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40
Set partitioning via inclusionexclusion
 SIAM J. Comput
"... Abstract. Given a set N with n elements and a family F of subsets, we show how to partition N into k such subsets in 2nnO(1) time. We also consider variations of this problem where the subsets may overlap or are weighted, and we solve the decision, counting, summation, and optimisation versions of t ..."
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Cited by 59 (7 self)
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Abstract. Given a set N with n elements and a family F of subsets, we show how to partition N into k such subsets in 2nnO(1) time. We also consider variations of this problem where the subsets may overlap or are weighted, and we solve the decision, counting, summation, and optimisation versions of these problems. Our algorithms are based on the principle of inclusion–exclusion and the zeta transform. In effect we get exact algorithms in 2nnO(1) time for several wellstudied partition problems including Domatic Number, Chromatic Number, Maximum kCut, Bin Packing, List Colouring, and the Chromatic Polynomial. We also have applications to Bayesian learning with decision graphs and to modelbased data clustering. If only polynomial space is available, our algorithms run in time 3nnO(1) if membership in F can be decided in polynomial time. We solve Chromatic Number in O(2.2461n) time and Domatic Number in O(2.8718n) time. Finally, we present a family of polynomial space approximation algorithms that find a number between χ(G) and d(1 + )χ(G)e in time O(1.2209n + 2.2461e−n). 1. Introduction. Graph colouring, domatic partitioning, weighted kcut, and a
A measure & conquer approach for the analysis of exact algorithms
, 2007
"... For more than 40 years Branch & Reduce exponentialtime backtracking algorithms have been among the most common tools used for finding exact solutions of NPhard problems. Despite that, the way to analyze such recursive algorithms is still far from producing tight worstcase running time bounds. ..."
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Cited by 49 (11 self)
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For more than 40 years Branch & Reduce exponentialtime backtracking algorithms have been among the most common tools used for finding exact solutions of NPhard problems. Despite that, the way to analyze such recursive algorithms is still far from producing tight worstcase 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 worstcase time analysis. In order to show the potentialities of Measure & Conquer, we consider two wellstudied NPhard 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 competitive with the current best time bounds, achieved with far more complicated algorithms (and standard analysis). Our examples
Finding, minimizing, and counting weighted subgraphs
 In Proceedings of the FourtyFirst Annual ACM Symposium on the Theory of Computing
, 2009
"... For a pattern graph H on k nodes, we consider the problems of finding and counting the number of (not necessarily induced) copies of H in a given large graph G on n nodes, as well as finding minimum weight copies in both nodeweighted and edgeweighted graphs. Our results include: • The number of cop ..."
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Cited by 31 (4 self)
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For a pattern graph H on k nodes, we consider the problems of finding and counting the number of (not necessarily induced) copies of H in a given large graph G on n nodes, as well as finding minimum weight copies in both nodeweighted and edgeweighted graphs. Our results include: • The number of copies of an H with an independent set of size s can be computed exactly in O ∗ (2 s n k−s+3) time. A minimum weight copy of such an H (with arbitrary real weights on nodes and edges) can be found in O(4 s+o(s) n k−s+3) time. (The O ∗ notation omits poly(k) factors.) These algorithms rely on fast algorithms for computing the permanent of a k × n matrix, over rings and semirings. • The number of copies of any H having minimum (or maximum) nodeweight (with arbitrary real weights on nodes) can be found in O(n ωk/3 + n 2k/3+o(1) ) time, where ω < 2.4 is the matrix multiplication exponent and k is divisible by 3. Similar results hold for other values of k. Also, the number of copies having exactly a prescribed weight can be found within this time. These algorithms extend the technique of Czumaj and Lingas (SODA 2007) and give a new (algorithmic) application of multiparty communication complexity. • Finding an edgeweighted triangle of weight exactly 0 in general graphs requires Ω(n 2.5−ε) time for all ε> 0, unless the 3SUM problem on N numbers can be solved in O(N 2−ε) time. This suggests that the edgeweighted problem is much harder than its nodeweighted version. 1
An O∗(2 n ) algorithm for graph coloring and other partitioning problems via inclusionexclusion
 IN PROCEEDINGS OF THE 47TH ANNUAL IEEE SYMPOSIUM ON FOUNDATIONS OF COMPUTER SCIENCE (FOCS 2006), IEEE
, 2006
"... We use the principle of inclusion and exclusion, combined with polynomial time segmentation and fast Möbius transform, to solve the generic problem of summing or optimizing over the partitions of n elements into a given number of weighted subsets. This problem subsumes various classical graph partit ..."
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Cited by 23 (1 self)
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We use the principle of inclusion and exclusion, combined with polynomial time segmentation and fast Möbius transform, to solve the generic problem of summing or optimizing over the partitions of n elements into a given number of weighted subsets. This problem subsumes various classical graph partitioning problems, such as graph coloring, domatic partitioning, and MAX kCUT, aswell as machine learning problems like decision graph learning and modelbased data clustering. Our algorithm runs in O ∗ (2 n) time, thus substantially improving on the usual O ∗ (3 n)time dynamic programming algorithm; the notation O ∗ suppresses factors polynomial in n. This result improves, e.g., Byskov’s recent record for graph coloring from O ∗ (2.4023 n) to O ∗ (2 n). We note that twenty five years ago, R. M. Karp used inclusion–exclusion in a similar fashion to reduce the space requirement of the usual dynamic programming algorithms from exponential to polynomial.
Combinatorial bounds via measure and conquer: Boundings minimal dominating sets and applications
 PRELIM.VERSION IN PROC. 16TH ISAAC
, 2006
"... We provide an algorithm listing all minimal dominating sets of a graph on n vertices in time O(1.7159n). This result can be seen as an algorithmic proof of the fact that the number of minimal dominating sets in a graph on n vertices is at most 1.7159n, thus improving on the trivial O(2n / √ n) boun ..."
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Cited by 18 (4 self)
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We provide an algorithm listing all minimal dominating sets of a graph on n vertices in time O(1.7159n). This result can be seen as an algorithmic proof of the fact that the number of minimal dominating sets in a graph on n vertices is at most 1.7159n, thus improving on the trivial O(2n / √ n) bound. Our result makes use of the measure and conquer technique which was recently developed in the area of exact algorithms. Based on this result, we derive an O(2.8718n) algorithm for the domatic number problem.
The Time Complexity of Constraint Satisfaction
"... Abstract. We study the time complexity of (d, k)CSP, the problem of deciding satisfiability of a constraint system C with n variables, domain size d, and at most k variables per constraint. We are interested in the question how the domain size d influences the complexity of deciding satisfiability. ..."
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Cited by 16 (0 self)
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Abstract. We study the time complexity of (d, k)CSP, the problem of deciding satisfiability of a constraint system C with n variables, domain size d, and at most k variables per constraint. We are interested in the question how the domain size d influences the complexity of deciding satisfiability. We show, assuming the Exponential Time Hypothesis, that two special cases, namely (d, 2)CSP with bounded variable frequency and dUNIQUECSP, already require exponential time Ω(d c·n) for some c> 0 independent of d. UNIQUECSP is the special case for which it is guaranteed that every input constraint system has at most 1 satisfying assignment. 1
Algorithms and Resource Requirements for Fundamental Problems
, 2007
"... no. DGE0234630. The views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of any sponsoring institution, the U.S. government or any other entity. ..."
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Cited by 11 (5 self)
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no. DGE0234630. The views and conclusions contained in this document are those of the author and should not be interpreted as representing the official policies, either expressed or implied, of any sponsoring institution, the U.S. government or any other entity.
Popular conjectures imply strong lower bounds for dynamic problems
 CoRR
"... Abstract—We consider several wellstudied problems in dynamic algorithms and prove that sufficient progress on any of them would imply a breakthrough on one of five major open problems in the theory of algorithms: 1) Is the 3SUM problem on n numbers in O(n2−ε) time for some ε> 0? 2) Can one dete ..."
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Cited by 11 (3 self)
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Abstract—We consider several wellstudied problems in dynamic algorithms and prove that sufficient progress on any of them would imply a breakthrough on one of five major open problems in the theory of algorithms: 1) Is the 3SUM problem on n numbers in O(n2−ε) time for some ε> 0? 2) Can one determine the satisfiability of a CNF formula on n variables and poly n clauses in O((2 − ε)npoly n) time for some ε> 0? 3) Is the All Pairs Shortest Paths problem for graphs on n vertices in O(n3−ε) time for some ε> 0? 4) Is there a linear time algorithm that detects whether a given graph contains a triangle? 5) Is there an O(n3−ε) time combinatorial algorithm for n×n Boolean matrix multiplication? The problems we consider include dynamic versions of bipartite perfect matching, bipartite maximum weight matching, single source reachability, single source shortest paths, strong connectivity, subgraph connectivity, diameter approximation and some nongraph problems such as Pagh’s problem defined in a recent paper by Pǎtraşcu[STOC 2010]. Index Terms—dynamic algorithms; all pairs shortest paths; 3SUM; lower bounds; I.
Exponential Time Algorithms: Structures, Measures, and Bounds
, 2008
"... This thesis studies exponential time algorithms, more precisely, algorithms exactly solving problems for which no polynomial time algorithm is known and likely to exist. Interested in worst–case upper bounds on the running times, several known techniques to design and analyze such algorithms are sur ..."
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Cited by 10 (6 self)
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This thesis studies exponential time algorithms, more precisely, algorithms exactly solving problems for which no polynomial time algorithm is known and likely to exist. Interested in worst–case upper bounds on the running times, several known techniques to design and analyze such algorithms are surveyed. A detailed presentation of the design and especially the analysis of branching algorithms is given. Then, the branching paradigm is used to design faster algorithms for various problems, including the Feedback Vertex Set problem, #Maximal Independent Sets, Max 2Sat, Max 2CSP, and mixed instances of the latter two problems. The analysis of these algorithms heavily relies on problem–specific measures of the instances. These measures capture the structure of the instances, not merely their size. This makes them more appropriate to quantify the progress an algorithm makes in the process of solving a problem for an instance. Upper bounds on mathematical objects are also proved in this thesis. A bound on the maximum number of minimal feedback vertex sets is derived via the same
Improved fixed parameter tractable algorithms for two “edge” problems
 MAXCUT and MAXDAG. Information Processing Letters
"... This article was published in an Elsevier journal. The attached copy is furnished to the author for noncommercial research and education use, including for instruction at the author’s institution, sharing with colleagues and providing to institution administration. Other uses, including reproductio ..."
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Cited by 7 (1 self)
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This article was published in an Elsevier journal. The attached copy is furnished to the author for noncommercial research and education use, including for instruction at the author’s institution, sharing with colleagues and providing to institution administration. Other uses, including reproduction and distribution, or selling or licensing copies, or posting to personal, institutional or third party websites are prohibited. In most cases authors are permitted to post their version of the article (e.g. in Word or Tex form) to their personal website or institutional repository. Authors requiring further information regarding Elsevier’s archiving and manuscript policies are encouraged to visit: