The Experts below are selected from a list of 108621 Experts worldwide ranked by ideXlab platform
Nicolas Nisse - One of the best experts on this subject based on the ideXlab platform.
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Connected Graph searching
Information & Computation, 2012Co-Authors: Lali Barriere, Pierre Fraigniaud, Nicolas Nisse, Paola Flocchini, Fedor V Fomin, Nicola Santoro, Dimitrios M ThilikosAbstract:In the Graph searching game the opponents are a set of searchers and a fugitive in a Graph. The searchers try to capture the fugitive by applying some sequence of moves that include placement, removal, or sliding of a searcher along an edge. The fugitive tries to avoid capture by moving along unguarded paths. The search number of a Graph is the minimum number of searchers required to guarantee the capture of the fugitive. In this paper, we initiate the study of this game under the natural restriction of connectivity where we demand that in each step of the search the locations of the Graph that are clean (i.e. non-accessible to the fugitive) remain Connected. We give evidence that many of the standard mathematical tools used so far in classic Graph searching fail under the connectivity requirement. We also settle the question on ''the price of connectivity'', that is, how many searchers more are required for searching a Graph when the connectivity demand is imposed. We make estimations of the price of connectivity on general Graphs and we provide tight bounds for the case of trees. In particular, for an n-vertex Graph the ratio between the Connected searching number and the non-Connected one is O(logn) while for trees this ratio is always at most 2. We also conjecture that this constant-ratio upper bound for trees holds also for all Graphs. Our combinatorial results imply a complete characterization of Connected Graph searching on trees. It is based on a forbidden-Graph characterization of the Connected search number. We prove that the Connected search game is monotone for trees, i.e. restricting search strategies to only those where the clean territories increase monotonically does not require more searchers. A consequence of our results is that the Connected search number can be computed in polynomial time on trees, moreover, we show how to make this algorithm distributed. Finally, we reveal connections of this parameter to other invariants on trees such as the Horton-Strahler number.
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Connected Graph Searching in Chordal Graphs
Discrete Applied Mathematics, 2009Co-Authors: Nicolas NisseAbstract:Graph searching was introduced by Parson [T. Parson, Pursuit-evasion in a Graph, in: Theory and Applications of Graphs, in: Lecture Notes in Mathematics, Springer-Verlag, 1976, pp. 426--441]: given a “contaminated†Graph G (e.g., a network containing a hostile intruder), the search number View the MathML source of the Graph G is the minimum number of searchers needed to “clear†the Graph (or to capture the intruder). A search strategy is Connected if, at every step of the strategy, the set of cleared edges induces a Connected subGraph. The Connected search number View the MathML source of a Graph G is the minimum k such that there exists a Connected search strategy for the Graph G using at most k searchers. This paper is concerned with the ratio between the Connected search number and the search number. We prove that, for any chordal Graph G of treewidth View the MathML source, View the MathML source. More precisely, we propose a polynomial-time algorithm that, given any chordal Graph G, computes a Connected search strategy for G using at most View the MathML source searchers. Our main tool is the notion of Connected tree-decomposition. We show that, for any Connected Graph G of chordality k, there exists a Connected search strategy using at most View the MathML source searchers where T is an optimal tree-decomposition of G.
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LATIN - Connected treewidth and Connected Graph searching
LATIN 2006: Theoretical Informatics, 2006Co-Authors: Pierre Fraigniaud, Nicolas NisseAbstract:We give a constructive proof of the equality between treewidth and Connected treewidth. More precisely, we describe an O(nk3)-time algorithm that, given any n-node width-k tree-decomposition of a Connected Graph G, returns a Connected tree-decomposition of G of width ≤ k. The equality between treewidth and Connected treewidth finds applications in Graph searching problems. First, using equality between treewidth and Connected treewidth, we prove that the Connected search number cs(G) of a Connected Graph G is at most logn+1 times larger than its search number. Second, using our constructive proof of equality between treewidth and Connected treewidth, we design an $O(log n\sqrt{log OPT}$)-approximation algorithm for Connected search, running in time O(t(n)+nk3log3/2k+mlog n) for n-node m-edge Connected Graphs of treewidth at most k, where t(n) is the time-complexity of the fastest algorithm for approximating the treewidth, up to a factor $O(\sqrt{log OPT}$).
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Connected treewidth and Connected Graph searching
Lecture Notes in Computer Science, 2006Co-Authors: Pierre Fraigniaud, Nicolas NisseAbstract:We give a constructive proof of the equality between treewidth and Connected treewidth. More precisely, we describe an O(nk 3 )-time algorithm that, given any n-node width-k tree-decomposition of a Connected Graph G, returns a Connected tree-decomposition of G of width < k. The equality between treewidth and Connected treewidth finds applications in Graph searching problems. First, using equality between treewidth and Connected treewidth, we prove that the Connected search number cs(G) of a Connected Graph G is at most logn + 1 times larger than its search number. Second, using our constructive proof of equality between treewidth and Connected treewidth, we design an O(log n√log OPT)-approximation algorithm for Connected search, running in time O(t(n) + nk 3 log 3/2 k + m log n) for n-node m-edge Connected Graphs of treewidth at most k, where t(n) is the time-complexity of the fastest algorithm for approximating the treewidth, up to a factor O(√log OPT).
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Connected Treewidth and Connected Graph Searching
2006Co-Authors: Pierre Fraigniaud, Nicolas NisseAbstract:We give a constructive proof of the equality between \emph{treewidth} and \emph{Connected treewidth}. More precisely, we describe an $O(nk^3)$-time algorithm that, given any $n$-node width-$k$ tree-decomposition of a Connected Graph $G$, returns a Connected tree-decomposition of $G$ of width $\leq k$. The equality between treewidth and Connected treewidth finds applications in \emph{Graph searching} problems. First, using equality between treewidth and Connected treewidth, we prove that the \emph{Connected} search number $\cs(G)$ of a Connected Graph $G$ is at most $\log{n}+1$ times larger than its search number. Second, using our constructive proof of equality between treewidth and Connected treewidth, we design an \\$O(\log n\sqrt{\log OPT})$-approximation algorithm for Connected search, running in time $O(t(n)+nk^3\log^{3/2}k+m\log n)$ for $n$-node $m$-edge Connected Graphs of treewidth at most $k$, where $t(n)$ is the time-complexity of the fastest algorithm for approximating the treewidth, up to a factor $O(\sqrt{\log OPT})$.
Zhongzhi Zhang - One of the best experts on this subject based on the ideXlab platform.
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maximizing the number of spanning trees in a Connected Graph
IEEE Transactions on Information Theory, 2020Co-Authors: Stacy Patterson, Zhongzhi ZhangAbstract:We study the problem of maximizing the number of spanning trees in a Connected Graph with $n$ vertices and $m$ edges, by adding at most $k$ edges from a given set of $q$ candidate edges, a problem that has applications in many domains. We give both algorithmic and hardness results for this problem: 1) We give a greedy algorithm that obtains an approximation ratio of $(1 - 1/e - \epsilon)$ in the exponent of the number of spanning trees for any $\epsilon > 0$ in time $\widetilde {O}(m \epsilon ^{-1} + (n + q) \epsilon ^{-3})$ , where $\widetilde {O}(\cdot)$ hides ${\mathrm{ poly}}\log (n)$ factors. Our running time is optimal with respect to the input size, up to logarithmic factors, and improves on the $O(n^{3})$ running time of the previous proposed greedy algorithm with an approximation ratio $(1 - 1/e)$ in the exponent. Notably, the independence of our running time of $k$ is novel, compared to conventional top- $k$ selections on Graphs that usually run in $\Omega (mk)$ time. 2) We show the exponential inapproximability of this problem by proving that there exists a constant $c > 0$ such that it is NP-hard to approximate the optimum number of spanning trees in the exponent within $(1 - c)$ .
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maximizing the number of spanning trees in a Connected Graph
arXiv: Data Structures and Algorithms, 2018Co-Authors: Stacy Patterson, Zhongzhi ZhangAbstract:We study the problem of maximizing the number of spanning trees in a Connected Graph by adding at most $k$ edges from a given candidate edge set. We give both algorithmic and hardness results for this problem: - We give a greedy algorithm that, using submodularity, obtains an approximation ratio of $(1 - 1/e - \epsilon)$ in the exponent of the number of spanning trees for any $\epsilon > 0$ in time $\tilde{O}(m \epsilon^{-1} + (n + q) \epsilon^{-3})$, where $m$ and $q$ is the number of edges in the original Graph and the candidate edge set, respectively. Our running time is optimal with respect to the input size up to logarithmic factors, and substantially improves upon the $O(n^3)$ running time of the previous proposed greedy algorithm with approximation ratio $(1 - 1/e)$ in the exponent. Notably, the independence of our running time of $k$ is novel, comparing to conventional top-$k$ selections on Graphs that usually run in $\Omega(mk)$ time. A key ingredient of our greedy algorithm is a routine for maintaining effective resistances under edge additions in an online-offline hybrid setting. - We show the exponential inapproximability of this problem by proving that there exists a constant $c > 0$ such that it is NP-hard to approximate the optimum number of spanning trees in the exponent within $(1 - c)$. This inapproximability result follows from a reduction from the minimum path cover in undirected Graphs, whose hardness again follows from the constant inapproximability of the Traveling Salesman Problem (TSP) with distances 1 and 2. Thus, the approximation ratio of our algorithm is also optimal up to a constant factor in the exponent. To our knowledge, this is the first hardness of approximation result for maximizing the number of spanning trees in a Graph, or equivalently, by Kirchhoff's matrix-tree theorem, maximizing the determinant of an SDDM matrix.
Pierre Fraigniaud - One of the best experts on this subject based on the ideXlab platform.
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Connected Graph searching
Information & Computation, 2012Co-Authors: Lali Barriere, Pierre Fraigniaud, Nicolas Nisse, Paola Flocchini, Fedor V Fomin, Nicola Santoro, Dimitrios M ThilikosAbstract:In the Graph searching game the opponents are a set of searchers and a fugitive in a Graph. The searchers try to capture the fugitive by applying some sequence of moves that include placement, removal, or sliding of a searcher along an edge. The fugitive tries to avoid capture by moving along unguarded paths. The search number of a Graph is the minimum number of searchers required to guarantee the capture of the fugitive. In this paper, we initiate the study of this game under the natural restriction of connectivity where we demand that in each step of the search the locations of the Graph that are clean (i.e. non-accessible to the fugitive) remain Connected. We give evidence that many of the standard mathematical tools used so far in classic Graph searching fail under the connectivity requirement. We also settle the question on ''the price of connectivity'', that is, how many searchers more are required for searching a Graph when the connectivity demand is imposed. We make estimations of the price of connectivity on general Graphs and we provide tight bounds for the case of trees. In particular, for an n-vertex Graph the ratio between the Connected searching number and the non-Connected one is O(logn) while for trees this ratio is always at most 2. We also conjecture that this constant-ratio upper bound for trees holds also for all Graphs. Our combinatorial results imply a complete characterization of Connected Graph searching on trees. It is based on a forbidden-Graph characterization of the Connected search number. We prove that the Connected search game is monotone for trees, i.e. restricting search strategies to only those where the clean territories increase monotonically does not require more searchers. A consequence of our results is that the Connected search number can be computed in polynomial time on trees, moreover, we show how to make this algorithm distributed. Finally, we reveal connections of this parameter to other invariants on trees such as the Horton-Strahler number.
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LATIN - Connected treewidth and Connected Graph searching
LATIN 2006: Theoretical Informatics, 2006Co-Authors: Pierre Fraigniaud, Nicolas NisseAbstract:We give a constructive proof of the equality between treewidth and Connected treewidth. More precisely, we describe an O(nk3)-time algorithm that, given any n-node width-k tree-decomposition of a Connected Graph G, returns a Connected tree-decomposition of G of width ≤ k. The equality between treewidth and Connected treewidth finds applications in Graph searching problems. First, using equality between treewidth and Connected treewidth, we prove that the Connected search number cs(G) of a Connected Graph G is at most logn+1 times larger than its search number. Second, using our constructive proof of equality between treewidth and Connected treewidth, we design an $O(log n\sqrt{log OPT}$)-approximation algorithm for Connected search, running in time O(t(n)+nk3log3/2k+mlog n) for n-node m-edge Connected Graphs of treewidth at most k, where t(n) is the time-complexity of the fastest algorithm for approximating the treewidth, up to a factor $O(\sqrt{log OPT}$).
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Connected treewidth and Connected Graph searching
Lecture Notes in Computer Science, 2006Co-Authors: Pierre Fraigniaud, Nicolas NisseAbstract:We give a constructive proof of the equality between treewidth and Connected treewidth. More precisely, we describe an O(nk 3 )-time algorithm that, given any n-node width-k tree-decomposition of a Connected Graph G, returns a Connected tree-decomposition of G of width < k. The equality between treewidth and Connected treewidth finds applications in Graph searching problems. First, using equality between treewidth and Connected treewidth, we prove that the Connected search number cs(G) of a Connected Graph G is at most logn + 1 times larger than its search number. Second, using our constructive proof of equality between treewidth and Connected treewidth, we design an O(log n√log OPT)-approximation algorithm for Connected search, running in time O(t(n) + nk 3 log 3/2 k + m log n) for n-node m-edge Connected Graphs of treewidth at most k, where t(n) is the time-complexity of the fastest algorithm for approximating the treewidth, up to a factor O(√log OPT).
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Connected Treewidth and Connected Graph Searching
2006Co-Authors: Pierre Fraigniaud, Nicolas NisseAbstract:We give a constructive proof of the equality between \emph{treewidth} and \emph{Connected treewidth}. More precisely, we describe an $O(nk^3)$-time algorithm that, given any $n$-node width-$k$ tree-decomposition of a Connected Graph $G$, returns a Connected tree-decomposition of $G$ of width $\leq k$. The equality between treewidth and Connected treewidth finds applications in \emph{Graph searching} problems. First, using equality between treewidth and Connected treewidth, we prove that the \emph{Connected} search number $\cs(G)$ of a Connected Graph $G$ is at most $\log{n}+1$ times larger than its search number. Second, using our constructive proof of equality between treewidth and Connected treewidth, we design an \\$O(\log n\sqrt{\log OPT})$-approximation algorithm for Connected search, running in time $O(t(n)+nk^3\log^{3/2}k+m\log n)$ for $n$-node $m$-edge Connected Graphs of treewidth at most $k$, where $t(n)$ is the time-complexity of the fastest algorithm for approximating the treewidth, up to a factor $O(\sqrt{\log OPT})$.
Yan-quan Feng - One of the best experts on this subject based on the ideXlab platform.
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Super-Connected but not super edge-Connected Graphs
Information Processing Letters, 2010Co-Authors: Jin-xin Zhou, Yan-quan FengAbstract:A Connected Graph G is super-Connected (resp. super edge-Connected) if every minimum vertex-cut (resp. edge-cut) isolates a vertex of G. In [Super connectivity of line Graphs, Inform. Process. Lett. 94 (2005) 191-195], Xu et al. shows that a super-Connected Graph with minimum degree at least 4 is also super edge-Connected. In this paper, a characterization of all super-Connected but not super edge-Connected Graphs is given. It follows from this result that there is a unique super-Connected but not super edge-Connected Graph with minimum degree 3, that is, the Ladder Graph L"3 of order 6, and that there are infinitely many super-Connected but not super edge-Connected Graphs with minimum degree 1 or 2.
Stacy Patterson - One of the best experts on this subject based on the ideXlab platform.
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maximizing the number of spanning trees in a Connected Graph
IEEE Transactions on Information Theory, 2020Co-Authors: Stacy Patterson, Zhongzhi ZhangAbstract:We study the problem of maximizing the number of spanning trees in a Connected Graph with $n$ vertices and $m$ edges, by adding at most $k$ edges from a given set of $q$ candidate edges, a problem that has applications in many domains. We give both algorithmic and hardness results for this problem: 1) We give a greedy algorithm that obtains an approximation ratio of $(1 - 1/e - \epsilon)$ in the exponent of the number of spanning trees for any $\epsilon > 0$ in time $\widetilde {O}(m \epsilon ^{-1} + (n + q) \epsilon ^{-3})$ , where $\widetilde {O}(\cdot)$ hides ${\mathrm{ poly}}\log (n)$ factors. Our running time is optimal with respect to the input size, up to logarithmic factors, and improves on the $O(n^{3})$ running time of the previous proposed greedy algorithm with an approximation ratio $(1 - 1/e)$ in the exponent. Notably, the independence of our running time of $k$ is novel, compared to conventional top- $k$ selections on Graphs that usually run in $\Omega (mk)$ time. 2) We show the exponential inapproximability of this problem by proving that there exists a constant $c > 0$ such that it is NP-hard to approximate the optimum number of spanning trees in the exponent within $(1 - c)$ .
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maximizing the number of spanning trees in a Connected Graph
arXiv: Data Structures and Algorithms, 2018Co-Authors: Stacy Patterson, Zhongzhi ZhangAbstract:We study the problem of maximizing the number of spanning trees in a Connected Graph by adding at most $k$ edges from a given candidate edge set. We give both algorithmic and hardness results for this problem: - We give a greedy algorithm that, using submodularity, obtains an approximation ratio of $(1 - 1/e - \epsilon)$ in the exponent of the number of spanning trees for any $\epsilon > 0$ in time $\tilde{O}(m \epsilon^{-1} + (n + q) \epsilon^{-3})$, where $m$ and $q$ is the number of edges in the original Graph and the candidate edge set, respectively. Our running time is optimal with respect to the input size up to logarithmic factors, and substantially improves upon the $O(n^3)$ running time of the previous proposed greedy algorithm with approximation ratio $(1 - 1/e)$ in the exponent. Notably, the independence of our running time of $k$ is novel, comparing to conventional top-$k$ selections on Graphs that usually run in $\Omega(mk)$ time. A key ingredient of our greedy algorithm is a routine for maintaining effective resistances under edge additions in an online-offline hybrid setting. - We show the exponential inapproximability of this problem by proving that there exists a constant $c > 0$ such that it is NP-hard to approximate the optimum number of spanning trees in the exponent within $(1 - c)$. This inapproximability result follows from a reduction from the minimum path cover in undirected Graphs, whose hardness again follows from the constant inapproximability of the Traveling Salesman Problem (TSP) with distances 1 and 2. Thus, the approximation ratio of our algorithm is also optimal up to a constant factor in the exponent. To our knowledge, this is the first hardness of approximation result for maximizing the number of spanning trees in a Graph, or equivalently, by Kirchhoff's matrix-tree theorem, maximizing the determinant of an SDDM matrix.
