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Christian Kudahl - One of the best experts on this subject based on the ideXlab platform.

  • advice complexity of the online induced subgraph problem
    Mathematical Foundations of Computer Science, 2016
    Co-Authors: Dennis Komm, Rastislav Kralovic, Christian Kudahl
    Abstract:

    Several well-studied graph problems aim to select a largest (or smallest) induced subgraph with a given Property of the input graph. Examples include maximum independent set, maximum planar graph, maximum clique, minimum feedback vertex set, and many others. In online versions of these problems, the vertices of the graph are presented in an adversarial order, and with each vertex, the online algorithm must irreversibly decide whether to include it into the constructed subgraph, based only on the subgraph induced by the vertices presented so far. We study the properties that are common to all these problems by investigating a generalized problem: for an arbitrary but fixed Hereditary Property pi, find some maximal induced subgraph having pi. We investigate this problem from the point of view of advice complexity, i.e., we ask how some additional information about the yet unrevealed parts of the input can influence the solution quality. We evaluate the information in a quantitative way by considering the best possible advice of given size that describes the unknown input. Using a result from Boyar et al. [STACS 2015, LIPIcs 30], we give a tight trade-off relationship stating that, for inputs of length n, roughly n/c bits of advice are both needed and sufficient to obtain a solution with competitive ratio c, regardless of the choice of pi, for any c (possibly a function of n). This complements the results from Bartal et al. [SIAM Journal on Computing 36(2), 2006] stating that, without any advice, even a randomized algorithm cannot achieve a competitive ratio better than Omega(n^{1-log_{4}3-o(1)}). Surprisingly, for a given coHereditary Property pi and the objective to find a minimum subgraph having pi, the advice complexity varies significantly with the choice of pi. We also consider a preemptive online model, inspired by some applications mainly in networking and scheduling, where the decision of the algorithm is not completely irreversible. In particular, the algorithm may discard some vertices previously assigned to the constructed set, but discarded vertices cannot be reinserted into the set. We show that, for the maximum induced subgraph problem, preemption does not significantly help by giving a lower bound of Omega(n/(c^2log c)) on the bits of advice that are needed to obtain competitive ratio c, where c is any increasing function bounded from above by sqrt(n/log n). We also give a linear lower bound for c close to 1.

  • advice complexity of the online induced subgraph problem
    arXiv: Computational Complexity, 2015
    Co-Authors: Dennis Komm, Rastislav Kralovic, Christian Kudahl
    Abstract:

    Several well-studied graph problems aim to select a largest (or smallest) induced subgraph with a given Property of the input graph. Examples of such problems include maximum independent set, maximum planar graph, and many others. We consider these problems, where the vertices are presented online. With each vertex, the online algorithm must decide whether to include it into the constructed subgraph, based only on the subgraph induced by the vertices presented so far. We study the properties that are common to all these problems by investigating the generalized problem: for a Hereditary Property \pty, find some maximal induced subgraph having \pty. We study this problem from the point of view of advice complexity. Using a result from Boyar et al. [STACS 2015], we give a tight trade-off relationship stating that for inputs of length n roughly n/c bits of advice are both needed and sufficient to obtain a solution with competitive ratio c, regardless of the choice of \pty, for any c (possibly a function of n). Surprisingly, a similar result cannot be obtained for the symmetric problem: for a given coHereditary Property \pty, find a minimum subgraph having \pty. We show that the advice complexity of this problem varies significantly with the choice of \pty. We also consider preemptive online model, where the decision of the algorithm is not completely irreversible. In particular, the algorithm may discard some vertices previously assigned to the constructed set, but discarded vertices cannot be reinserted into the set again. We show that, for the maximum induced subgraph problem, preemption cannot help much, giving a lower bound of $\Omega(n/(c^2\log c))$ bits of advice needed to obtain competitive ratio $c$, where $c$ is any increasing function bounded by \sqrt{n/log n}. We also give a linear lower bound for c close to 1.

Rastislav Kralovic - One of the best experts on this subject based on the ideXlab platform.

  • advice complexity of the online induced subgraph problem
    Mathematical Foundations of Computer Science, 2016
    Co-Authors: Dennis Komm, Rastislav Kralovic, Christian Kudahl
    Abstract:

    Several well-studied graph problems aim to select a largest (or smallest) induced subgraph with a given Property of the input graph. Examples include maximum independent set, maximum planar graph, maximum clique, minimum feedback vertex set, and many others. In online versions of these problems, the vertices of the graph are presented in an adversarial order, and with each vertex, the online algorithm must irreversibly decide whether to include it into the constructed subgraph, based only on the subgraph induced by the vertices presented so far. We study the properties that are common to all these problems by investigating a generalized problem: for an arbitrary but fixed Hereditary Property pi, find some maximal induced subgraph having pi. We investigate this problem from the point of view of advice complexity, i.e., we ask how some additional information about the yet unrevealed parts of the input can influence the solution quality. We evaluate the information in a quantitative way by considering the best possible advice of given size that describes the unknown input. Using a result from Boyar et al. [STACS 2015, LIPIcs 30], we give a tight trade-off relationship stating that, for inputs of length n, roughly n/c bits of advice are both needed and sufficient to obtain a solution with competitive ratio c, regardless of the choice of pi, for any c (possibly a function of n). This complements the results from Bartal et al. [SIAM Journal on Computing 36(2), 2006] stating that, without any advice, even a randomized algorithm cannot achieve a competitive ratio better than Omega(n^{1-log_{4}3-o(1)}). Surprisingly, for a given coHereditary Property pi and the objective to find a minimum subgraph having pi, the advice complexity varies significantly with the choice of pi. We also consider a preemptive online model, inspired by some applications mainly in networking and scheduling, where the decision of the algorithm is not completely irreversible. In particular, the algorithm may discard some vertices previously assigned to the constructed set, but discarded vertices cannot be reinserted into the set. We show that, for the maximum induced subgraph problem, preemption does not significantly help by giving a lower bound of Omega(n/(c^2log c)) on the bits of advice that are needed to obtain competitive ratio c, where c is any increasing function bounded from above by sqrt(n/log n). We also give a linear lower bound for c close to 1.

  • advice complexity of the online induced subgraph problem
    arXiv: Computational Complexity, 2015
    Co-Authors: Dennis Komm, Rastislav Kralovic, Christian Kudahl
    Abstract:

    Several well-studied graph problems aim to select a largest (or smallest) induced subgraph with a given Property of the input graph. Examples of such problems include maximum independent set, maximum planar graph, and many others. We consider these problems, where the vertices are presented online. With each vertex, the online algorithm must decide whether to include it into the constructed subgraph, based only on the subgraph induced by the vertices presented so far. We study the properties that are common to all these problems by investigating the generalized problem: for a Hereditary Property \pty, find some maximal induced subgraph having \pty. We study this problem from the point of view of advice complexity. Using a result from Boyar et al. [STACS 2015], we give a tight trade-off relationship stating that for inputs of length n roughly n/c bits of advice are both needed and sufficient to obtain a solution with competitive ratio c, regardless of the choice of \pty, for any c (possibly a function of n). Surprisingly, a similar result cannot be obtained for the symmetric problem: for a given coHereditary Property \pty, find a minimum subgraph having \pty. We show that the advice complexity of this problem varies significantly with the choice of \pty. We also consider preemptive online model, where the decision of the algorithm is not completely irreversible. In particular, the algorithm may discard some vertices previously assigned to the constructed set, but discarded vertices cannot be reinserted into the set again. We show that, for the maximum induced subgraph problem, preemption cannot help much, giving a lower bound of $\Omega(n/(c^2\log c))$ bits of advice needed to obtain competitive ratio $c$, where $c$ is any increasing function bounded by \sqrt{n/log n}. We also give a linear lower bound for c close to 1.

Dennis Komm - One of the best experts on this subject based on the ideXlab platform.

  • advice complexity of the online induced subgraph problem
    Mathematical Foundations of Computer Science, 2016
    Co-Authors: Dennis Komm, Rastislav Kralovic, Christian Kudahl
    Abstract:

    Several well-studied graph problems aim to select a largest (or smallest) induced subgraph with a given Property of the input graph. Examples include maximum independent set, maximum planar graph, maximum clique, minimum feedback vertex set, and many others. In online versions of these problems, the vertices of the graph are presented in an adversarial order, and with each vertex, the online algorithm must irreversibly decide whether to include it into the constructed subgraph, based only on the subgraph induced by the vertices presented so far. We study the properties that are common to all these problems by investigating a generalized problem: for an arbitrary but fixed Hereditary Property pi, find some maximal induced subgraph having pi. We investigate this problem from the point of view of advice complexity, i.e., we ask how some additional information about the yet unrevealed parts of the input can influence the solution quality. We evaluate the information in a quantitative way by considering the best possible advice of given size that describes the unknown input. Using a result from Boyar et al. [STACS 2015, LIPIcs 30], we give a tight trade-off relationship stating that, for inputs of length n, roughly n/c bits of advice are both needed and sufficient to obtain a solution with competitive ratio c, regardless of the choice of pi, for any c (possibly a function of n). This complements the results from Bartal et al. [SIAM Journal on Computing 36(2), 2006] stating that, without any advice, even a randomized algorithm cannot achieve a competitive ratio better than Omega(n^{1-log_{4}3-o(1)}). Surprisingly, for a given coHereditary Property pi and the objective to find a minimum subgraph having pi, the advice complexity varies significantly with the choice of pi. We also consider a preemptive online model, inspired by some applications mainly in networking and scheduling, where the decision of the algorithm is not completely irreversible. In particular, the algorithm may discard some vertices previously assigned to the constructed set, but discarded vertices cannot be reinserted into the set. We show that, for the maximum induced subgraph problem, preemption does not significantly help by giving a lower bound of Omega(n/(c^2log c)) on the bits of advice that are needed to obtain competitive ratio c, where c is any increasing function bounded from above by sqrt(n/log n). We also give a linear lower bound for c close to 1.

  • advice complexity of the online induced subgraph problem
    arXiv: Computational Complexity, 2015
    Co-Authors: Dennis Komm, Rastislav Kralovic, Christian Kudahl
    Abstract:

    Several well-studied graph problems aim to select a largest (or smallest) induced subgraph with a given Property of the input graph. Examples of such problems include maximum independent set, maximum planar graph, and many others. We consider these problems, where the vertices are presented online. With each vertex, the online algorithm must decide whether to include it into the constructed subgraph, based only on the subgraph induced by the vertices presented so far. We study the properties that are common to all these problems by investigating the generalized problem: for a Hereditary Property \pty, find some maximal induced subgraph having \pty. We study this problem from the point of view of advice complexity. Using a result from Boyar et al. [STACS 2015], we give a tight trade-off relationship stating that for inputs of length n roughly n/c bits of advice are both needed and sufficient to obtain a solution with competitive ratio c, regardless of the choice of \pty, for any c (possibly a function of n). Surprisingly, a similar result cannot be obtained for the symmetric problem: for a given coHereditary Property \pty, find a minimum subgraph having \pty. We show that the advice complexity of this problem varies significantly with the choice of \pty. We also consider preemptive online model, where the decision of the algorithm is not completely irreversible. In particular, the algorithm may discard some vertices previously assigned to the constructed set, but discarded vertices cannot be reinserted into the set again. We show that, for the maximum induced subgraph problem, preemption cannot help much, giving a lower bound of $\Omega(n/(c^2\log c))$ bits of advice needed to obtain competitive ratio $c$, where $c$ is any increasing function bounded by \sqrt{n/log n}. We also give a linear lower bound for c close to 1.

Suk Andrew - One of the best experts on this subject based on the ideXlab platform.

  • Erdős-Hajnal Conjecture for Graphs with Bounded VC-Dimension
    'Springer Science and Business Media LLC', 2019
    Co-Authors: Fox Jacob, Pach János, Suk Andrew
    Abstract:

    The Vapnik-Chervonenkis dimension (in short, VC-dimension) of a graph is defined as the VC-dimension of the set system induced by the neighborhoods of its vertices. We show that every n-vertex graph with bounded VC-dimension contains a clique or an independent set of size at least e(logn)1-o(1). The dependence on the VC-dimension is hidden in the o(1) term. This improves the general lower bound, eclogn, due to Erds and Hajnal, which is valid in the class of graphs satisfying any fixed nontrivial Hereditary Property. Our result is almost optimal and nearly matches the celebrated Erds-Hajnal conjecture, according to which one can always find a clique or an independent set of size at least e(logn). Our results partially explain why most geometric intersection graphs arising in discrete and computational geometry have exceptionally favorable Ramsey-type properties. Our main tool is a partitioning result found by Lovasz-Szegedy and Alon-Fischer-Newman, which is called the ultra-strong regularity lemma for graphs with bounded VC-dimension. We extend this lemma to k-uniform hypergraphs, and prove that the number of parts in the partition can be taken to be (1/epsilon)O(d), improving the original bound of (1/epsilon)O(d2) in the graph setting. We show that this bound is tight up to an absolute constant factor in the exponent. Moreover, we give an O(nk)-time algorithm for finding a partition meeting the requirements. Finally, we establish tight bounds on Ramsey-Turan numbers for graphs with bounded VC-dimension

  • Erdos-Hajnal conjecture for graphs with bounded VC-dimension
    2017
    Co-Authors: Fox Jacob, Pach János, Suk Andrew
    Abstract:

    The Vapnik-Chervonenkis dimension (in short, VC-dimension) of a graph is defined as the VC-dimension of the set system induced by the neighborhoods of its vertices. We show that every $n$-vertex graph with bounded VC-dimension contains a clique or an independent set of size at least $e^{(\log n)^{1 - o(1)}}$. The dependence on the VC-dimension is hidden in the $o(1)$ term. This improves the general lower bound, $e^{c\sqrt{\log n}}$, due to Erdos and Hajnal, which is valid in the class of graphs satisfying any fixed nontrivial Hereditary Property. Our result is almost optimal and nearly matches the celebrated Erdos-Hajnal conjecture, according to which one can always find a clique or an independent set of size at least $e^{\Omega(\log n)}$. Our results partially explain why most geometric intersection graphs arising in discrete and computational geometry have exceptionally favorable Ramsey-type properties. Our main tool is a partitioning result found by Lov\'asz-Szegedy and Alon-Fischer-Newman, which is called the "ultra-strong regularity lemma" for graphs with bounded VC-dimension. We extend this lemma to $k$-uniform hypergraphs, and prove that the number of parts in the partition can be taken to be $(1/\varepsilon)^{O(d)}$, improving the original bound of $(1/\varepsilon)^{O(d^2)}$ in the graph setting. We show that this bound is tight up to an absolute constant factor in the exponent. Moreover, we give an $O(n^k)$-time algorithm for finding a partition meeting the requirements. Finally, we establish tight bounds on Ramsey-Tur\'an numbers for graphs with bounded VC-dimension

  • Erdos-hajnal conjecture for graphs with bounded VC-Dimension
    Schloss Dagstuhl Leibniz-Zentrum für Informatik, 2017
    Co-Authors: Fox Jacob, Pach János, Suk Andrew
    Abstract:

    The Vapnik-Chervonenkis dimension (in short, VC-dimension) of a graph is defined as the VC-dimension of the set system induced by the neighborhoods of its vertices. We show that every n-vertex graph with bounded VC-dimension contains a clique or an independent set of size at least e(log n)1-σ(1). The dependence on the VC-dimension is hidden in the o(1) term. This improves the general lower bound, ec√log n due to Erdos and Hajnal, which is valid in the class of graphs satisfying any fixed nontrivial Hereditary Property. Our result is almost optimal and nearly matches the celebrated Erdos-Hajnal conjecture, according to which one can always find a clique or an independent set of size at least eΩ(log n). Our results partially explain why most geometric intersection graphs arising in discrete and computational geometry have exceptionally favorable Ramsey-type properties. Our main tool is a partitioning result found by Lovász-Szegedy and Alon-Fischer-Newman, which is called the "ultra-strong regularity lemma" for graphs with bounded VC-dimension. We extend this lemma to k-uniform hypergraphs, and prove that the number of parts in the partition can be taken to be (1/ϵ)O(d) improving the original bound of (1/ϵ) O(d2) in the graph setting. We show that this bound is tight up to an absolute constant factor in the exponent. Moreover, we give an O(nk)-time algorithm for finding a partition meeting the requirements in the k-uniform setting. © Jacob Fox, János Pach, and Andrew Suk

Fox Jacob - One of the best experts on this subject based on the ideXlab platform.

  • Erdős-Hajnal Conjecture for Graphs with Bounded VC-Dimension
    'Springer Science and Business Media LLC', 2019
    Co-Authors: Fox Jacob, Pach János, Suk Andrew
    Abstract:

    The Vapnik-Chervonenkis dimension (in short, VC-dimension) of a graph is defined as the VC-dimension of the set system induced by the neighborhoods of its vertices. We show that every n-vertex graph with bounded VC-dimension contains a clique or an independent set of size at least e(logn)1-o(1). The dependence on the VC-dimension is hidden in the o(1) term. This improves the general lower bound, eclogn, due to Erds and Hajnal, which is valid in the class of graphs satisfying any fixed nontrivial Hereditary Property. Our result is almost optimal and nearly matches the celebrated Erds-Hajnal conjecture, according to which one can always find a clique or an independent set of size at least e(logn). Our results partially explain why most geometric intersection graphs arising in discrete and computational geometry have exceptionally favorable Ramsey-type properties. Our main tool is a partitioning result found by Lovasz-Szegedy and Alon-Fischer-Newman, which is called the ultra-strong regularity lemma for graphs with bounded VC-dimension. We extend this lemma to k-uniform hypergraphs, and prove that the number of parts in the partition can be taken to be (1/epsilon)O(d), improving the original bound of (1/epsilon)O(d2) in the graph setting. We show that this bound is tight up to an absolute constant factor in the exponent. Moreover, we give an O(nk)-time algorithm for finding a partition meeting the requirements. Finally, we establish tight bounds on Ramsey-Turan numbers for graphs with bounded VC-dimension

  • Erdos-Hajnal conjecture for graphs with bounded VC-dimension
    2017
    Co-Authors: Fox Jacob, Pach János, Suk Andrew
    Abstract:

    The Vapnik-Chervonenkis dimension (in short, VC-dimension) of a graph is defined as the VC-dimension of the set system induced by the neighborhoods of its vertices. We show that every $n$-vertex graph with bounded VC-dimension contains a clique or an independent set of size at least $e^{(\log n)^{1 - o(1)}}$. The dependence on the VC-dimension is hidden in the $o(1)$ term. This improves the general lower bound, $e^{c\sqrt{\log n}}$, due to Erdos and Hajnal, which is valid in the class of graphs satisfying any fixed nontrivial Hereditary Property. Our result is almost optimal and nearly matches the celebrated Erdos-Hajnal conjecture, according to which one can always find a clique or an independent set of size at least $e^{\Omega(\log n)}$. Our results partially explain why most geometric intersection graphs arising in discrete and computational geometry have exceptionally favorable Ramsey-type properties. Our main tool is a partitioning result found by Lov\'asz-Szegedy and Alon-Fischer-Newman, which is called the "ultra-strong regularity lemma" for graphs with bounded VC-dimension. We extend this lemma to $k$-uniform hypergraphs, and prove that the number of parts in the partition can be taken to be $(1/\varepsilon)^{O(d)}$, improving the original bound of $(1/\varepsilon)^{O(d^2)}$ in the graph setting. We show that this bound is tight up to an absolute constant factor in the exponent. Moreover, we give an $O(n^k)$-time algorithm for finding a partition meeting the requirements. Finally, we establish tight bounds on Ramsey-Tur\'an numbers for graphs with bounded VC-dimension

  • Erdos-hajnal conjecture for graphs with bounded VC-Dimension
    Schloss Dagstuhl Leibniz-Zentrum für Informatik, 2017
    Co-Authors: Fox Jacob, Pach János, Suk Andrew
    Abstract:

    The Vapnik-Chervonenkis dimension (in short, VC-dimension) of a graph is defined as the VC-dimension of the set system induced by the neighborhoods of its vertices. We show that every n-vertex graph with bounded VC-dimension contains a clique or an independent set of size at least e(log n)1-σ(1). The dependence on the VC-dimension is hidden in the o(1) term. This improves the general lower bound, ec√log n due to Erdos and Hajnal, which is valid in the class of graphs satisfying any fixed nontrivial Hereditary Property. Our result is almost optimal and nearly matches the celebrated Erdos-Hajnal conjecture, according to which one can always find a clique or an independent set of size at least eΩ(log n). Our results partially explain why most geometric intersection graphs arising in discrete and computational geometry have exceptionally favorable Ramsey-type properties. Our main tool is a partitioning result found by Lovász-Szegedy and Alon-Fischer-Newman, which is called the "ultra-strong regularity lemma" for graphs with bounded VC-dimension. We extend this lemma to k-uniform hypergraphs, and prove that the number of parts in the partition can be taken to be (1/ϵ)O(d) improving the original bound of (1/ϵ) O(d2) in the graph setting. We show that this bound is tight up to an absolute constant factor in the exponent. Moreover, we give an O(nk)-time algorithm for finding a partition meeting the requirements in the k-uniform setting. © Jacob Fox, János Pach, and Andrew Suk