽  
Heuristic Algorithms ◽  
State Of The Art ◽  
Exact Algorithm ◽  
Exact Algorithms ◽  
Maximum Weight ◽  
Large Graphs ◽  
Sparse Graphs ◽  
Short Time ◽  
Maximum Weight Clique

This paper explores techniques to quickly solve the maximum weight clique problem (MWCP) in very large scale sparse graphs. Due to their size, and the hardness of MWCP, it is infeasible to solve many of these graphs with exact algorithms. Although recent heuristic algorithms make progress in solving MWCP in large graphs, they still need considerable time to get a high-quality solution. In this work, we focus on solving MWCP for large sparse graphs within a short time limit. We propose a new method for MWCP which interleaves clique finding with data reduction rules. We propose novel ideas to make this process efficient, and develop an algorithm called FastWClq. Experiments on a broad range of large sparse graphs show that FastWClq finds better solutions than state-of-the-art algorithms while the running time of FastWClq is much shorter than the competitors for most instances. Further, FastWClq proves the optimality of its solutions for roughly half of the graphs, all with at least 105 vertices, with an average time of 21 seconds.

scholarly journals Formulation of the Problem of Maximum Clique Determination in Non-Oriented Graphs

2018 ◽  
Vol 7 (4.3) ◽  
pp. 293
Author(s):  
S. V. Listrovoy ◽  
O. V. Golovko ◽  
V. M. Butenko ◽  
M. V. Ushakov ◽  
. .
Keyword(s):  
Control Systems ◽  
Real Time ◽  
Maximum Clique ◽  
Maximum Size ◽  
Maximum Clique Problem ◽  
Sparse Graph ◽  
Large Graphs ◽  
Exponential Time ◽  
Use Of Resources ◽  
Traffic Organization

In the train traffic organization, large amount of information should be processed in real time. Thereby in complicated dispatching systems, rational use of resources is needed. To perform this work a model of a management system is constructed and it is represented as a sparse graph. Further optimization of the model requires solving the Maximum Clique Problem (MCP) for the less time than the exponential time. This article contains two procedures for solving the task for subexponential time. Procedure B accurately estimates the size of the maximum clique in the graph and performs the sorting of the vertices, in such a way that the vertices that are not exactly in the maximum clique can be rejected. Procedure A, in fact, finds cliques of the maximum size in a given graph, using the procedure B. The main advantage of this method is that using it is expedient in real time for sufficiently large graphs, which in turn is important for the construction of control systems.  

scholarly journals Using Machine Learning for Quantum Annealing Accuracy Prediction

Algorithms ◽  
2021 ◽  
Vol 14 (6) ◽  
pp. 187
Author(s):  
Aaron Barbosa ◽  
Elijah Pelofske ◽  
Georg Hahn ◽  
Hristo N. Djidjev
Keyword(s):  
Machine Learning ◽  
Maximum Clique ◽  
Classification Model ◽  
Maximum Clique Problem ◽  
Problem Instance ◽  
Np Hard ◽  
Machine Learning Classification ◽  
Hard Problems ◽  
Problem Instances ◽  
D Wave

Quantum annealers, such as the device built by D-Wave Systems, Inc., offer a way to compute solutions of NP-hard problems that can be expressed in Ising or quadratic unconstrained binary optimization (QUBO) form. Although such solutions are typically of very high quality, problem instances are usually not solved to optimality due to imperfections of the current generations quantum annealers. In this contribution, we aim to understand some of the factors contributing to the hardness of a problem instance, and to use machine learning models to predict the accuracy of the D-Wave 2000Q annealer for solving specific problems. We focus on the maximum clique problem, a classic NP-hard problem with important applications in network analysis, bioinformatics, and computational chemistry. By training a machine learning classification model on basic problem characteristics such as the number of edges in the graph, or annealing parameters, such as the D-Wave’s chain strength, we are able to rank certain features in the order of their contribution to the solution hardness, and present a simple decision tree which allows to predict whether a problem will be solvable to optimality with the D-Wave 2000Q. We extend these results by training a machine learning regression model that predicts the clique size found by D-Wave.

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