We study the problem of list-decodable (robust) Gaussian mean estimation and the related problem of learning mixtures of separated spherical Gaussians. In the former problem, we are given a set T of points in n with the promise that an α-fraction of points in T, where 0< α < 1/2, are drawn from an unknown mean identity covariance Gaussian G, and no assumptions are made about the remaining points. The goal is to output a small list of candidate vectors with the guarantee that at least one of the candidates is close to the mean of G. In the latter problem, we are given samples from a k-mixture of spherical Gaussians on n and the goal is to estimate the unknown model parameters up to small accuracy. We develop a set of techniques that yield new efficient algorithms with significantly improved guarantees for these problems. Specifically, our main contributions are as follows: List-Decodable Mean Estimation. Fix any d ∈ + and 0< α <1/2. We design an algorithm with sample complexity Od ((nd/α)) and runtime Od ((n/α)d) that outputs a list of O(1/α) many candidate vectors such that with high probability one of the candidates is within ℓ2-distance Od(α−1/(2d)) from the mean of G. The only previous algorithm for this problem achieved error Õ(α−1/2) under second moment conditions. For d = O(1/), where >0 is a constant, our algorithm runs in polynomial time and achieves error O(α). For d = Θ(log(1/α)), our algorithm runs in time (n/α)O(log(1/α)) and achieves error O(log3/2(1/α)), almost matching the information-theoretically optimal bound of Θ(log1/2(1/α)) that we establish. We also give a Statistical Query (SQ) lower bound suggesting that the complexity of our algorithm is qualitatively close to best possible. Learning Mixtures of Spherical Gaussians. We give a learning algorithm for mixtures of spherical Gaussians, with unknown spherical covariances, that succeeds under significantly weaker separation assumptions compared to prior work. For the prototypical case of a uniform k-mixture of identity covariance Gaussians we obtain the following: For any >0, if the pairwise separation between the means is at least Ω(k+√log(1/δ)), our algorithm learns the unknown parameters within accuracy δ with sample complexity and running time (n, 1/δ, (k/)1/). Moreover, our algorithm is robust to a small dimension-independent fraction of corrupted data. The previously best known polynomial time algorithm required separation at least k1/4 (k/δ). Finally, our algorithm works under separation of Õ(log3/2(k)+√log(1/δ)) with sample complexity and running time (n, 1/δ, klogk). This bound is close to the information-theoretically minimum separation of Ω(√logk). Our main technical contribution is a new technique, using degree-d multivariate polynomials, to remove outliers from high-dimensional datasets where the majority of the points are corrupted.
{"title":"List-decodable robust mean estimation and learning mixtures of spherical gaussians","authors":"Ilias Diakonikolas, D. Kane, Alistair Stewart","doi":"10.1145/3188745.3188758","DOIUrl":"https://doi.org/10.1145/3188745.3188758","url":null,"abstract":"We study the problem of list-decodable (robust) Gaussian mean estimation and the related problem of learning mixtures of separated spherical Gaussians. In the former problem, we are given a set T of points in n with the promise that an α-fraction of points in T, where 0< α < 1/2, are drawn from an unknown mean identity covariance Gaussian G, and no assumptions are made about the remaining points. The goal is to output a small list of candidate vectors with the guarantee that at least one of the candidates is close to the mean of G. In the latter problem, we are given samples from a k-mixture of spherical Gaussians on n and the goal is to estimate the unknown model parameters up to small accuracy. We develop a set of techniques that yield new efficient algorithms with significantly improved guarantees for these problems. Specifically, our main contributions are as follows: List-Decodable Mean Estimation. Fix any d ∈ + and 0< α <1/2. We design an algorithm with sample complexity Od ((nd/α)) and runtime Od ((n/α)d) that outputs a list of O(1/α) many candidate vectors such that with high probability one of the candidates is within ℓ2-distance Od(α−1/(2d)) from the mean of G. The only previous algorithm for this problem achieved error Õ(α−1/2) under second moment conditions. For d = O(1/), where >0 is a constant, our algorithm runs in polynomial time and achieves error O(α). For d = Θ(log(1/α)), our algorithm runs in time (n/α)O(log(1/α)) and achieves error O(log3/2(1/α)), almost matching the information-theoretically optimal bound of Θ(log1/2(1/α)) that we establish. We also give a Statistical Query (SQ) lower bound suggesting that the complexity of our algorithm is qualitatively close to best possible. Learning Mixtures of Spherical Gaussians. We give a learning algorithm for mixtures of spherical Gaussians, with unknown spherical covariances, that succeeds under significantly weaker separation assumptions compared to prior work. For the prototypical case of a uniform k-mixture of identity covariance Gaussians we obtain the following: For any >0, if the pairwise separation between the means is at least Ω(k+√log(1/δ)), our algorithm learns the unknown parameters within accuracy δ with sample complexity and running time (n, 1/δ, (k/)1/). Moreover, our algorithm is robust to a small dimension-independent fraction of corrupted data. The previously best known polynomial time algorithm required separation at least k1/4 (k/δ). Finally, our algorithm works under separation of Õ(log3/2(k)+√log(1/δ)) with sample complexity and running time (n, 1/δ, klogk). This bound is close to the information-theoretically minimum separation of Ω(√logk). Our main technical contribution is a new technique, using degree-d multivariate polynomials, to remove outliers from high-dimensional datasets where the majority of the points are corrupted.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81194119","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}
We give an m1+o(1)βo(1)-time algorithm for generating uniformly random spanning trees in weighted graphs with max-to-min weight ratio β. In the process, we illustrate how fundamental tradeoffs in graph partitioning can be overcome by eliminating vertices from a graph using Schur complements of the associated Laplacian matrix. Our starting point is the Aldous-Broder algorithm, which samples a random spanning tree using a random walk. As in prior work, we use fast Laplacian linear system solvers to shortcut the random walk from a vertex v to the boundary of a set of vertices assigned to v called a “shortcutter.” We depart from prior work by introducing a new way of employing Laplacian solvers to shortcut the walk. To bound the amount of shortcutting work, we show that most random walk steps occur far away from an unvisited vertex. We apply this observation by charging uses of a shortcutter S to random walk steps in the Schur complement obtained by eliminating all vertices in S that are not assigned to it.
{"title":"An almost-linear time algorithm for uniform random spanning tree generation","authors":"Aaron Schild","doi":"10.1145/3188745.3188852","DOIUrl":"https://doi.org/10.1145/3188745.3188852","url":null,"abstract":"We give an m1+o(1)βo(1)-time algorithm for generating uniformly random spanning trees in weighted graphs with max-to-min weight ratio β. In the process, we illustrate how fundamental tradeoffs in graph partitioning can be overcome by eliminating vertices from a graph using Schur complements of the associated Laplacian matrix. Our starting point is the Aldous-Broder algorithm, which samples a random spanning tree using a random walk. As in prior work, we use fast Laplacian linear system solvers to shortcut the random walk from a vertex v to the boundary of a set of vertices assigned to v called a “shortcutter.” We depart from prior work by introducing a new way of employing Laplacian solvers to shortcut the walk. To bound the amount of shortcutting work, we show that most random walk steps occur far away from an unvisited vertex. We apply this observation by charging uses of a shortcutter S to random walk steps in the Schur complement obtained by eliminating all vertices in S that are not assigned to it.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75446527","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}
We present a deterministic distributed algorithm, in the LOCAL model, that computes a (1+o(1))Δ-edge-coloring in polylogarithmic-time, so long as the maximum degree Δ=Ω(logn). For smaller Δ, we give a polylogarithmic-time 3Δ/2-edge-coloring. These are the first deterministic algorithms to go below the natural barrier of 2Δ−1 colors, and they improve significantly on the recent polylogarithmic-time (2Δ−1)(1+o(1))-edge-coloring of Ghaffari and Su [SODA’17] and the (2Δ−1)-edge-coloring of Fischer, Ghaffari, and Kuhn [FOCS’17], positively answering the main open question of the latter. The key technical ingredient of our algorithm is a simple and novel gradual packing of judiciously chosen near-maximum matchings, each of which becomes one of the color classes.
{"title":"Deterministic distributed edge-coloring with fewer colors","authors":"M. Ghaffari, F. Kuhn, Yannic Maus, Jara Uitto","doi":"10.1145/3188745.3188906","DOIUrl":"https://doi.org/10.1145/3188745.3188906","url":null,"abstract":"We present a deterministic distributed algorithm, in the LOCAL model, that computes a (1+o(1))Δ-edge-coloring in polylogarithmic-time, so long as the maximum degree Δ=Ω(logn). For smaller Δ, we give a polylogarithmic-time 3Δ/2-edge-coloring. These are the first deterministic algorithms to go below the natural barrier of 2Δ−1 colors, and they improve significantly on the recent polylogarithmic-time (2Δ−1)(1+o(1))-edge-coloring of Ghaffari and Su [SODA’17] and the (2Δ−1)-edge-coloring of Fischer, Ghaffari, and Kuhn [FOCS’17], positively answering the main open question of the latter. The key technical ingredient of our algorithm is a simple and novel gradual packing of judiciously chosen near-maximum matchings, each of which becomes one of the color classes.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88542319","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}
Explaining the excellent practical performance of the simplex method for linear programming has been a major topic of research for over 50 years. One of the most successful frameworks for understanding the simplex method was given by Spielman and Teng (JACM ‘04), who the developed the notion of smoothed analysis. Starting from an arbitrary linear program with d variables and n constraints, Spielman and Teng analyzed the expected runtime over random perturbations of the LP (smoothed LP), where variance σ Gaussian noise is added to the LP data. In particular, they gave a two-stage shadow vertex simplex algorithm which uses an expected O(n86 d55 σ−30) number of simplex pivots to solve the smoothed LP. Their analysis and runtime was substantially improved by SpielmanDeshpande (FOCS ‘05) and later Vershynin (SICOMP ‘09). The fastest current algorithm, due to Vershynin, solves the smoothed LP using an expected O(d3 log3 n σ−4 + d9log7 n) number of pivots, improving the dependence on n from polynomial to logarithmic. While the original proof of SpielmanTeng has now been substantially simplified, the resulting analyses are still quite long and complex and the parameter dependencies far from optimal. In this work, we make substantial progress on this front, providing an improved and simpler analysis of shadow simplex methods, where our main algorithm requires an expected O(d2 √logn σ−2 + d5 log3/2 n) number of simplex pivots. We obtain our results via an improved shadow bound, key to earlier analyses as well, combined with algorithmic techniques of Borgwardt (ZOR ‘82) and Vershynin. As an added bonus, our analysis is completely modular, allowing us to obtain non-trivial bounds for perturbations beyond Gaussians, such as Laplace perturbations.
{"title":"A friendly smoothed analysis of the simplex method","authors":"D. Dadush, Sophie Huiberts","doi":"10.1145/3188745.3188826","DOIUrl":"https://doi.org/10.1145/3188745.3188826","url":null,"abstract":"Explaining the excellent practical performance of the simplex method for linear programming has been a major topic of research for over 50 years. One of the most successful frameworks for understanding the simplex method was given by Spielman and Teng (JACM ‘04), who the developed the notion of smoothed analysis. Starting from an arbitrary linear program with d variables and n constraints, Spielman and Teng analyzed the expected runtime over random perturbations of the LP (smoothed LP), where variance σ Gaussian noise is added to the LP data. In particular, they gave a two-stage shadow vertex simplex algorithm which uses an expected O(n86 d55 σ−30) number of simplex pivots to solve the smoothed LP. Their analysis and runtime was substantially improved by SpielmanDeshpande (FOCS ‘05) and later Vershynin (SICOMP ‘09). The fastest current algorithm, due to Vershynin, solves the smoothed LP using an expected O(d3 log3 n σ−4 + d9log7 n) number of pivots, improving the dependence on n from polynomial to logarithmic. While the original proof of SpielmanTeng has now been substantially simplified, the resulting analyses are still quite long and complex and the parameter dependencies far from optimal. In this work, we make substantial progress on this front, providing an improved and simpler analysis of shadow simplex methods, where our main algorithm requires an expected O(d2 √logn σ−2 + d5 log3/2 n) number of simplex pivots. We obtain our results via an improved shadow bound, key to earlier analyses as well, combined with algorithmic techniques of Borgwardt (ZOR ‘82) and Vershynin. As an added bonus, our analysis is completely modular, allowing us to obtain non-trivial bounds for perturbations beyond Gaussians, such as Laplace perturbations.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81892914","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}
We show that the computational problem Consensus Halving is PPA-Complete, the first PPA-Completeness result for a problem whose definition does not involve an explicit circuit. We also show that an approximate version of this problem is polynomial-time equivalent to Necklace Splitting, which establishes PPAD-hardness for Necklace Splitting and suggests that it is also PPA-Complete.
{"title":"Consensus halving is PPA-complete","authors":"Aris Filos-Ratsikas, P. Goldberg","doi":"10.1145/3188745.3188880","DOIUrl":"https://doi.org/10.1145/3188745.3188880","url":null,"abstract":"We show that the computational problem Consensus Halving is PPA-Complete, the first PPA-Completeness result for a problem whose definition does not involve an explicit circuit. We also show that an approximate version of this problem is polynomial-time equivalent to Necklace Splitting, which establishes PPAD-hardness for Necklace Splitting and suggests that it is also PPA-Complete.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84001275","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}
Diptarka Chakraborty, Lior Kamma, Kasper Green Larsen
The conjectured hardness of Boolean matrix-vector multiplication has been used with great success to prove conditional lower bounds for numerous important data structure problems, see Henzinger et al. [STOC’15]. In recent work, Larsen and Williams [SODA’17] attacked the problem from the upper bound side and gave a surprising cell probe data structure (that is, we only charge for memory accesses, while computation is free). Their cell probe data structure answers queries in Õ(n7/4) time and is succinct in the sense that it stores the input matrix in read-only memory, plus an additional Õ(n7/4) bits on the side. In this paper, we essentially settle the cell probe complexity of succinct Boolean matrix-vector multiplication. We present a new cell probe data structure with query time Õ(n3/2) storing just Õ(n3/2) bits on the side. We then complement our data structure with a lower bound showing that any data structure storing r bits on the side, with n < r < n2 must have query time t satisfying t r = Ω(n3). For r ≤ n, any data structure must have t = Ω(n2). Since lower bounds in the cell probe model also apply to classic word-RAM data structures, the lower bounds naturally carry over. We also prove similar lower bounds for matrix-vector multiplication over F2.
{"title":"Tight cell probe bounds for succinct Boolean matrix-vector multiplication","authors":"Diptarka Chakraborty, Lior Kamma, Kasper Green Larsen","doi":"10.1145/3188745.3188830","DOIUrl":"https://doi.org/10.1145/3188745.3188830","url":null,"abstract":"The conjectured hardness of Boolean matrix-vector multiplication has been used with great success to prove conditional lower bounds for numerous important data structure problems, see Henzinger et al. [STOC’15]. In recent work, Larsen and Williams [SODA’17] attacked the problem from the upper bound side and gave a surprising cell probe data structure (that is, we only charge for memory accesses, while computation is free). Their cell probe data structure answers queries in Õ(n7/4) time and is succinct in the sense that it stores the input matrix in read-only memory, plus an additional Õ(n7/4) bits on the side. In this paper, we essentially settle the cell probe complexity of succinct Boolean matrix-vector multiplication. We present a new cell probe data structure with query time Õ(n3/2) storing just Õ(n3/2) bits on the side. We then complement our data structure with a lower bound showing that any data structure storing r bits on the side, with n < r < n2 must have query time t satisfying t r = Ω(n3). For r ≤ n, any data structure must have t = Ω(n2). Since lower bounds in the cell probe model also apply to classic word-RAM data structures, the lower bounds naturally carry over. We also prove similar lower bounds for matrix-vector multiplication over F2.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73872652","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}
Omar Fawzi, Antoine Grospellier, Anthony Leverrier
We show that quantum expander codes, a constant-rate family of quantum low-density parity check (LDPC) codes, with the quasi-linear time decoding algorithm of Leverrier, Tillich and Zémor can correct a constant fraction of random errors with very high probability. This is the first construction of a constant-rate quantum LDPC code with an efficient decoding algorithm that can correct a linear number of random errors with a negligible failure probability. Finding codes with these properties is also motivated by Gottesman’s construction of fault tolerant schemes with constant space overhead. In order to obtain this result, we study a notion of α-percolation: for a random subset E of vertices of a given graph, we consider the size of the largest connected α-subset of E, where X is an α-subset of E if |X ∩ E| ≥ α |X|.
{"title":"Efficient decoding of random errors for quantum expander codes","authors":"Omar Fawzi, Antoine Grospellier, Anthony Leverrier","doi":"10.1145/3188745.3188886","DOIUrl":"https://doi.org/10.1145/3188745.3188886","url":null,"abstract":"We show that quantum expander codes, a constant-rate family of quantum low-density parity check (LDPC) codes, with the quasi-linear time decoding algorithm of Leverrier, Tillich and Zémor can correct a constant fraction of random errors with very high probability. This is the first construction of a constant-rate quantum LDPC code with an efficient decoding algorithm that can correct a linear number of random errors with a negligible failure probability. Finding codes with these properties is also motivated by Gottesman’s construction of fault tolerant schemes with constant space overhead. In order to obtain this result, we study a notion of α-percolation: for a random subset E of vertices of a given graph, we consider the size of the largest connected α-subset of E, where X is an α-subset of E if |X ∩ E| ≥ α |X|.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74539888","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}
In the classical Node-Disjoint Paths (NDP) problem, we are given an n-vertex graph G=(V,E), and a collection M={(s1,t1),…,(sk,tk)} of pairs of its vertices, called source-destination, or demand pairs. The goal is to route as many of the demand pairs as possible, where to route a pair we need to select a path connecting it, so that all selected paths are disjoint in their vertices. The best current algorithm for NDP achieves an O(√n)-approximation, while, until recently, the best negative result was a factor Ω(log1/2−єn)-hardness of approximation, for any constant є, unless NP ⊆ ZPTIME(npoly logn). In a recent work, the authors have shown an improved 2Ω(√logn)-hardness of approximation for NDP, unless NP⊆ DTIME(nO(logn)), even if the underlying graph is a subgraph of a grid graph, and all source vertices lie on the boundary of the grid. Unfortunately, this result does not extend to grid graphs. The approximability of the NDP problem on grid graphs has remained a tantalizing open question, with the best current upper bound of Õ(n1/4), and the best current lower bound of APX-hardness. In a recent work, the authors showed a 2Õ(√logn)-approximation algorithm for NDP in grid graphs, if all source vertices lie on the boundary of the grid – a result that can be seen as suggesting that a sub-polynomial approximation may be achievable for NDP in grids. In this paper we show that this is unlikely to be the case, and come close to resolving the approximability of NDP in general, and of NDP in grids in particular. Our main result is that NDP is 2Ω(log1−є n)-hard to approximate for any constant є, assuming that NP⊈RTIME(npoly logn), and that it is nΩ (1/(loglogn)2)-hard to approximate, assuming that for some constant δ>0, NP ⊈RTIME(2nδ). These results hold even for grid graphs and wall graphs, and extend to the closely related Edge-Disjoint Paths problem, even in wall graphs. Our hardness proof performs a reduction from the 3COL(5) problem to NDP, using a new graph partitioning problem as a proxy. Unlike the more standard approach of employing Karp reductions to prove hardness of approximation, our proof is a Cook-type reduction, where, given an input instance of 3COL(5), we produce a large number of instances of NDP, and apply an approximation algorithm for NDP to each of them. The construction of each new instance of NDP crucially depends on the solutions to the previous instances that were found by the approximation algorithm.
{"title":"Almost polynomial hardness of node-disjoint paths in grids","authors":"Julia Chuzhoy, David H. K. Kim, Rachit Nimavat","doi":"10.1145/3188745.3188772","DOIUrl":"https://doi.org/10.1145/3188745.3188772","url":null,"abstract":"In the classical Node-Disjoint Paths (NDP) problem, we are given an n-vertex graph G=(V,E), and a collection M={(s1,t1),…,(sk,tk)} of pairs of its vertices, called source-destination, or demand pairs. The goal is to route as many of the demand pairs as possible, where to route a pair we need to select a path connecting it, so that all selected paths are disjoint in their vertices. The best current algorithm for NDP achieves an O(√n)-approximation, while, until recently, the best negative result was a factor Ω(log1/2−єn)-hardness of approximation, for any constant є, unless NP ⊆ ZPTIME(npoly logn). In a recent work, the authors have shown an improved 2Ω(√logn)-hardness of approximation for NDP, unless NP⊆ DTIME(nO(logn)), even if the underlying graph is a subgraph of a grid graph, and all source vertices lie on the boundary of the grid. Unfortunately, this result does not extend to grid graphs. The approximability of the NDP problem on grid graphs has remained a tantalizing open question, with the best current upper bound of Õ(n1/4), and the best current lower bound of APX-hardness. In a recent work, the authors showed a 2Õ(√logn)-approximation algorithm for NDP in grid graphs, if all source vertices lie on the boundary of the grid – a result that can be seen as suggesting that a sub-polynomial approximation may be achievable for NDP in grids. In this paper we show that this is unlikely to be the case, and come close to resolving the approximability of NDP in general, and of NDP in grids in particular. Our main result is that NDP is 2Ω(log1−є n)-hard to approximate for any constant є, assuming that NP⊈RTIME(npoly logn), and that it is nΩ (1/(loglogn)2)-hard to approximate, assuming that for some constant δ>0, NP ⊈RTIME(2nδ). These results hold even for grid graphs and wall graphs, and extend to the closely related Edge-Disjoint Paths problem, even in wall graphs. Our hardness proof performs a reduction from the 3COL(5) problem to NDP, using a new graph partitioning problem as a proxy. Unlike the more standard approach of employing Karp reductions to prove hardness of approximation, our proof is a Cook-type reduction, where, given an input instance of 3COL(5), we produce a large number of instances of NDP, and apply an approximation algorithm for NDP to each of them. The construction of each new instance of NDP crucially depends on the solutions to the previous instances that were found by the approximation algorithm.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91248654","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}
A. Balliu, J. Hirvonen, Janne H. Korhonen, Tuomo Lempiäinen, D. Olivetti, J. Suomela
A number of recent papers – e.g. Brandt et al. (STOC 2016), Chang et al. (FOCS 2016), Ghaffari & Su (SODA 2017), Brandt et al. (PODC 2017), and Chang & Pettie (FOCS 2017) – have advanced our understanding of one of the most fundamental questions in theory of distributed computing: what are the possible time complexity classes of LCL problems in the LOCAL model? In essence, we have a graph problem Π in which a solution can be verified by checking all radius-O(1) neighbourhoods, and the question is what is the smallest T such that a solution can be computed so that each node chooses its own output based on its radius-T neighbourhood. Here T is the distributed time complexity of Π. The time complexity classes for deterministic algorithms in bounded-degree graphs that are known to exist by prior work are Θ(1), Θ(log* n), Θ(logn), Θ(n1/k), and Θ(n). It is also known that there are two gaps: one between ω(1) and o(loglog* n), and another between ω(log* n) and o(logn). It has been conjectured that many more gaps exist, and that the overall time hierarchy is relatively simple – indeed, this is known to be the case in restricted graph families such as cycles and grids. We show that the picture is much more diverse than previously expected. We present a general technique for engineering LCL problems with numerous different deterministic time complexities, including Θ(logα n) for any α ≥ 1, 2Θ(logα n) for any α ≤ 1, and Θ(nα) for any α < 1/2 in the high end of the complexity spectrum, and Θ(logα log* n) for any α ≥ 1, 2Θ(logα log* n) for any α ≤ 1, and Θ((log* n)α) for any α ≤ 1 in the low end of the complexity spectrum; here α is a positive rational number.
{"title":"New classes of distributed time complexity","authors":"A. Balliu, J. Hirvonen, Janne H. Korhonen, Tuomo Lempiäinen, D. Olivetti, J. Suomela","doi":"10.1145/3188745.3188860","DOIUrl":"https://doi.org/10.1145/3188745.3188860","url":null,"abstract":"A number of recent papers – e.g. Brandt et al. (STOC 2016), Chang et al. (FOCS 2016), Ghaffari & Su (SODA 2017), Brandt et al. (PODC 2017), and Chang & Pettie (FOCS 2017) – have advanced our understanding of one of the most fundamental questions in theory of distributed computing: what are the possible time complexity classes of LCL problems in the LOCAL model? In essence, we have a graph problem Π in which a solution can be verified by checking all radius-O(1) neighbourhoods, and the question is what is the smallest T such that a solution can be computed so that each node chooses its own output based on its radius-T neighbourhood. Here T is the distributed time complexity of Π. The time complexity classes for deterministic algorithms in bounded-degree graphs that are known to exist by prior work are Θ(1), Θ(log* n), Θ(logn), Θ(n1/k), and Θ(n). It is also known that there are two gaps: one between ω(1) and o(loglog* n), and another between ω(log* n) and o(logn). It has been conjectured that many more gaps exist, and that the overall time hierarchy is relatively simple – indeed, this is known to be the case in restricted graph families such as cycles and grids. We show that the picture is much more diverse than previously expected. We present a general technique for engineering LCL problems with numerous different deterministic time complexities, including Θ(logα n) for any α ≥ 1, 2Θ(logα n) for any α ≤ 1, and Θ(nα) for any α < 1/2 in the high end of the complexity spectrum, and Θ(logα log* n) for any α ≥ 1, 2Θ(logα log* n) for any α ≤ 1, and Θ((log* n)α) for any α ≤ 1 in the low end of the complexity spectrum; here α is a positive rational number.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87292397","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}
We study the Ordered k-Median problem, in which the solution is evaluated by first sorting the client connection costs and then multiplying them with a predefined non-increasing weight vector (higher connection costs are taken with larger weights). Since the 1990s, this problem has been studied extensively in the discrete optimization and operations research communities and has emerged as a framework unifying many fundamental clustering and location problems such as k-Median and k-Center. Obtaining non-trivial approximation algorithms was an open problem even for simple topologies such as trees. Recently, Aouad and Segev (2017) were able to obtain an O(log n) approximation algorithm for Ordered k-Median using a sophisticated local-search approach. The existence of a constant-factor approximation algorithm, however, remained open even for the rectangular weight vector. In this paper, we provide an LP-rounding constant-factor approximation algorithm for the Ordered k-Median problem. We achieve this result by revealing an interesting connection to the classic k-Median problem. In particular, we propose a novel LP relaxation that uses the constraints of the natural LP relaxation for k-Median but minimizes over a non-metric, distorted cost vector. This cost function (approximately) emulates the weighting of distances in an optimum solution and can be guessed by means of a clever enumeration scheme of Aouad and Segev. Although the resulting LP has an unbounded integrality gap, we can show that the LP rounding process by Charikar and Li (2012) for k-Median, operating on the original, metric space, gives a constant-factor approximation when relating not only to the LP value but also to a combinatorial bound derived from the guessing phase. To analyze the rounding process under the non-linear, ranking-based objective of Ordered k-Median, we employ several new ideas and technical ingredients that we believe could be of interest in some of the numerous other settings related to ordered, weighted cost functions.
{"title":"Constant-factor approximation for ordered k-median","authors":"J. Byrka, Krzysztof Sornat, J. Spoerhase","doi":"10.1145/3188745.3188930","DOIUrl":"https://doi.org/10.1145/3188745.3188930","url":null,"abstract":"We study the Ordered k-Median problem, in which the solution is evaluated by first sorting the client connection costs and then multiplying them with a predefined non-increasing weight vector (higher connection costs are taken with larger weights). Since the 1990s, this problem has been studied extensively in the discrete optimization and operations research communities and has emerged as a framework unifying many fundamental clustering and location problems such as k-Median and k-Center. Obtaining non-trivial approximation algorithms was an open problem even for simple topologies such as trees. Recently, Aouad and Segev (2017) were able to obtain an O(log n) approximation algorithm for Ordered k-Median using a sophisticated local-search approach. The existence of a constant-factor approximation algorithm, however, remained open even for the rectangular weight vector. In this paper, we provide an LP-rounding constant-factor approximation algorithm for the Ordered k-Median problem. We achieve this result by revealing an interesting connection to the classic k-Median problem. In particular, we propose a novel LP relaxation that uses the constraints of the natural LP relaxation for k-Median but minimizes over a non-metric, distorted cost vector. This cost function (approximately) emulates the weighting of distances in an optimum solution and can be guessed by means of a clever enumeration scheme of Aouad and Segev. Although the resulting LP has an unbounded integrality gap, we can show that the LP rounding process by Charikar and Li (2012) for k-Median, operating on the original, metric space, gives a constant-factor approximation when relating not only to the LP value but also to a combinatorial bound derived from the guessing phase. To analyze the rounding process under the non-linear, ranking-based objective of Ordered k-Median, we employ several new ideas and technical ingredients that we believe could be of interest in some of the numerous other settings related to ordered, weighted cost functions.","PeriodicalId":20593,"journal":{"name":"Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91458810","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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