Author

# Mathieu Liedloff

Other affiliations: Metz

Bio: Mathieu Liedloff is an academic researcher from University of Orléans. The author has contributed to research in topics: Dominating set & Parameterized complexity. The author has an hindex of 18, co-authored 76 publications receiving 981 citations. Previous affiliations of Mathieu Liedloff include Metz.

##### Papers

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TL;DR: This paper presents a branching algorithm whose running time has been analyzed using the Measure-and-Conquer technique and provides a lower bound of ?

73 citations

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TL;DR: It is shown that the maximum number of maximal bicliques in a graph on n vertices is Θ(3n/3), and an exact exponential-time algorithm is used that computes the number of distinct maximal independent sets in a graphs in time O(1.3642n), where n is thenumber of vertices of the input graph.

Abstract: Bicliques of graphs have been studied extensively, partially motivated by the large number of applications. In this paper we improve Prisner’s upper bound on the number of maximal bicliques (Combinatorica, 20, 109–117, 2000) and show that the maximum number of maximal bicliques in a graph on n vertices is Θ(3 n/3). Our major contribution is an exact exponential-time algorithm. This branching algorithm computes the number of distinct maximal independent sets in a graph in time O(1.3642 n ), where n is the number of vertices of the input graph. We use this counting algorithm and previously known algorithms for various other problems related to independent sets to derive algorithms for problems related to bicliques via polynomial-time reductions.

64 citations

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TL;DR: Cockayne et al. as discussed by the authors showed that the Roman domination number of an interval graph can be computed in linear time, and they also gave polynomial-time algorithms for computing Roman domination numbers of AT-free graphs and graphs with a d-octopus.

57 citations

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TL;DR: This work considers the $\mathcal{NP}$-hard problem of finding a spanning tree with a maximum number of internal vertices, and develops a branching algorithm for graphs with maximum degree three that only needs polynomial space and has a running time of 1.8612 n when analyzed with respect to the number of vertices.

Abstract: We consider the $\mathcal{NP}$-hard problem of finding a spanning tree with a maximum number of internal vertices. This problem is a generalization of the famous Hamiltonian Path problem. Our dynamic-programming algorithms for general and degree-bounded graphs have running times of the form $\mathcal{O}^{*}(c^{n})$ with c≤2. For graphs with bounded degree, c<2. The main result, however, is a branching algorithm for graphs with maximum degree three. It only needs polynomial space and has a running time of $\mathcal{O}(1.8612^{n})$ when analyzed with respect to the number of vertices. We also show that its running time is $2.1364^{k} n^{\mathcal{O}(1)}$ when the goal is to find a spanning tree with at least k internal vertices. Both running time bounds are obtained via a Measure & Conquer analysis, the latter one being a novel use of this kind of analysis for parameterized algorithms.

50 citations

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25 Aug 2008

TL;DR: Iterative compression has recently led to a number of breakthroughs in parameterized complexity and it is shown that the technique can also be useful in the design of exact exponential time algorithms to solve NP-hard problems.

Abstract: Iterative compression has recently led to a number of breakthroughs in parameterized complexity. Here, we show that the technique can also be useful in the design of exact exponential time algorithms to solve NP-hard problems. We exemplify our findings with algorithms for the Maximum Independent Set problem, a parameterized and a counting version of d-Hitting Set and the Maximum Induced Cluster Subgraph problem.

48 citations

##### Cited by

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TL;DR: 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 step beyond limitations of current algorithms design.

Abstract: 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 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.

284 citations

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22 Nov 2006

TL;DR: Koivisto et al. as discussed by the authors presented an O(2k n2 + n m) algorithm for the Steiner tree problem in graphs with n vertices, k terminals, and m edges with bounded integer weights.

Abstract: We present a fast algorithm for the subset convolution problem:given functions f and g defined on the lattice of subsets of ann-element set n, compute their subset convolution f*g, defined for S⊆ N by [ (f * g)(S) = [T ⊆ S] f(T) g(S/T),,]where addition and multiplication is carried out in an arbitrary ring. Via Mobius transform and inversion, our algorithm evaluates the subset convolution in O(n2 2n) additions and multiplications, substanti y improving upon the straightforward O(3n) algorithm. Specifically, if the input functions have aninteger range [-M,-M+1,...,M], their subset convolution over the ordinary sum--product ring can be computed in O(2n log M) time; the notation O suppresses polylogarithmic factors.Furthermore, using a standard embedding technique we can compute the subset convolution over the max--sum or min--sum semiring in O(2n M) time.To demonstrate the applicability of fast subset convolution, wepresent the first O(2k n2 + n m) algorithm for the Steiner tree problem in graphs with n vertices, k terminals, and m edges with bounded integer weights, improving upon the O(3kn + 2k n2 + n m) time bound of the classical Dreyfus-Wagner algorithm. We also discuss extensions to recent O(2n)-time algorithms for covering and partitioning problems (Bjorklund and Husfeldt, FOCS 2006; Koivisto, FOCS 2006).

280 citations

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TL;DR: Work in the area encompasses both structural questions (Is the reconfiguration graph connected?) and algorithmic ones (How can one find the shortest sequence of steps between two solutions?)

Abstract: Reconfiguration is concerned with relationships among solutions to a problem instance, where the reconfiguration of one solution to another is a sequence of steps such that each step produces an intermediate feasible solution. The solution space can be represented as a reconfiguration graph, where two vertices representing solutions are adjacent if one can be formed from the other in a single step. Work in the area encompasses both structural questions (Is the reconfiguration graph connected?) and algorithmic ones (How can one find the shortest sequence of steps between two solutions?) This survey discusses techniques, results, and future directions in the area.

174 citations

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TL;DR: The goal of the problem is to find out an L(h, k)-labelling with a minimum span, concerning both the values of h and k and the considered classes of graphs.

Abstract: Given any fixed non-negative integer values h and k, the L(h, k)-labelling problem consists in an assignment of non-negative integers to the nodes of a graph such that adjacent nodes receive values which differ by at least h, and nodes connected by a 2-length path receive values which differ by at least k. The span of an L(h, k)-labelling is the difference between the largest and the smallest assigned frequency. The goal of the problem is to find out an L(h, k)-labelling with a minimum span. The L(h, k)-labelling problem has intensively been studied following many approaches and restricted to many special cases, concerning both the values of h and k and the considered classes of graphs. This paper reviews the results from previously published literature, looking at the problem with a graph algorithmic approach. It is an update of a previous survey written by the same author.

173 citations

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TL;DR: It is proved that $\gamma_R(G))\leq8n/11$ when $\delta(G)\geq2$ and $n\geq9$, and this is sharp.

Abstract: A Roman dominating function of a graph $G$ is a labeling $f\colon\,V(G)\to\{0,1,2\}$ such that every vertex with label 0 has a neighbor with label 2. The Roman domination number $\gamma_R(G)$ of $G$ is the minimum of $\sum_{v\in V(G)}f(v)$ over such functions. Let $G$ be a connected $n$-vertex graph. We prove that $\gamma_R(G)\leq4n/5$, and we characterize the graphs achieving equality. We obtain sharp upper and lower bounds for $\gamma_R(G)+\gamma_R(\overline{G})$ and $\gamma_R(G)\gamma_R(\overline{G})$, improving known results for domination number. We prove that $\gamma_R(G)\leq8n/11$ when $\delta(G)\geq2$ and $n\geq9$, and this is sharp.

163 citations