Vertex Descent is a local search algorithm which forms the basis of a wide spectrum of tabu search, simulated annealing and hybrid evolutionary algorithms for graph colouring. These algorithms are usually treated as experimental and provide strong results on established benchmarks. As a step towards studying these heuristics analytically, an analysis of the behaviour of Vertex Descent is provided. It is shown that Vertex Descent is able to find feasible colourings for several types of instances in expected polynomial time. This includes 2-colouring of paths and 3-colouring of graphs with maximum degree 3. The same also holds for 3-colouring of a subset of 3-colourable graphs with maximum degree 4. As a consequence, Vertex Descent finds a 3-colouring in expected polynomial time for the smallest graph for which Br 'elaz's heuristic DSATUR needs 4 colours. On the other hand, Vertex Descent may fail for forests with maximum degree 3 with high probability.
Numerous analytical results have also been obtained for other combinatorial optimisation problems. For the maximum matching problem, behaviours of local search and evolutionary algorithms for paths and worst-case approximations were investigated @cite_15 . For the vertex cover problem, behaviour of hybrid algorithms was analysed for specific types of graphs @cite_37 and an iterated local search algorithm was investigated in the context of sparse random graphs @cite_51 . Fixed-parameter evolutionary algorithms were also analysed @cite_14 . Other studied problems include makespan scheduling @cite_32 , the Euclidean travelling salesperson problem @cite_2 , the Eulerian cycle problem @cite_22 , the minimum spanning tree problem @cite_45 or the minimum cut problem @cite_43 .
@cite_37 @cite_14 @cite_22 @cite_32 @cite_43 @cite_45 @cite_2 @cite_15 @cite_51
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We present a general approximation technique for a large class of graph problems. Our technique mostly applies to problems of covering, at minimum cost, the vertices of a graph with trees, cycles or paths satisfying certain requirements. In particular, many basic combinatorial optimization problems fit in this framework, including the shortest path, minimum-cost spanning tree, minimum-weight perfect matching, traveling salesman and Steiner tree problems. Our technique produces approximation algorithms that run in @math time and come within a factor of 2 of optimal for most of these problems. For instance, we obtain a 2-approximation algorithm for the minimum-weight perfect matching problem under the triangle inequality. Our running time of @math time compares favorably with the best strongly polynomial exact algorithms running in @math time for dense graphs. A similar result is obtained for the 2-matching problem and its variants. We also derive the first approximation algorithms for many NP-complete problems, including the non-fixed point-to-point connection problem, the exact path partitioning problem and complex location-design problems. Moreover, for the prize-collecting traveling salesman or Steiner tree problems, we obtain 2-approximation algorithms, therefore improving the previously best-known performance guarantees of 2.5 and 3, respectively [Math. Programming, 59 (1993), pp. 413--420]. For some positive constant , we give a (3 2- )-approximation algorithm for the following problem: given a graph G_0=(V,E_0), find the shortest tour that visits every vertex at least once. This is a special case of the metric traveling salesman problem when the underlying metric is defined by shortest path distances in G_0. The result improves on the 3 2-approximation algorithm due to Christofides [C76] for this special case. Similar to Christofides, our algorithm finds a spanning tree whose cost is upper bounded by the optimum, then it finds the minimum cost Eulerian augmentation (or T-join) of that tree. The main difference is in the selection of the spanning tree. Except in certain cases where the solution of LP is nearly integral, we select the spanning tree randomly by sampling from a maximum entropy distribution defined by the linear programming relaxation. Despite the simplicity of the algorithm, the analysis builds on a variety of ideas such as properties of strongly Rayleigh measures from probability theory, graph theoretical results on the structure of near minimum cuts, and the integrality of the T-join polytope from polyhedral theory. Also, as a byproduct of our result, we show new properties of the near minimum cuts of any graph, which may be of independent interest. For a given graph G over n vertices, let OPT"G denote the size of an optimal solution in G of a particular minimization problem (e.g., the size of a minimum vertex cover). A randomized algorithm will be called an @a-approximation algorithm with an additive error for this minimization problem if for any given additive error parameter @e>0 it computes a value [email protected]? such that, with probability at least 2 3, it holds that OPT"[email protected][email protected][email protected][email protected]@?OPT"[email protected] Assume that the maximum degree or average degree of G is bounded. In this case, we show a reduction from local distributed approximation algorithms for the vertex cover problem to sublinear approximation algorithms for this problem. This reduction can be modified easily and applied to other optimization problems that have local distributed approximation algorithms, such as the dominating set problem. We also show that for the minimum vertex cover problem, the query complexity of such approximation algorithms must grow at least linearly with the average degree [email protected]? of the graph. This lower bound holds for every multiplicative factor @a and small constant @e as long as [email protected]?=O(n @a). In particular this means that for dense graphs it is not possible to design an algorithm whose complexity is o(n). In this chapter we review the main results known on local search algorithms with worst case guarantees. We consider classical combinatorial optimization problems: satisfiability problems, traveling salesman and quadratic assignment problems, set packing and set covering problems, maximum independent set, maximum cut, several facility location related problems and finally several scheduling problems. A replica placement problem in a distributed file systems is also considered as an example of the use of a local search algorithm in a distributed environment. For each problem we have provided the neighborhoods used along with approximation results. Proofs when too technical are omitted, but often sketch of proofs are provided.