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Balanced families of perfect hash functions and their applications
 Proc. ICALP
, 2007
"... Abstract. The construction of perfect hash functions is a wellstudied topic. In this paper, this concept is generalized with the following definition. We say that a family of functions from [n] to[k] isaδbalanced (n, k)family of perfect hash functions if for every S ⊆ [n], S  = k, the number o ..."
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Cited by 12 (3 self)
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Abstract. The construction of perfect hash functions is a wellstudied topic. In this paper, this concept is generalized with the following definition. We say that a family of functions from [n] to[k] isaδbalanced (n, k)family of perfect hash functions if for every S ⊆ [n], S  = k, the number of functions that are 11 on S is between T/δ and δT for some constant T>0. The standard definition of a family of perfect hash functions requires that there will be at least one function that is 11 on S,for each S of size k. In the new notion of balanced families, we require the number of 11 functions to be almost the same (taking δ to be close to 1) for every such S. Our main result is that for any constant δ>1, a δbalanced (n, k)family of perfect hash functions of size 2 O(k log log k) log n can be constructed in time 2 O(k log log k) nlog n. Using the technique of colorcoding we can apply our explicit constructions to devise approximation algorithms for various counting problems in graphs. In particular, we exhibit a deterministic polynomial time algorithm for approximating both the number of simple paths of length k and the number of simple log n cycles of size k for any k ≤ O() in a graph with n vertices. The log log log n approximation is up to any fixed desirable relative error.
Connected coloring completion for general graphs: algorithms and complexity
 IN: PROCEEDINGS COCOON 2007, LECTURE NOTES IN COMPUTER SCIENCE
, 2007
"... An rcomponent connected coloring of a graph is a coloring of the vertices so that each color class induces a subgraph having at most r connected components. The concept has been wellstudied for r = 1, in the case of trees, under the rubric of convex coloring, used in modeling perfect phylogenies. ..."
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Cited by 9 (6 self)
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An rcomponent connected coloring of a graph is a coloring of the vertices so that each color class induces a subgraph having at most r connected components. The concept has been wellstudied for r = 1, in the case of trees, under the rubric of convex coloring, used in modeling perfect phylogenies. Several applications in bioinformatics of connected coloring problems on general graphs are discussed, including analysis of proteinprotein interaction networks and protein structure graphs, and of phylogenetic relationships modeled by splits trees. We investigate the rCOMPONENT CONNECTED COLORING COMPLETION (rCCC) problem, that takes as input a partially colored graph, having k uncolored vertices, and asks whether the partial coloring can be completed to an rcomponent connected coloring. For r = 1 this problem is shown to be NPhard, but fixedparameter tractable when parameterized by the number of uncolored vertices, solvable in time O∗(8k). We also show that the 1CCC problem, parameterized (only) by the treewidth t of the graph, is fixedparameter tractable; we show this by a method that is of independent interest. The rCCC problem is shown to be W [1]hard, when parameterized by the treewidth bound t, for any r ≥ 2. Our proof also shows that the problem is NPcomplete for r = 2, for general graphs.
AGAPE (ANR09BLAN0159) and the LanguedocRoussillon “Chercheur d’avenir ” project
"... the date of receipt and acceptance should be inserted later Abstract Vertex deletion and edge deletion problems play a central role in parameterized complexity. Examples include classical problems like Feedback Vertex Set, Odd Cycle Transversal, and Chordal Deletion. The study of analogous edge con ..."
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the date of receipt and acceptance should be inserted later Abstract Vertex deletion and edge deletion problems play a central role in parameterized complexity. Examples include classical problems like Feedback Vertex Set, Odd Cycle Transversal, and Chordal Deletion. The study of analogous edge contraction problems has so far been left largely unexplored from a parameterized perspective. We consider two basic problems of this type: Tree Contraction and Path Contraction. These two problems take as input an undirected graph G on n vertices and an integer k, and the task is to determine whether we can obtain a tree or a path, respectively, by a sequence of at most k edge contractions in G. For Tree Contraction, we present a randomized 4k nO(1) time polynomialspace algorithm, as well as a deterministic 4.98k nO(1) time algorithm, based on a variant of the color coding technique of Alon, Yuster and Zwick. We also present a deterministic 2k+o(k) + nO(1) time algorithm for Path Contraction. Furthermore, we show that Path Contraction has a kernel with at most 5k + 3 vertices, while Tree Contraction does not have a polynomial kernel unless NP ⊆ coNP/poly. We find the latter result surprising because of the connection be
The density maximization problem in graphs
 J COMB OPTIM
, 2013
"... We consider a framework for biobjective network construction problems where one objective is to be maximized while the other is to be minimized. Given a host graph G = (V,E) with edge weights we ∈ Z and edge lengths e ∈ N for e ∈ E we define the density of a pattern subgraph H = (V ′,E′) ⊆ G as t ..."
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We consider a framework for biobjective network construction problems where one objective is to be maximized while the other is to be minimized. Given a host graph G = (V,E) with edge weights we ∈ Z and edge lengths e ∈ N for e ∈ E we define the density of a pattern subgraph H = (V ′,E′) ⊆ G as the ratio (H) = ∑e∈E ′ we/ e∈E ′ e. We consider the problem of computing a maximum density pattern H under various additional constraints. In doing so, we compute a single Paretooptimal solution with the best weight per cost ratio subject to additional constraints further narrowing down feasible solutions for the underlying biobjective network construction problem. First, we consider the problem of computing a maximum density pattern with weight at least W and length at most L in a host G. We call this problem the bi
1On the Incompatibility of Connectivity and Local Pooling in Random Graphs
"... Abstract—For a wireless communications network, Local Pooling (LoP) is a desirable property due to its sufficiency for the optimality of lowcomplexity greedy scheduling techniques. However, LoP in network graphs with a primary interference model requires an edge sparsity that may be prohibitive to ..."
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Abstract—For a wireless communications network, Local Pooling (LoP) is a desirable property due to its sufficiency for the optimality of lowcomplexity greedy scheduling techniques. However, LoP in network graphs with a primary interference model requires an edge sparsity that may be prohibitive to other desirable properties in wireless networks, such as connectivity. In this paper, we investigate the impact of the edge density on both LoP and the size of the largest component under the primary interference model, as the number of nodes in the network grows large. For both ErdősRényi (ER) and random geometric (RG) graphs, we employ threshold functions to establish critical values for either the edge probability or communication radius necessary for these properties to hold. These thresholds demonstrate that LoP and connectivity (or even the presence of a giant component) cannot both hold asymptotically as the network grows in size for a large class of edge probability or communication radius functions. We then use simulation to explore this problem in the regime of small network sizes, which suggests the probability that an ER or RG graph satisfies LoP and contains a giant component decays quickly with the size of the network. Index Terms—local pooling; greedy maximal scheduling; primary interference; random graphs; connectivity; giant component. I.
Universite ́ libre de Bruxelles and
"... Abstract. Given two finite posets P and Q, P is a chain minor of Q if there exists a partial function f from the elements of Q to the elements of P such that for every chain in P there is a chain CQ in Q with the property that f restricted to CQ is an isomorphism of chains. We give an algorithm to d ..."
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Abstract. Given two finite posets P and Q, P is a chain minor of Q if there exists a partial function f from the elements of Q to the elements of P such that for every chain in P there is a chain CQ in Q with the property that f restricted to CQ is an isomorphism of chains. We give an algorithm to decide whether a poset P is a chain minor of a poset Q that runs in time O(Q  log Q) for every fixed poset P. This solves an open problem from the monograph by Downey and Fellows [Parameterized Complexity, 1999] who asked whether the problem was fixed parameter tractable.