We describe a general approach to optimization which we term "Squeaky Wheel" Optimization (swo). In swo, a greedy algorithm is used to construct a solution which is then analyzed to find the trouble spots, i.e., those elements, that, if improved, are likely to improve the objective function score. The results of the analysis are used to generate new priorities that determine the order in which the greedy algorithm constructs the next solution. This Construct/Analyze/Prioritize cycle continues until some limit is reached, or an acceptable solution is found. SWO can be viewed as operating on two search spaces: solutions and prioritizations. Successive solutions are only indirectly related, via the re-prioritization that results from analyzing the prior solution. Similarly, successive prioritizations are generated by constructing and analyzing solutions. This "coupled search" has some interesting properties, which we discuss. We report encouraging experimental results on two ...

We dene the `frozen development' of coloring random graphs. We identify two nodes in a graph as frozen if they are the same color in all legal colorings. This is analogous to studies of the development of a backbone or spine in SAT (the Satisability problem). We rst describe in detail the algorithmic techniques used to study frozen development. We present strong empirical evidence that freezing in 3-coloring is sudden. A single edge typically causes the size of the graph to collapse in size by 28%. We also use the frozen development to calculate unbiased estimates of probability of colorability in random graphs, even where this probability is as low as 10 300 . We investigate the links between frozen development and the solution cost of graph coloring. In SAT, a discontinuity in the order parameter has been correlated with the hardness of SAT instances, and our data for coloring is suggestive of an asymptotic discontinuity. The uncolorability threshold is known to give rise to har...

Abstract. We consider the multilevel paradigm and its potential to aid the solution of combinatorial optimisation problems. The multilevel paradigm is a simple one, which involves recursive coarsening to create a hierarchy of approximations to the original problem. An initial solution is found (sometimes for the original problem, sometimes the coarsest) and then iteratively refined at each level. As a general solution strategy, the multilevel paradigm has been in use for many years and has been applied to many problem areas (most notably in the form of multigrid techniques). However, with the exception of the graph partitioning problem, multilevel techniques have not been widely applied to combinatorial optimisation problems. In this paper we address the issue of multilevel refinement for such problems and, with the aid of examples and results in graph partitioning, graph colouring and the travelling salesman problem, make a case for its use as a metaheuristic. The results provide compelling evidence that, although the multilevel framework cannot be considered as a panacea for combinatorial problems, it can provide an extremely useful addition to the combinatorial optimisation toolkit. We also give a possible explanation for the underlying process and extract some generic guidelines for its future use on other combinatorial problems.

In a previous work, we proposed a new integer programming formulation for the graph coloring problem which, to a certain extent, avoids symmetry. We studied the facet structure of the 0/1-polytope associated with it. Based on these theoretical results, we present now a Branch-and-Cut algorithm for the graph coloring problem. Our computational experiences compare favorably with the well-known exact graph coloring algorithm DSATUR. Keyword: Graph Coloring; Integer Programming; Branch-and-Cut algorithms 1

Employee timetabling is the operation of assigning employees to tasks in a set of shifts during a xed period of time, typically a week. We present a general de nition of employee timetabling problems (ETPs) that captures many real-world problem formulations and includes complex constraints. The proposed model of ETPs can be represented in a tabular form that is both intuitive and ecient for constraint representation and processing. The constraint networks of ETPs include non-binary constraints and are dicult to formulate in terms of simple constraint solvers. We investigate the use of local search techniques for solving ETPs. In particular, we propose several versions of hill-climbing that make use of a novel search space that includes also partial assignments.

Two contrasting search paradigms for solving combinatorial problems are systematic backtracking and local search. The former is often eective on highly structured problems because of its ability to exploit consistency techniques, while the latter tends to scale better on very large problems. Neither approach is ideal for all problems, and a current trend in arti cial intelligence is the hybridisation of search techniques. This paper describes a use of forward checking in local search: pruning coloration neighbourhoods for graph colouring. The approach is evaluated on standard benchmarks and compared with several other algorithms. Good results are obtained; in particular, one variant nds improved colourings on geometric graphs, while another is very eective on equipartite graphs. Its application to other combinatorial problems is discussed.

We motivate, derive and implement a multilevel approach to the graph colouring problem. The resulting algorithm progressively coarsens the problem, initialises a colouring and then employs either Culberson 's iterated greedy algorithm or tabu search to refine the solution on each of the coarsened problems in reverse order. Tests on a large suite of problem instances indicate that for low-density graphs (up to around 30% edge density) the multilevel paradigm can either speed up (iterated greedy) or even improve (tabu search) the asymptotic convergence. This augments existing evidence that, although the multilevel framework cannot be considered as a panacea for combinatorial optimisation problems, it can provide a useful addition to the combinatorial optimisation toolkit.

This is a brief description of the Smallk coloring program. We define a list coloring problem[1, 4] as a graph G = (V; E) where each vertex has a set of available colors, A(v), also called its availset. For our algorithm, we first formulate a k-coloring problem as a list coloring, initializing the availsets A(v) = f1; : : : ; kg; 8v. In the description of the algorithm we will frequently need to express the fact that for every coloring c of G, for a pair of vertices x; y, either c[x] = a or c[y] = b. We use the notation hx = a. y = bi. A set of such clauses is maintained for every pair of vertices in the graph. As a notational convenience, for U ` V, we define A(U) = [x2UA(x), the union of the available colors over U. Thus, A(N(v)) is the set of available colors of the neighbors of v. The algorithm proceeds in two stages. Stage 1: In the first stage a recursive backtrack search reduces the graph by the deletion of edges and vertices and available colors, where each reduction step is recorded in a stack. When backtrack is forced, the stack is popped and the graph restored to its state at the previous search depth. If the graph becomes empty, the algorithm proceeds to the second stage, otherwise when no further options are available it reports failure. Stage 2: In the second stage the stack information is used to reconstruct and color the graph, with each vertex assigned its final color as it is popped from the stack. In a final step, the coloring is verified for correctness on a copy of the original graph. The graph reductions are generalizations and extensions of those in [2, 3]. Many of the following reductions are applied recursively at each step of the search. Some may only be applied to the initial graph due to efficiency reasons, as early experiments inicated the overhead exceeded the savings in practice.

Abstract. In this paper, we study the perturbation operator of Iterated Local Search. To guide more efficiently the search to move towards new promising regions of the search space, we introduce a Critical Element-Guided Perturbation strategy (CEGP). This perturbation approach consists of the identification of critical elements and then focusing on these critical elements within the perturbation operator. Computational experiments on two case studies—graph coloring and course timetabling—give evidence that this critical element-guided perturbation strategy helps reinforce the performance of Iterated Local Search.