Tutorial on Surrogate Constraint Approaches for Optimization in Graphs

Surrogate constraint methods have been embedded in a variety of mathematical programming applications over the past thirty years, yet their potential uses and underlying principles remain incompletely understood by a large segment of the optimization community. In a number of significant domains of combinatorial optimization, researchers have produced solution strategies without recognizing that they can be derived as special instances of surrogate constraint methods. Once the connection to surrogate constraint ideas is exposed, additional ways to exploit this framework become visible, frequently offering opportunities for improvement.We provide a tutorial on surrogate constraint approaches for optimization in graphs, illustrating the key ideas by reference to independent set and graph coloring problems, including constructions for weighted independent sets which have applications to associated covering and weighted maximum clique problems. In these settings, the surrogate constraints can be generated relative to well-known packing and covering formulations that are convenient for exposing key notions. The surrogate constraint approaches yield widely used heuristics for identifying independent sets as simple special cases, and also afford previously unidentified heuristics that have greater power in these settings. Our tutorial also shows how the use of surrogate constraints can be placed within the context of vocabulary building strategies for independent set and coloring problems, providing a framework for applying surrogate constraints that can be used in other applications.At a higher level, we show how to make use of surrogate constraint information, together with specialized algorithms for solving associated sub-problems, to obtain stronger objective function bounds and improved choice rules for heuristic or exact methods. The theorems that support these developments yield further strategies for exploiting surrogate constraint relaxations, both in graph optimization and integer programming generally.     

Efficient Constraint Propagation for Graph Coloring

In this paper, we investigate constraint propagation, a mechanism that is run at each basic step of a backtrack search algorithm such as the popular MAC. From a statistical analysis of some relevant features concerning propagation on a large set of graph coloring instances, we show that it is possible to make reasonable predictions about the capability of constraint propagation to detect inconsistency. Using this observation in order to control propagation effort, we show its practical effectiveness.     

Graph unique-maximum and conflict-free colorings

We investigate the relationship between two kinds of vertex colorings of graphs: unique-maximum colorings and conflict-free colorings. In a unique-maximum coloring, the colors are ordered, and in every path of the graph the maximum color appears only once. In a conflict-free coloring, in every path of the graph there is a color that appears only once. We also study computational complexity aspects of conflict-free colorings and prove a completeness result. Finally, we improve lower bounds for those chromatic numbers of the grid graph.     

Circular-arc graph coloring: On chords and circuits in the meeting graph

In compilers register allocation in loops is usually performed by coloring a corresponding circular-arc graph. Generally, the problem of finding the chromatic number of circular-arc graphs is known to be NP-complete. Thus, approximation algorithms should be considered. In this paper we propose heuristics based on decomposition of a so called meeting graph into a set of circuits. We explain the importance of the meeting graph for our solutions and prove properties of our decomposition of the graph into circuits. We derive inequalities relating the number of circuits in the decomposition to the size of the maximum stable set of chords, and present experimental results. Finally, we discuss the quality of our heuristics for circular-arc graph coloring.     

Coloring the edges of a directed graph

Finding large cliques in a graph is an important problem in applied discrete mathematics. In directed graph a possible corresponding problem is finding large transitive subtournaments. It is well-known that coloring the graph speeds up the clique search in the undirected case. In this paper we propose coloring schemes to speed up the tournament search in the directed case. We prove two complexity results about the proposed colorings. A consequence of these results is that in practical computations we have to be content with approximate greedy coloring algorithms.     

调度问题中两类分离约束传播算法的比较及一种改进算法

约束传播算法是求解约束满足问题的一种重要方法。调度问题是一种特殊的约束满足问题。本介绍了调度问题中的Edge-Finding和Energy-Reasoning两种分离约束传播算法，并对它们进行了比较，中最后给出了一种结合Energy-Reasoning的Edge-Finding改进算法。     

Colorful monochromatic connectivity

An edge-coloring of a connected graph is monochromatically-connecting if there is a monochromatic path joining any two vertices. How “colorful” can a monochromatically-connecting coloring be? Let mc(G) denote the maximum number of colors used in a monochromatically-connecting coloring of a graph G. We prove some nontrivial upper and lower bounds for mc(G) and relate it to other graph parameters such as the chromatic number, the connectivity, the maximum degree, and the diameter.     

On global warming: Flow-based soft global constraints

In case a CSP is over-constrained, it is natural to allow some constraints, called soft constraints, to be violated. We propose a generic method to soften global constraints that can be represented by a flow in a graph. Such constraints are softened by adding violation arcs to the graph and then computing a minimum-weight flow in the extended graph to measure the violation. We present efficient propagation algorithms, based on different violation measures, achieving domain consistency for the alldifferent constraint, the global cardinality constraint, the regular constraint and the same constraint. This work was for a large part carried out while the author was at the Centrum voor Wiskunde en Informatica, Amsterdam, The Netherlands     

Sampling subproblems of heterogeneous Max‐Cut problems and approximation algorithms

Recent work in the analysis of randomized approximation algorithms for NP‐hard optimization problems has involved approximating the solution to a problem by the solution of a related subproblem of constant size, where the subproblem is constructed by sampling elements of the original problem uniformly at random. In light of interest in problems with a heterogeneous structure, for which uniform sampling might be expected to yield suboptimal results, we investigate the use of nonuniform sampling probabilities. We develop and analyze an algorithm which uses a novel sampling method to obtain improved bounds for approximating the Max‐Cut of a graph. In particular, we show that by judicious choice of sampling probabilities one can obtain error bounds that are superior to the ones obtained by uniform sampling, both for unweighted and weighted versions of Max‐Cut. Of at least as much interest as the results we derive are the techniques we use. The first technique is a method to compute a compressed approximate decomposition of a matrix as the product of three smaller matrices, each of which has several appealing properties. The second technique is a method to approximate the feasibility or infeasibility of a large linear program by checking the feasibility or infeasibility of a nonuniformly randomly chosen subprogram of the original linear program. We expect that these and related techniques will prove fruitful for the future development of randomized approximation algorithms for problems whose input instances contain heterogeneities. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

An Efficient Branch-and-bound Algorithm for Finding a Maximum Clique with Computational Experiments

We present an exact and efficient branch-and-bound algorithm MCR for finding a maximum clique in an arbitrary graph. The algorithm is not specialized for any particular type of graph. It employs approximate coloring to obtain an upper bound on the size of a maximum clique along with an improved appropriate sorting of vertices. We demonstrate by computational experiments on random graphs with up to 15,000 vertices and on DIMACS benchmark graphs that in general, our algorithm decidedly outperforms other existing algorithms. The algorithm has been successfully applied to interesting problems in bioinformatics, image processing, design of quantum circuits, and design of DNA and RNA sequences for biomolecular computation.     

Exact Max-SAT solvers for over-constrained problems

We present a new generic problem solving approach for over-constrained problems based on Max-SAT. We first define a Boolean clausal form formalism, called soft CNF formulas, that deals with blocks of clauses instead of individual clauses, and that allows one to declare each block either as hard (i.e., must be satisfied by any solution) or soft (i.e., can be violated by some solution). We then present two Max-SAT solvers that find a truth assignment that satisfies all the hard blocks of clauses and the maximum number of soft blocks of clauses. Our solvers are branch and bound algorithms equipped with original lazy data structures, powerful inference techniques, good quality lower bounds, and original variable selection heuristics. Finally, we report an experimental investigation on a representative sample of instances (random 2-SAT, Max-CSP, graph coloring, pigeon hole and quasigroup completion) which provides experimental evidence that our approach is very competitive compared with the state-of-the-art approaches developed in the CSP and SAT communities. Research partially supported by projects TIN2004-07933-C03-03 and TIC2003-00950 funded by the Ministerio de Educación y Ciencia. The second author is supported by a grant Ramón y Cajal.     

Dominator Colorings in Some Classes of Graphs

A dominator coloring is a coloring of the vertices of a graph such that every vertex is either alone in its color class or adjacent to all vertices of at least one other class. We present new bounds on the dominator coloring number of a graph, with applications to chordal graphs. We show how to compute the dominator coloring number in polynomial time for P ₄-free graphs, and we give a polynomial-time characterization of graphs with dominator coloring number at most 3.     

Lower Bounds for the Rate of Convergence of Dynamic Reconstruction Algorithms for Distributed-Parameter Systems

We obtain lower bounds for the rate of convergence of reconstruction algorithms for distributed-parameter systems of parabolic type. In the case of a pointwise constraint on control for known reconstruction algorithms, we establish a lower bound on the rate of convergence, which shows that, given certain conditions, for each solution of the system one can choose such a collection of measurements so that the reconstruction error will not be less than a certain value. In the case of unbounded controls, we obtain lower bounds for a possible reconstruction error for each trajectory as well as for a given set of trajectories. For a system of special form, we construct an algorithm for which we obtain upper and lower bounds for accuracy having identical order for a specific choice of matching of the parameters.     

Optimal approximation of sparse hessians and its equivalence to a graph coloring problem

We consider the problem of approximating the Hessian matrix of a smooth non-linear function using a minimum number of gradient evaluations, particularly in the case that the Hessian has a known, fixed sparsity pattern. We study the class of Direct Methods for this problem, and propose two new ways of classifying Direct Methods. Examples are given that show the relationships among optimal methods from each class. The problem of finding a non-overlapping direct cover is shown to be equivalent to a generalized graph coloring problem—the distance-2 graph coloring problem. A theorem is proved showing that the general distance-k graph coloring problem is NP-Complete for all fixedk≥2, and hence that the optimal non-overlapping direct cover problem is also NP-Complete. Some worst-case bounds on the performance of a simple coloring heuristic are given. An appendix proves a well-known folklore result, which gives lower bounds on the number of gradient evaluations needed in any possible approximation method. This research was partially supported by the Department of Energy Contract AM03-76SF00326. PA#DE-AT03-76ER72018; Army Research Office Contract DAA29-79-C-0110; Office of Naval Research Contract N00014-74-C-0267; National Science Foundation Grants MCS76-81259, MCS-79260099 and ECS-8012974.     

Constraint satisfaction problems: Algorithms and applications

A constraint satisfaction problem (CSP) requires a value, selected from a given finite domain, to be assigned to each variable in the problem, so that all constraints relating the variables are satisfied. Many combinatorial problems in operational research, such as scheduling and timetabling, can be formulated as CSPs. Researchers in artificial intelligence (AI) usually adopt a constraint satisfaction approach as their preferred method when tackling such problems. However, constraint satisfaction approaches are not widely known amongst operational researchers. The aim of this paper is to introduce constraint satisfaction to the operational researcher. We start by defining CSPs, and describing the basic techniques for solving them. We then show how various combinatorial optimization problems are solved using a constraint satisfaction approach. Based on computational experience in the literature, constraint satisfaction approaches are compared with well-known operational research (OR) techniques such as integer programming, branch and bound, and simulated annealing.     

Compact cyclic edge-colorings of graphs

The paper is devoted to a model of compact cyclic edge-coloring of graphs. This variant of edge-coloring finds its applications in modeling schedules in production systems, in which production proceeds in a cyclic way. We point out optimal colorings for some graph classes and we construct graphs which cannot be colored in a compact cyclic manner. Moreover, we prove some theoretical properties of considered coloring model such as upper bounds on the number of colors in optimal compact cyclic coloring.     

Minimum sum edge colorings of multicycles

In the minimum sum edge coloring problem, we aim to assign natural numbers to edges of a graph, so that adjacent edges receive different numbers, and the sum of the numbers assigned to the edges is minimum. The chromatic edge strength of a graph is the minimum number of colors required in a minimum sum edge coloring of this graph. We study the case of multicycles, defined as cycles with parallel edges, and give a closed-form expression for the chromatic edge strength of a multicycle, thereby extending a theorem due to Berge. It is shown that the minimum sum can be achieved with a number of colors equal to the chromatic index. We also propose simple algorithms for finding a minimum sum edge coloring of a multicycle. Finally, these results are generalized to a large family of minimum cost coloring problems.     

关于图的星色数的一点注记

A star coloring of an undirected graph G is a proper coloring of G such that no path of length 3 in G is bicolored.The star chromatic number of an undirected graph G,denoted by χs(G),is the smallest integer k for which G admits a star coloring with k colors.In this paper,we show that if G is a graph with maximum degree △,then χs(G) ≤ [7△3/2],which gets better bound than those of Fertin,Raspaud and Reed.     

A branch and bound algorithm for the maximum clique problem

We present a branch and bound algorithm for the maximum clique problem in arbitrary graphs. The main part of the algorithm consists in the determination of upper bounds by graph colorings. Using a modification of a known graph coloring method called DSATUR we simultaneously derive lower and upper bounds for the clique number.

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Zusammenfassung Wir stellen einen Branch and Bound Algorithmus für das Maximum Clique Problem in einem beliebigen Graphen vor. Das Hauptaugenmerk richtet sich dabei auf die Bestimmung oberer Schranken mit Hilfe von Färbungen von Graphen. Es wird eine Modifikation einer bekannten Färbungsmethode, genannt DSATUR, verwendet, mit der sich gleichzeitig obere und untere Schranken für die Cliquezahl erstellen lassen.

     

Improved upper bounds on acyclic edge colorings

An acyclic edge coloring of a graph is a proper edge coloring such that every cycle contains edges of at least three distinct colors.The acyclic chromatic index of a graph G,denoted by a′(G),is the minimum number k such that there is an acyclic edge coloring using k colors.It is known that a′(G)≤16△for every graph G where △denotes the maximum degree of G.We prove that a′(G)13.8△for an arbitrary graph G.We also reduce the upper bounds of a′(G)to 9.8△and 9△with girth 5 and 7,respectively.