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Measure and Conquer: Domination -- A case study
"... 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 ..."
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Cited by 57 (22 self)
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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 the 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
Measure and Conquer: A Simple O(2^0.288n) Independent Set Algorithm
"... 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 ..."
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Cited by 41 (4 self)
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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.
Space and time complexity of exact algorithms: Some open problems
, 2004
"... We discuss open questions around worst case time and space bounds for NP-hard problems. We are interested in exponential time solutions for these problems with a relatively good worst case behavior. ..."
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Cited by 30 (0 self)
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We discuss open questions around worst case time and space bounds for NP-hard problems. We are interested in exponential time solutions for these problems with a relatively good worst case behavior.
Exact algorithms for treewidth and minimum fill-in
- In Proceedings of the 31st International Colloquium on Automata, Languages and Programming (ICALP 2004). Lecture Notes in Comput. Sci
, 2004
"... 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 g ..."
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Cited by 28 (17 self)
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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).
Solving Connected Dominating Set Faster than 2^n
, 2006
"... In the connected dominating set problem we are given an n-node undirected graph, and we are asked to find a minimum cardinality connected subset S of nodes such that each node not in S is adjacent to some node in S. This problem is also equivalent to finding a spanning tree with maximum number of le ..."
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Cited by 25 (9 self)
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In the connected dominating set problem we are given an n-node undirected graph, and we are asked to find a minimum cardinality connected subset S of nodes such that each node not in S is adjacent to some node in S. This problem is also equivalent to finding a spanning tree with maximum number of leaves. Despite its relevance in applications, the best known exact algorithm for the problem is the trivial Ω(2 n) algorithm that enumerates all the subsets of nodes. This is not the case for the general (unconnected) version of the problem, for which much faster algorithms are available. Such a difference is not surprising, since connectivity is a global property, and non-local problems are typically much harder to solve exactly. In this paper we break the 2 n barrier, by presenting a simple O(1.9407 n) algorithm for the connected dominating set problem. The algorithm makes use of new domination rules, and its analysis is based on the Measure and Conquer technique.
Exact algorithms for exact satisfiability and number of perfect matchings
- In Proc. 33rd ICALP
, 2006
"... Abstract. We present exact algorithms with exponential running times for variants of n-element set cover problems, based on divide-and-conquer and on inclusion–exclusion characterisations. We show that the Exact Satisfiability problem of size l with m clauses can be solved in time 2 m l O(1) and pol ..."
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Cited by 22 (7 self)
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Abstract. We present exact algorithms with exponential running times for variants of n-element set cover problems, based on divide-and-conquer and on inclusion–exclusion characterisations. We show that the Exact Satisfiability problem of size l with m clauses can be solved in time 2 m l O(1) and polynomial space. The same bounds hold for counting the number of solutions. As a special case, we can count the number of perfect matchings in an n-vertex graph in time 2 n n O(1) and polynomial space. We also show how to count the number of perfect matchings in time O(1.732 n) and exponential space. Using the same techniques we show how to compute Chromatic Number of an n-vertex graph in time O(2.4423 n) and polynomial space, or time O(2.3236 n) and exponential space. 1
Algorithms for counting 2-SAT solutions and colorings with applications
- TR05-033, Electronic Colloquium on Computational Complexity
, 2005
"... An algorithm is presented for exactly solving (in fact, counting) the number of maximum weight satisfying assignments of a 2-Cnf formula. The worst case running time of ..."
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Cited by 19 (2 self)
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An algorithm is presented for exactly solving (in fact, counting) the number of maximum weight satisfying assignments of a 2-Cnf formula. The worst case running time of
Algorithms Based on the Treewidth Of Sparse Graphs
- IN PROCEEDINGS OF THE 31ST INTERNATIONAL WORKSHOP ON GRAPH-THEORETIC CONCEPTS IN COMPUTER SCIENCE (WG 2005), LNCS
, 2005
"... We prove that given a graph, one can efficiently find a set of no more than m/5.217 + 1 nodes whose removal yields a partial two-tree. As an ..."
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Cited by 16 (0 self)
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We prove that given a graph, one can efficiently find a set of no more than m/5.217 + 1 nodes whose removal yields a partial two-tree. As an
