Pub Date : 2022-11-07DOI: 10.48550/arXiv.2211.03893
Yu Chen, S. Khanna, Zihan Tan
We study the query complexity of the metric Steiner Tree problem, where we are given an $n times n$ metric on a set $V$ of vertices along with a set $T subseteq V$ of $k$ terminals, and the goal is to find a tree of minimum cost that contains all terminals in $T$. The query complexity for the related minimum spanning tree (MST) problem is well-understood: for any fixed $varepsilon>0$, one can estimate the MST cost to within a $(1+varepsilon)$-factor using only $tilde{O}(n)$ queries, and this is known to be tight. This implies that a $(2 + varepsilon)$-approximate estimate of Steiner Tree cost can be obtained with $tilde{O}(k)$ queries by simply applying the MST cost estimation algorithm on the metric induced by the terminals. Our first result shows that any (randomized) algorithm that estimates the Steiner Tree cost to within a $(5/3 - varepsilon)$-factor requires $Omega(n^2)$ queries, even if $k$ is a constant. This lower bound is in sharp contrast to an upper bound of $O(nk)$ queries for computing a $(5/3)$-approximate Steiner Tree, which follows from previous work by Du and Zelikovsky. Our second main result, and the main technical contribution of this work, is a sublinear query algorithm for estimating the Steiner Tree cost to within a strictly better-than-$2$ factor, with query complexity $tilde{O}(n^{12/7} + n^{6/7}cdot k)=tilde{O}(n^{13/7})=o(n^2)$. We complement this result by showing an $tilde{Omega}(n + k^{6/5})$ query lower bound for any algorithm that estimates Steiner Tree cost to a strictly better than $2$ factor. Thus $tilde{Omega}(n^{6/5})$ queries are needed to just beat $2$-approximation when $k = Omega(n)$; a sharp contrast to MST cost estimation where a $(1+o(1))$-approximate estimate of cost is achievable with only $tilde{O}(n)$ queries.
{"title":"Query Complexity of the Metric Steiner Tree Problem","authors":"Yu Chen, S. Khanna, Zihan Tan","doi":"10.48550/arXiv.2211.03893","DOIUrl":"https://doi.org/10.48550/arXiv.2211.03893","url":null,"abstract":"We study the query complexity of the metric Steiner Tree problem, where we are given an $n times n$ metric on a set $V$ of vertices along with a set $T subseteq V$ of $k$ terminals, and the goal is to find a tree of minimum cost that contains all terminals in $T$. The query complexity for the related minimum spanning tree (MST) problem is well-understood: for any fixed $varepsilon>0$, one can estimate the MST cost to within a $(1+varepsilon)$-factor using only $tilde{O}(n)$ queries, and this is known to be tight. This implies that a $(2 + varepsilon)$-approximate estimate of Steiner Tree cost can be obtained with $tilde{O}(k)$ queries by simply applying the MST cost estimation algorithm on the metric induced by the terminals. Our first result shows that any (randomized) algorithm that estimates the Steiner Tree cost to within a $(5/3 - varepsilon)$-factor requires $Omega(n^2)$ queries, even if $k$ is a constant. This lower bound is in sharp contrast to an upper bound of $O(nk)$ queries for computing a $(5/3)$-approximate Steiner Tree, which follows from previous work by Du and Zelikovsky. Our second main result, and the main technical contribution of this work, is a sublinear query algorithm for estimating the Steiner Tree cost to within a strictly better-than-$2$ factor, with query complexity $tilde{O}(n^{12/7} + n^{6/7}cdot k)=tilde{O}(n^{13/7})=o(n^2)$. We complement this result by showing an $tilde{Omega}(n + k^{6/5})$ query lower bound for any algorithm that estimates Steiner Tree cost to a strictly better than $2$ factor. Thus $tilde{Omega}(n^{6/5})$ queries are needed to just beat $2$-approximation when $k = Omega(n)$; a sharp contrast to MST cost estimation where a $(1+o(1))$-approximate estimate of cost is achievable with only $tilde{O}(n)$ queries.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"os-2 1","pages":"4893-4935"},"PeriodicalIF":0.0,"publicationDate":"2022-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87682891","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-06DOI: 10.48550/arXiv.2211.03161
Yakov Nekrich, S. Rahul
In the orthogonal range reporting problem we must pre-process a set $P$ of multi-dimensional points, so that for any axis-parallel query rectangle $q$ all points from $qcap P$ can be reported efficiently. In this paper we study the query complexity of multi-dimensional orthogonal range reporting in the pointer machine model. We present a data structure that answers four-dimensional orthogonal range reporting queries in almost-optimal time $O(log nloglog n + k)$ and uses $O(nlog^4 n)$ space, where $n$ is the number of points in $P$ and $k$ is the number of points in $qcap P$ . This is the first data structure with nearly-linear space usage that achieves almost-optimal query time in 4d. This result can be immediately generalized to $dge 4$ dimensions: we show that there is a data structure supporting $d$-dimensional range reporting queries in time $O(log^{d-3} nloglog n+k)$ for any constant $dge 4$.
{"title":"4D Range Reporting in the Pointer Machine Model in Almost-Optimal Time","authors":"Yakov Nekrich, S. Rahul","doi":"10.48550/arXiv.2211.03161","DOIUrl":"https://doi.org/10.48550/arXiv.2211.03161","url":null,"abstract":"In the orthogonal range reporting problem we must pre-process a set $P$ of multi-dimensional points, so that for any axis-parallel query rectangle $q$ all points from $qcap P$ can be reported efficiently. In this paper we study the query complexity of multi-dimensional orthogonal range reporting in the pointer machine model. We present a data structure that answers four-dimensional orthogonal range reporting queries in almost-optimal time $O(log nloglog n + k)$ and uses $O(nlog^4 n)$ space, where $n$ is the number of points in $P$ and $k$ is the number of points in $qcap P$ . This is the first data structure with nearly-linear space usage that achieves almost-optimal query time in 4d. This result can be immediately generalized to $dge 4$ dimensions: we show that there is a data structure supporting $d$-dimensional range reporting queries in time $O(log^{d-3} nloglog n+k)$ for any constant $dge 4$.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"40 1","pages":"1862-1876"},"PeriodicalIF":0.0,"publicationDate":"2022-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77720792","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-05DOI: 10.48550/arXiv.2211.02951
Joachim Gudmundsson, Martin P. Seybold, Sampson Wong
Map matching is a common preprocessing step for analysing vehicle trajectories. In the theory community, the most popular approach for map matching is to compute a path on the road network that is the most spatially similar to the trajectory, where spatial similarity is measured using the Fr'echet distance. A shortcoming of existing map matching algorithms under the Fr'echet distance is that every time a trajectory is matched, the entire road network needs to be reprocessed from scratch. An open problem is whether one can preprocess the road network into a data structure, so that map matching queries can be answered in sublinear time. In this paper, we investigate map matching queries under the Fr'echet distance. We provide a negative result for geometric planar graphs. We show that, unless SETH fails, there is no data structure that can be constructed in polynomial time that answers map matching queries in $O((pq)^{1-delta})$ query time for any $delta>0$, where $p$ and $q$ are the complexities of the geometric planar graph and the query trajectory, respectively. We provide a positive result for realistic input graphs, which we regard as the main result of this paper. We show that for $c$-packed graphs, one can construct a data structure of $tilde O(cp)$ size that can answer $(1+varepsilon)$-approximate map matching queries in $tilde O(c^4 q log^4 p)$ time, where $tilde O(cdot)$ hides lower-order factors and dependence of $varepsilon$.
{"title":"Map matching queries on realistic input graphs under the Fréchet distance","authors":"Joachim Gudmundsson, Martin P. Seybold, Sampson Wong","doi":"10.48550/arXiv.2211.02951","DOIUrl":"https://doi.org/10.48550/arXiv.2211.02951","url":null,"abstract":"Map matching is a common preprocessing step for analysing vehicle trajectories. In the theory community, the most popular approach for map matching is to compute a path on the road network that is the most spatially similar to the trajectory, where spatial similarity is measured using the Fr'echet distance. A shortcoming of existing map matching algorithms under the Fr'echet distance is that every time a trajectory is matched, the entire road network needs to be reprocessed from scratch. An open problem is whether one can preprocess the road network into a data structure, so that map matching queries can be answered in sublinear time. In this paper, we investigate map matching queries under the Fr'echet distance. We provide a negative result for geometric planar graphs. We show that, unless SETH fails, there is no data structure that can be constructed in polynomial time that answers map matching queries in $O((pq)^{1-delta})$ query time for any $delta>0$, where $p$ and $q$ are the complexities of the geometric planar graph and the query trajectory, respectively. We provide a positive result for realistic input graphs, which we regard as the main result of this paper. We show that for $c$-packed graphs, one can construct a data structure of $tilde O(cp)$ size that can answer $(1+varepsilon)$-approximate map matching queries in $tilde O(c^4 q log^4 p)$ time, where $tilde O(cdot)$ hides lower-order factors and dependence of $varepsilon$.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"25 1","pages":"1464-1492"},"PeriodicalIF":0.0,"publicationDate":"2022-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84540991","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-04DOI: 10.48550/arXiv.2211.02717
D. Lokshtanov, Fahad Panolan, Saket Saurabh, Jie Xue, M. Zehavi
We initiate a systematic study of approximation schemes for fundamental optimization problems on disk graphs, a common generalization of both planar graphs and unit-disk graphs. Our main contribution is a general framework for designing efficient polynomial-time approximation schemes (EPTASes) for vertex-deletion problems on disk graphs, which results in EPTASes for many problems including Vertex Cover, Feedback Vertex Set, Small Cycle Hitting (in particular, Triangle Hitting), $P_k$-Hitting for $kin{3,4,5}$, Path Deletion, Pathwidth $1$-Deletion, Component Order Connectivity, Bounded Degree Deletion, Pseudoforest Deletion, Finite-Type Component Deletion, etc. All EPTASes obtained using our framework are robust in the sense that they do not require a realization of the input graph. To the best of our knowledge, prior to this work, the only problems known to admit (E)PTASes on disk graphs are Maximum Clique, Independent Set, Dominating set, and Vertex Cover, among which the existing PTAS [Erlebach et al., SICOMP'05] and EPTAS [Leeuwen, SWAT'06] for Vertex Cover require a realization of the input disk graph (while ours does not). The core of our framework is a reduction for a broad class of (approximation) vertex-deletion problems from (general) disk graphs to disk graphs of bounded local radius, which is a new invariant of disk graphs introduced in this work. Disk graphs of bounded local radius can be viewed as a mild generalization of planar graphs, which preserves certain nice properties of planar graphs. Specifically, we prove that disk graphs of bounded local radius admit the Excluded Grid Minor property and have locally bounded treewidth. This allows existing techniques for designing approximation schemes on planar graphs (e.g., bidimensionality and Baker's technique) to be directly applied to disk graphs of bounded local radius.
{"title":"A Framework for Approximation Schemes on Disk Graphs","authors":"D. Lokshtanov, Fahad Panolan, Saket Saurabh, Jie Xue, M. Zehavi","doi":"10.48550/arXiv.2211.02717","DOIUrl":"https://doi.org/10.48550/arXiv.2211.02717","url":null,"abstract":"We initiate a systematic study of approximation schemes for fundamental optimization problems on disk graphs, a common generalization of both planar graphs and unit-disk graphs. Our main contribution is a general framework for designing efficient polynomial-time approximation schemes (EPTASes) for vertex-deletion problems on disk graphs, which results in EPTASes for many problems including Vertex Cover, Feedback Vertex Set, Small Cycle Hitting (in particular, Triangle Hitting), $P_k$-Hitting for $kin{3,4,5}$, Path Deletion, Pathwidth $1$-Deletion, Component Order Connectivity, Bounded Degree Deletion, Pseudoforest Deletion, Finite-Type Component Deletion, etc. All EPTASes obtained using our framework are robust in the sense that they do not require a realization of the input graph. To the best of our knowledge, prior to this work, the only problems known to admit (E)PTASes on disk graphs are Maximum Clique, Independent Set, Dominating set, and Vertex Cover, among which the existing PTAS [Erlebach et al., SICOMP'05] and EPTAS [Leeuwen, SWAT'06] for Vertex Cover require a realization of the input disk graph (while ours does not). The core of our framework is a reduction for a broad class of (approximation) vertex-deletion problems from (general) disk graphs to disk graphs of bounded local radius, which is a new invariant of disk graphs introduced in this work. Disk graphs of bounded local radius can be viewed as a mild generalization of planar graphs, which preserves certain nice properties of planar graphs. Specifically, we prove that disk graphs of bounded local radius admit the Excluded Grid Minor property and have locally bounded treewidth. This allows existing techniques for designing approximation schemes on planar graphs (e.g., bidimensionality and Baker's technique) to be directly applied to disk graphs of bounded local radius.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"8 1","pages":"2228-2241"},"PeriodicalIF":0.0,"publicationDate":"2022-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84156613","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-03DOI: 10.48550/arXiv.2211.01945
A. Balliu, S. Brandt, F. Kuhn, D. Olivetti
We investigate the distributed complexity of maximal matching and maximal independent set (MIS) in hypergraphs in the LOCAL model. A maximal matching of a hypergraph $H=(V_H,E_H)$ is a maximal disjoint set $Msubseteq E_H$ of hyperedges and an MIS $Ssubseteq V_H$ is a maximal set of nodes such that no hyperedge is fully contained in $S$. Both problems can be solved by a simple sequential greedy algorithm, which can be implemented naively in $O(Delta r + log^* n)$ rounds, where $Delta$ is the maximum degree, $r$ is the rank, and $n$ is the number of nodes. We show that for maximal matching, this naive algorithm is optimal in the following sense. Any deterministic algorithm for solving the problem requires $Omega(min{Delta r, log_{Delta r} n})$ rounds, and any randomized one requires $Omega(min{Delta r, log_{Delta r} log n})$ rounds. Hence, for any algorithm with a complexity of the form $O(f(Delta, r) + g(n))$, we have $f(Delta, r) in Omega(Delta r)$ if $g(n)$ is not too large, and in particular if $g(n) = log^* n$ (which is the optimal asymptotic dependency on $n$ due to Linial's lower bound [FOCS'87]). Our lower bound proof is based on the round elimination framework, and its structure is inspired by a new round elimination fixed point that we give for the $Delta$-vertex coloring problem in hypergraphs. For the MIS problem on hypergraphs, we show that for $Deltall r$, there are significant improvements over the naive $O(Delta r + log^* n)$-round algorithm. We give two deterministic algorithms for the problem. We show that a hypergraph MIS can be computed in $O(Delta^2cdotlog r + Deltacdotlog rcdot log^* r + log^* n)$ rounds. We further show that at the cost of a worse dependency on $Delta$, the dependency on $r$ can be removed almost entirely, by giving an algorithm with complexity $Delta^{O(Delta)}cdotlog^* r + O(log^* n)$.
研究了局部模型超图中最大匹配和最大独立集的分布复杂度。超图$H=(V_H,E_H)$的最大匹配是超边的最大不相交集$Msubseteq E_H$，而MIS $Ssubseteq V_H$是使超边不完全包含在$S$中的最大节点集。这两个问题都可以通过一个简单的顺序贪婪算法来解决，该算法可以在$O(Delta r + log^* n)$轮中简单地实现，其中$Delta$是最大度，$r$是秩，$n$是节点数。我们证明了对于最大匹配，这种朴素算法在以下意义上是最优的。任何解决问题的确定性算法都需要$Omega(min{Delta r, log_{Delta r} n})$轮，任何随机算法都需要$Omega(min{Delta r, log_{Delta r} log n})$轮。因此，对于任何复杂度为$O(f(Delta, r) + g(n))$形式的算法，如果$g(n)$不是太大，特别是如果$g(n) = log^* n$(由于Linial的下界[FOCS'87]，这是对$n$的最优渐近依赖)，我们有$f(Delta, r) in Omega(Delta r)$。我们的下界证明是基于循环消去框架的，其结构灵感来自于我们对超图中$Delta$ -顶点着色问题给出的一个新的循环消去不动点。对于超图上的MIS问题，我们表明对于$Deltall r$，相对于朴素的$O(Delta r + log^* n)$ -round算法有显著的改进。针对这一问题，给出了两种确定性算法。我们证明了超图MIS可以在$O(Delta^2cdotlog r + Deltacdotlog rcdot log^* r + log^* n)$轮内计算。我们进一步表明，通过给出复杂度为$Delta^{O(Delta)}cdotlog^* r + O(log^* n)$的算法，以对$Delta$的更严重依赖为代价，可以几乎完全消除对$r$的依赖。
{"title":"Distributed Maximal Matching and Maximal Independent Set on Hypergraphs","authors":"A. Balliu, S. Brandt, F. Kuhn, D. Olivetti","doi":"10.48550/arXiv.2211.01945","DOIUrl":"https://doi.org/10.48550/arXiv.2211.01945","url":null,"abstract":"We investigate the distributed complexity of maximal matching and maximal independent set (MIS) in hypergraphs in the LOCAL model. A maximal matching of a hypergraph $H=(V_H,E_H)$ is a maximal disjoint set $Msubseteq E_H$ of hyperedges and an MIS $Ssubseteq V_H$ is a maximal set of nodes such that no hyperedge is fully contained in $S$. Both problems can be solved by a simple sequential greedy algorithm, which can be implemented naively in $O(Delta r + log^* n)$ rounds, where $Delta$ is the maximum degree, $r$ is the rank, and $n$ is the number of nodes. We show that for maximal matching, this naive algorithm is optimal in the following sense. Any deterministic algorithm for solving the problem requires $Omega(min{Delta r, log_{Delta r} n})$ rounds, and any randomized one requires $Omega(min{Delta r, log_{Delta r} log n})$ rounds. Hence, for any algorithm with a complexity of the form $O(f(Delta, r) + g(n))$, we have $f(Delta, r) in Omega(Delta r)$ if $g(n)$ is not too large, and in particular if $g(n) = log^* n$ (which is the optimal asymptotic dependency on $n$ due to Linial's lower bound [FOCS'87]). Our lower bound proof is based on the round elimination framework, and its structure is inspired by a new round elimination fixed point that we give for the $Delta$-vertex coloring problem in hypergraphs. For the MIS problem on hypergraphs, we show that for $Deltall r$, there are significant improvements over the naive $O(Delta r + log^* n)$-round algorithm. We give two deterministic algorithms for the problem. We show that a hypergraph MIS can be computed in $O(Delta^2cdotlog r + Deltacdotlog rcdot log^* r + log^* n)$ rounds. We further show that at the cost of a worse dependency on $Delta$, the dependency on $r$ can be removed almost entirely, by giving an algorithm with complexity $Delta^{O(Delta)}cdotlog^* r + O(log^* n)$.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"30 1","pages":"2632-2676"},"PeriodicalIF":0.0,"publicationDate":"2022-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81779022","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-11-02DOI: 10.48550/arXiv.2211.01468
L. Li, Sushant Sachdeva
We demonstrate that for expander graphs, for all $epsilon>0,$ there exists a data structure of size $widetilde{O}(nepsilon^{-1})$ which can be used to return $(1 + epsilon)$-approximations to effective resistances in $widetilde{O}(1)$ time per query. Short of storing all effective resistances, previous best approaches could achieve $widetilde{O}(nepsilon^{-2})$ size and $widetilde{O}(epsilon^{-2})$ time per query by storing Johnson-Lindenstrauss vectors for each vertex, or $widetilde{O}(nepsilon^{-1})$ size and $widetilde{O}(nepsilon^{-1})$ time per query by storing a spectral sketch. Our construction is based on two key ideas: 1) $epsilon^{-1}$-sparse, $epsilon$-additive approximations to $DL^+1_u$ for all $u,$ can be used to recover $(1 + epsilon)$-approximations to the effective resistances, 2) In expander graphs, only $widetilde{O}(epsilon^{-1})$ coordinates of a vector similar to $DL^+1_u$ are larger than $epsilon.$ We give an efficient construction for such a data structure in $widetilde{O}(m + nepsilon^{-2})$ time via random walks. This results in an algorithm for computing $(1+epsilon)$-approximate effective resistances for $s$ vertex pairs in expanders that runs in $widetilde{O}(m + nepsilon^{-2} + s)$ time, improving over the previously best known running time of $m^{1 + o(1)} + (n + s)n^{o(1)}epsilon^{-1.5}$ for $s = omega(nepsilon^{-0.5}).$ We employ the above algorithm to compute a $(1+delta)$-approximation to the number of spanning trees in an expander graph, or equivalently, approximating the (pseudo)determinant of its Laplacian in $widetilde{O}(m + n^{1.5}delta^{-1})$ time. This improves on the previously best known result of $m^{1+o(1)} + n^{1.875+o(1)}delta^{-1.75}$ time, and matches the best known size of determinant sparsifiers.
{"title":"A New Approach to Estimating Effective Resistances and Counting Spanning Trees in Expander Graphs","authors":"L. Li, Sushant Sachdeva","doi":"10.48550/arXiv.2211.01468","DOIUrl":"https://doi.org/10.48550/arXiv.2211.01468","url":null,"abstract":"We demonstrate that for expander graphs, for all $epsilon>0,$ there exists a data structure of size $widetilde{O}(nepsilon^{-1})$ which can be used to return $(1 + epsilon)$-approximations to effective resistances in $widetilde{O}(1)$ time per query. Short of storing all effective resistances, previous best approaches could achieve $widetilde{O}(nepsilon^{-2})$ size and $widetilde{O}(epsilon^{-2})$ time per query by storing Johnson-Lindenstrauss vectors for each vertex, or $widetilde{O}(nepsilon^{-1})$ size and $widetilde{O}(nepsilon^{-1})$ time per query by storing a spectral sketch. Our construction is based on two key ideas: 1) $epsilon^{-1}$-sparse, $epsilon$-additive approximations to $DL^+1_u$ for all $u,$ can be used to recover $(1 + epsilon)$-approximations to the effective resistances, 2) In expander graphs, only $widetilde{O}(epsilon^{-1})$ coordinates of a vector similar to $DL^+1_u$ are larger than $epsilon.$ We give an efficient construction for such a data structure in $widetilde{O}(m + nepsilon^{-2})$ time via random walks. This results in an algorithm for computing $(1+epsilon)$-approximate effective resistances for $s$ vertex pairs in expanders that runs in $widetilde{O}(m + nepsilon^{-2} + s)$ time, improving over the previously best known running time of $m^{1 + o(1)} + (n + s)n^{o(1)}epsilon^{-1.5}$ for $s = omega(nepsilon^{-0.5}).$ We employ the above algorithm to compute a $(1+delta)$-approximation to the number of spanning trees in an expander graph, or equivalently, approximating the (pseudo)determinant of its Laplacian in $widetilde{O}(m + n^{1.5}delta^{-1})$ time. This improves on the previously best known result of $m^{1+o(1)} + n^{1.875+o(1)}delta^{-1.75}$ time, and matches the best known size of determinant sparsifiers.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"20 1","pages":"2728-2745"},"PeriodicalIF":0.0,"publicationDate":"2022-11-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81463624","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-31DOI: 10.48550/arXiv.2210.17515
M. Derakhshan, Alireza Farhadi
In the stochastic weighted matching problem, the goal is to find a large-weight matching of a graph when we are uncertain about the existence of its edges. In particular, each edge $e$ has a known weight $w_e$ but is realized independently with some probability $p_e$. The algorithm may query an edge to see whether it is realized. We consider the well-studied query commit version of the problem, in which any queried edge that happens to be realized must be included in the solution. Gamlath, Kale, and Svensson showed that when the input graph is bipartite, the problem admits a $(1-1/e)$-approximation. In this paper, we give an algorithm that for an absolute constant $delta>0.0014$ obtains a $(1-1/e+delta)$-approximation, therefore breaking this prevalent bound.
{"title":"Beating (1-1/e)-Approximation for Weighted Stochastic Matching","authors":"M. Derakhshan, Alireza Farhadi","doi":"10.48550/arXiv.2210.17515","DOIUrl":"https://doi.org/10.48550/arXiv.2210.17515","url":null,"abstract":"In the stochastic weighted matching problem, the goal is to find a large-weight matching of a graph when we are uncertain about the existence of its edges. In particular, each edge $e$ has a known weight $w_e$ but is realized independently with some probability $p_e$. The algorithm may query an edge to see whether it is realized. We consider the well-studied query commit version of the problem, in which any queried edge that happens to be realized must be included in the solution. Gamlath, Kale, and Svensson showed that when the input graph is bipartite, the problem admits a $(1-1/e)$-approximation. In this paper, we give an algorithm that for an absolute constant $delta>0.0014$ obtains a $(1-1/e+delta)$-approximation, therefore breaking this prevalent bound.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"163 1","pages":"1931-1961"},"PeriodicalIF":0.0,"publicationDate":"2022-10-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72864924","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-24DOI: 10.48550/arXiv.2210.13395
Kishen N. Gowda, Thomas W. Pensyl, A. Srinivasan, Khoa Trinh
The current best approximation algorithms for $k$-median rely on first obtaining a structured fractional solution known as a bi-point solution, and then rounding it to an integer solution. We improve this second step by unifying and refining previous approaches. We describe a hierarchy of increasingly-complex partitioning schemes for the facilities, along with corresponding sets of algorithms and factor-revealing non-linear programs. We prove that the third layer of this hierarchy is a $2.613$-approximation, improving upon the current best ratio of $2.675$, while no layer can be proved better than $2.588$ under the proposed analysis. On the negative side, we give a family of bi-point solutions which cannot be approximated better than the square root of the golden ratio, even if allowed to open $k+o(k)$ facilities. This gives a barrier to current approaches for obtaining an approximation better than $2 sqrt{phi} approx 2.544$. Altogether we reduce the approximation gap of bi-point solutions by two thirds.
{"title":"Improved Bi-point Rounding Algorithms and a Golden Barrier for k-Median","authors":"Kishen N. Gowda, Thomas W. Pensyl, A. Srinivasan, Khoa Trinh","doi":"10.48550/arXiv.2210.13395","DOIUrl":"https://doi.org/10.48550/arXiv.2210.13395","url":null,"abstract":"The current best approximation algorithms for $k$-median rely on first obtaining a structured fractional solution known as a bi-point solution, and then rounding it to an integer solution. We improve this second step by unifying and refining previous approaches. We describe a hierarchy of increasingly-complex partitioning schemes for the facilities, along with corresponding sets of algorithms and factor-revealing non-linear programs. We prove that the third layer of this hierarchy is a $2.613$-approximation, improving upon the current best ratio of $2.675$, while no layer can be proved better than $2.588$ under the proposed analysis. On the negative side, we give a family of bi-point solutions which cannot be approximated better than the square root of the golden ratio, even if allowed to open $k+o(k)$ facilities. This gives a barrier to current approaches for obtaining an approximation better than $2 sqrt{phi} approx 2.544$. Altogether we reduce the approximation gap of bi-point solutions by two thirds.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"50 1","pages":"987-1011"},"PeriodicalIF":0.0,"publicationDate":"2022-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79763655","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-23DOI: 10.48550/arXiv.2210.12601
Pan Peng, Yuichi Yoshida
We show sublinear-time algorithms for Max Cut and Max E2Lin$(q)$ on expanders in the adjacency list model that distinguishes instances with the optimal value more than $1-varepsilon$ from those with the optimal value less than $1-rho$ for $rho gg varepsilon$. The time complexities for Max Cut and Max $2$Lin$(q)$ are $widetilde{O}(frac{1}{phi^2rho} cdot m^{1/2+O(varepsilon/(phi^2rho))})$ and $widetilde{O}(mathrm{poly}(frac{q}{phirho})cdot {(mq)}^{1/2+O(q^6varepsilon/phi^2rho^2)})$, respectively, where $m$ is the number of edges in the underlying graph and $phi$ is its conductance. Then, we show a sublinear-time algorithm for Unique Label Cover on expanders with $phi gg epsilon$ in the bounded-degree model. The time complexity of our algorithm is $widetilde{O}_d(2^{q^{O(1)}cdotphi^{1/q}cdot varepsilon^{-1/2}}cdot n^{1/2+q^{O(q)}cdot varepsilon^{4^{1.5-q}}cdot phi^{-2}})$, where $n$ is the number of variables. We complement these algorithmic results by showing that testing $3$-colorability requires $Omega(n)$ queries even on expanders.
{"title":"Sublinear-Time Algorithms for Max Cut, Max E2Lin(q), and Unique Label Cover on Expanders","authors":"Pan Peng, Yuichi Yoshida","doi":"10.48550/arXiv.2210.12601","DOIUrl":"https://doi.org/10.48550/arXiv.2210.12601","url":null,"abstract":"We show sublinear-time algorithms for Max Cut and Max E2Lin$(q)$ on expanders in the adjacency list model that distinguishes instances with the optimal value more than $1-varepsilon$ from those with the optimal value less than $1-rho$ for $rho gg varepsilon$. The time complexities for Max Cut and Max $2$Lin$(q)$ are $widetilde{O}(frac{1}{phi^2rho} cdot m^{1/2+O(varepsilon/(phi^2rho))})$ and $widetilde{O}(mathrm{poly}(frac{q}{phirho})cdot {(mq)}^{1/2+O(q^6varepsilon/phi^2rho^2)})$, respectively, where $m$ is the number of edges in the underlying graph and $phi$ is its conductance. Then, we show a sublinear-time algorithm for Unique Label Cover on expanders with $phi gg epsilon$ in the bounded-degree model. The time complexity of our algorithm is $widetilde{O}_d(2^{q^{O(1)}cdotphi^{1/q}cdot varepsilon^{-1/2}}cdot n^{1/2+q^{O(q)}cdot varepsilon^{4^{1.5-q}}cdot phi^{-2}})$, where $n$ is the number of variables. We complement these algorithmic results by showing that testing $3$-colorability requires $Omega(n)$ queries even on expanders.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"134 1","pages":"4936-4965"},"PeriodicalIF":0.0,"publicationDate":"2022-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77373990","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-10-15DOI: 10.48550/arXiv.2210.08293
Lorenzo Ciardo, Stanislav Živný
We show that approximate graph colouring is not solved by any level of the affine integer programming (AIP) hierarchy. To establish the result, we translate the problem of exhibiting a graph fooling a level of the AIP hierarchy into the problem of constructing a highly symmetric crystal tensor. In order to prove the existence of crystals in arbitrary dimension, we provide a combinatorial characterisation for realisable systems of tensors; i.e., sets of low-dimensional tensors that can be realised as the projections of a single high-dimensional tensor.
{"title":"Approximate Graph Colouring and Crystals","authors":"Lorenzo Ciardo, Stanislav Živný","doi":"10.48550/arXiv.2210.08293","DOIUrl":"https://doi.org/10.48550/arXiv.2210.08293","url":null,"abstract":"We show that approximate graph colouring is not solved by any level of the affine integer programming (AIP) hierarchy. To establish the result, we translate the problem of exhibiting a graph fooling a level of the AIP hierarchy into the problem of constructing a highly symmetric crystal tensor. In order to prove the existence of crystals in arbitrary dimension, we provide a combinatorial characterisation for realisable systems of tensors; i.e., sets of low-dimensional tensors that can be realised as the projections of a single high-dimensional tensor.","PeriodicalId":92709,"journal":{"name":"Proceedings of the ... Annual ACM-SIAM Symposium on Discrete Algorithms. ACM-SIAM Symposium on Discrete Algorithms","volume":"74 1","pages":"2256-2267"},"PeriodicalIF":0.0,"publicationDate":"2022-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86934184","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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