Recently, there have been increasing interests and progresses in lowering the worst case time complexity for well-known NP-hard problems, in particular for the Vertex Cover problem. In this paper, new properties for the Vertex Cover problem are indicated and several simple and new techniques are introduced, which lead to an improved algorithm of time O(kn + 1:271 k k 2 ) for the problem. Our algorithm also induces improvement on previous algorithms for the Independent Set problem on graphs of small degree. 1 Introduction Many optimization problems from industrial applications are NP-hard. According to the NPcompleteness theory [10], these problems cannot be solved in polynomial time unless P = NP. However, this fact does not obviate the need for solving these problems for their practical importance. There has been a number of approaches to attacking the NP-hardness of optimization problems, including approximation algorithms, heuristic algorithms, and average time analysis. Recent...

Abstract. We discuss fast exponential time solutions for NP-complete problems. We survey known results and approaches, we provide pointers to the literature, and we discuss several open problems in this area. The list of discussed NP-complete problems includes the travelling salesman problem, scheduling under precedence constraints, satisfiability, knapsack, graph coloring, independent sets in graphs, bandwidth of a graph, and many more. 1

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Davis-Putnam-style exponential-time backtracking algorithms are the most common algorithms used for finding exact solutions of NP-hard problems. The analysis of such recursive algorithms is based on the bounded search tree technique: a measure of the size of the subproblems is defined; this measure is used to lower bound the progress made by the algorithm at each branching step. For the last 30 years the research on exact algorithms has been mainly focused on the design of more and more sophisticated algorithms. However, measures used in the analysis of backtracking algorithms are usually very simple. In this paper we stress that a more careful choice of the measure can lead to significantly better worst case time analysis. As an example, we consider the minimum dominating set problem. The currently fastest algorithm for this problem has running time O(2 0.850n) on nnodes graphs. By measuring the progress of the (same) algorithm in a different way, we refine the time bound to O(2 0.598n). A good choice of the measure can provide such a (surprisingly big) improvement; this suggests that the running time of many other exponential-time recursive algorithms is largely overestimated because of a “bad” choice of the measure.

a fvg); fvg [ SparseFindMIS(G \Gamma fvg \Gamma N(v))) end Figure 1: An algorithm that runs in time 2 0:114e on e-edge graphs function FindMIS(G) begin if G contains at most 5n=2 edges then return SparseFindMIS(G) if G contains an edge fu; vg such that N(u) ae N(v) [ fvg then return FindMIS(G \Gamma fvg) let d be the maximum degree of any vertex in G (* so d 6 *) if d 8 then let v be a degree-d vertex return max(FindMIS(G \Gamm

In this paper, we present a new load balancing algorithm and its application to distributed branch & bound algorithms. We demonstrate the efficiency of this scheme by solving some NP-complete problems on a network of up to 256 Transputers. The parallelization of our branch & bound algorithm is fully distributed. Every processor performs the same algorithm but on a different part of the solution tree. In this case, it is necessary to distribute subproblems among the processors to achieve a well balanced workload. We present a load balancing method which overcomes the problem of search overhead and idle times by an appropriate load model and avoids trashing effects by a feedback control strategy. To show the performance of our strategy, we solved the Vertex Cover and the weighted Vertex Cover problem for graphs of up to 150 nodes, using highly efficient branch and bound algorithms. Although the computing times were very short on a 256 processor network, we were able to achieve a speedup ...

We show that the treewidth and the minimum fill-in of an n-vertex graph can be computed in time O(1.8899 n). Our results are based on combinatorial proofs that an n-vertex graph has O(1.7087 n) minimal separators and O(1.8135 n) potential maximal cliques. We also show that for the class of AT-free graphs the running time of our algorithms can be reduced to O(1.4142 n).

The rapidly increasing number of known protein structures has resulted in a situation where approximate structures corresponding to new sequences are often available from one of two

For more than 30 years Davis-Putnam-style exponentialtime backtracking algorithms have been the most common tools used for finding exact solutions of NP-hard problems. Despite of that, the way to analyze such recursive algorithms is still far from producing tight worst case running time bounds. The “Measure and Conquer” approach is one of the recent attempts to step beyond such limitations. The approach is based on the choice of the measure of the subproblems recursively generated by the algorithm considered; this measure is used to lower bound the progress made by the algorithm at each branching step. A good choice of the measure can lead to a significantly better worst case time analysis. In this paper we apply “Measure and Conquer ” to the analysis of a very simple backtracking algorithm solving the well-studied maximum independent set problem. The result of the analysis is striking: the running time of the algorithm is O(2 0.288n), which is competitive with the current best time bounds obtained with far more complicated algorithms (and naive analysis). Our example shows that a good choice of the measure, made in the very first stages of exact algorithms design, can have a tremendous impact on the running time bounds achievable.

In Pattern Recognition, the Graph Matching problem involves the matching of a sample data graph with the subgraph of a larger model graph where vertices and edges correspond to pattern parts and their relations. In this paper, we present Rulegraphs, a new method that combines the Graph Matching approach with Rule-Based approaches from Machine Learning. This new method reduces the cardinality of the (NP-Complete) Graph Matching problem by replacing model part, and their relational, attribute states by rules which depict attribute bounds and evidence for di erent classes. We show how rulegraphs, when combined with techniques for checking feature label-compatibilities, not only reduce the search space but also improve the uniqueness of the matching process.