For more than 40 years, Branch & Reduce exponential-time backtracking algorithms have been among the most common tools used for finding exact solutions of NP-hard problems. Despite that, the way to analyze such recursive algorithms is still far from producing tight worst-case running time bounds. Motivated by this, we use an approach, that we call "Measure & Conquer", as an attempt to step beyond such limitations. The approach is based on the careful design of a nonstandard measure of the subproblem size; this measure is then used to lower bound the progress made by the algorithm at each branching step. The idea is that a smarter measure may capture behaviors of the algorithm that a standard measure might not be able to exploit, and hence lead to a significantly better worst-case time analysis.In order to show the potentialities of Measure & Conquer, we consider two well-studied NP-hard problems: minimum dominating set and maximum independent set. For the first problem, we consider the current best algorithm, and prove (thanks to a better measure) a much tighter running time bound for it. For the second problem, we describe a new, simple algorithm, and show that its running time is Preliminary parts of this article appeared in competitive with the current best time bounds, achieved with far more complicated algorithms (and standard analysis).Our examples show 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.

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.

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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 ).

Abstract.A certifying algorithm for a problem is an algorithm that provides a certificate with each answer that it produces. The certificate is a piece of evidence that proves that the answer has not been compromised by a bug in the implementation. We give linear-time certifying algorithms for recognition of interval graphs and permutation graphs, and for a few other related problems. Previous algorithms fail to provide supporting evidence when they claim that the input graph is not a member of the class. We show that our certificates of nonmembership can be authenticated in O(|V |) time. 1. Introduction. A recognition algorithm is an algorithm that decides whether some given input (graph, geometrical object, picture, etc.) has a certain property. Such an algorithm accepts the input if it has the property or rejects it if it does not. A certifying algorithm for a decision problem is an algorithm that provides a certificate with each answer that it produces. The certificate is a piece of evidence that proves that the answer has not been compromised by a bug in the implementation.We give linear-time certifying algorithms for recognition of interval graphs and permutation graphs. Previous algorithms fail to provide supporting evidence of nonmembership. We show that our certificates of nonmembership can be authenticated in O(n) time, where n is the number of vertices.A familiar example of a certifying recognition algorithm is a recognition algorithm for bipartite graphs that computes a 2-coloring for bipartite input graphs and an odd cycle for nonbipartite input graphs. A more complex example is the linear-time planarity test which is part of the library of efficient data structures and algorithms (LEDA) system [19, section 8.7]. It computes a planar embedding for planar input graphs and a Kuratowski subgraph (a subdivision of K 5 or K 3,3 ) for nonplanar input graphs.Certifying versions of recognition algorithms are highly desirable in practice; see [28,20,21] and [19, section 2.14] for general discussions on result checking. Consider a planarity testing algorithm that produces a planar embedding if the graph is planar, and simply declares it nonplanar otherwise. Though the algorithm may have been proven correct, the implementation may contain bugs. When the algorithm

A v ertex edge coloring c : V ! f 1; 2; : : : ; t g c 0 : E ! f 1; 2; : : : ; t g of a graph G = V ;E i s a v ertex edge t-ranking if for any t wo v ertices edges of the same color every path between them contains a vertex edge of larger color. The vertex ranking number r G edge ranking number 0 r G is the smallest value of t such that G has a vertex edge t-ranking. In this paper we study the algorithmic complexity o f t h e vertex ranking and edge ranking problems. Among others it is shown that r G can be computed in polynomial time when restricted to graphs with treewidth at most k for any xed k. We c haracterize those graphs where the vertex ranking number r and the chromatic number coincide on all induced subgraphs, show that r G = G implies G = !G largest clique size and give a formula for 0 r K n .

We present O(n'R + n3R3) time algorithms to compute the treewidth, pathwidth, minimum fill-in and minimum interval graph completion of asteroidal triple-free graphs, where n is the number of vertices and R is the number of minimal separators of the input graph. This yields polynomial time algorithms for the four NP-complete graph problems on any subclass of the asteroidal triple-free graphs that has a polynomially bounded number of minimal separators, as e.g. cocomparability graphs of bounded dimension and d-trapezoid graphs for any fixed d > 1.

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