A graph G is a k-leaf power if there exists a tree T whose leaf set is V(G), and such that uv ∈ E(G) if and only if the distance between u and v in T is at most k (and u ≠ v). The graph classes of k-leaf powers have several applications in computational biology, but recognizing them has remained a challenging algorithmic problem for the past two decades. The best known result is that 6-leaf powers can be recognized in polynomial time. In this paper, we present an algorithm that decides whether a graph G is a k-leaf power in time O(nf(k)) for some function f that depends only on k (but has the growth rate of a power tower function). Our techniques are based on the fact that either a k-leaf power has a corresponding tree of low maximum degree, in which case finding it is easy, or every corresponding tree has large maximum degree. In the latter case, large degree vertices in the tree imply that G has redundant substructures which can be pruned from the graph. In addition to solving a longstanding open problem, we hope that the structural results presented in this work can lead to further results on k-leaf powers and related classes.
{"title":"Recognizing k-leaf powers in polynomial time, for constant k","authors":"Manuel Lafond","doi":"10.1145/3614094","DOIUrl":"https://doi.org/10.1145/3614094","url":null,"abstract":"A graph G is a k-leaf power if there exists a tree T whose leaf set is V(G), and such that uv ∈ E(G) if and only if the distance between u and v in T is at most k (and u ≠ v). The graph classes of k-leaf powers have several applications in computational biology, but recognizing them has remained a challenging algorithmic problem for the past two decades. The best known result is that 6-leaf powers can be recognized in polynomial time. In this paper, we present an algorithm that decides whether a graph G is a k-leaf power in time O(nf(k)) for some function f that depends only on k (but has the growth rate of a power tower function). Our techniques are based on the fact that either a k-leaf power has a corresponding tree of low maximum degree, in which case finding it is easy, or every corresponding tree has large maximum degree. In the latter case, large degree vertices in the tree imply that G has redundant substructures which can be pruned from the graph. In addition to solving a longstanding open problem, we hope that the structural results presented in this work can lead to further results on k-leaf powers and related classes.","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":"1 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41637755","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper we study the computational complexity of solving a class of block structured integer programs (IPs) - so called multistage stochastic IPs. A multistage stochastic IP is an IP of the form min {c⊺x∣Ax = b, x ≥ 0, x integral} where the constraint matrix A consists of small block matrices ordered on the diagonal line and for each stage there are larger blocks with few columns connecting the blocks in a tree like fashion. Over the last years there was enormous progress in the area of block structured IPs. For many of the known block IP classes - such as n-fold, tree-fold, and two-stage stochastic IPs, nearly matching upper and lower bounds are known concerning their computational complexity. One of the major gaps that remained however was the parameter dependency in the running time for an algorithm solving multistage stochastic IPs. Previous algorithms require a tower of t exponentials, where t is the number of stages. In contrast, only a double exponential lower bound was known based on the exponential time hypothesis. In this paper we show that the tower of t exponentials is actually not necessary. We show an improved running time of (2^{(dleftVert A rightVert _infty)^{mathcal {O}(d^{3t+1})}} cdot rnlog ^{mathcal {O}(2^d)}(rn) ) for the algorithm solving multistage stochastic IPs, where d is the sum of columns in the connecting blocks and rn is the number of rows. Hence, we obtain the first bound by an elementary function for the running time of an algorithm solving multistage stochastic IPs. In contrast to previous works, our algorithm has only a triple exponential dependency on the parameters and only doubly exponential for every constant t. By this we come very close to the known double exponential bound that holds already for two-stage stochastic IPs, i.e. multistage stochastic IPs with two stages. The improved running time of the algorithm is based on new bounds for the proximity of multistage stochastic IPs. The idea behind the bound is based on generalization of a structural lemma originally used for two-stage stochastic IPs. While the structural lemma requires iteration to be applied to multistage stochastic IPs, our generalization directly applies to inherent combinatorial properties of multiple stages. Already a special case of our lemma yields an improved bound for the Graver complexity of multistage stochastic IPs.
{"title":"Collapsing the Tower - On the Complexity of Multistage Stochastic IPs","authors":"Kim-Manuel Klein, J. Reuter","doi":"10.1145/3604554","DOIUrl":"https://doi.org/10.1145/3604554","url":null,"abstract":"In this paper we study the computational complexity of solving a class of block structured integer programs (IPs) - so called multistage stochastic IPs. A multistage stochastic IP is an IP of the form min {c⊺x∣Ax = b, x ≥ 0, x integral} where the constraint matrix A consists of small block matrices ordered on the diagonal line and for each stage there are larger blocks with few columns connecting the blocks in a tree like fashion. Over the last years there was enormous progress in the area of block structured IPs. For many of the known block IP classes - such as n-fold, tree-fold, and two-stage stochastic IPs, nearly matching upper and lower bounds are known concerning their computational complexity. One of the major gaps that remained however was the parameter dependency in the running time for an algorithm solving multistage stochastic IPs. Previous algorithms require a tower of t exponentials, where t is the number of stages. In contrast, only a double exponential lower bound was known based on the exponential time hypothesis. In this paper we show that the tower of t exponentials is actually not necessary. We show an improved running time of (2^{(dleftVert A rightVert _infty)^{mathcal {O}(d^{3t+1})}} cdot rnlog ^{mathcal {O}(2^d)}(rn) ) for the algorithm solving multistage stochastic IPs, where d is the sum of columns in the connecting blocks and rn is the number of rows. Hence, we obtain the first bound by an elementary function for the running time of an algorithm solving multistage stochastic IPs. In contrast to previous works, our algorithm has only a triple exponential dependency on the parameters and only doubly exponential for every constant t. By this we come very close to the known double exponential bound that holds already for two-stage stochastic IPs, i.e. multistage stochastic IPs with two stages. The improved running time of the algorithm is based on new bounds for the proximity of multistage stochastic IPs. The idea behind the bound is based on generalization of a structural lemma originally used for two-stage stochastic IPs. While the structural lemma requires iteration to be applied to multistage stochastic IPs, our generalization directly applies to inherent combinatorial properties of multiple stages. Already a special case of our lemma yields an improved bound for the Graver complexity of multistage stochastic IPs.","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45374238","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Takehiro Ito, Yuni Iwamasa, Naonori Kakimura, Naoyuki Kamiyama, Yusuke Kobayashi, Shun-ichi Maezawa, Yuta Nozaki, Y. Okamoto, K. Ozeki
We initiate the study of k-edge-connected orientations of undirected graphs through edge flips for k ≥ 2. We prove that in every orientation of an undirected 2k-edge-connected graph, there exists a sequence of edges such that flipping their directions one by one does not decrease the edge connectivity, and the final orientation is k-edge connected. This yields an “edge-flip based” new proof of Nash-Williams’ theorem: A undirected graph G has a k-edge-connected orientation if and only if G is 2k-edge connected. As another consequence of the theorem, we prove that the edge-flip graph of k-edge-connected orientations of an undirected graph G is connected if G is (2k+2)-edge connected. This has been known to be true only when k=1.
{"title":"Monotone Edge Flips to an Orientation of Maximum Edge-Connectivity à la Nash-Williams","authors":"Takehiro Ito, Yuni Iwamasa, Naonori Kakimura, Naoyuki Kamiyama, Yusuke Kobayashi, Shun-ichi Maezawa, Yuta Nozaki, Y. Okamoto, K. Ozeki","doi":"10.1145/3561302","DOIUrl":"https://doi.org/10.1145/3561302","url":null,"abstract":"We initiate the study of k-edge-connected orientations of undirected graphs through edge flips for k ≥ 2. We prove that in every orientation of an undirected 2k-edge-connected graph, there exists a sequence of edges such that flipping their directions one by one does not decrease the edge connectivity, and the final orientation is k-edge connected. This yields an “edge-flip based” new proof of Nash-Williams’ theorem: A undirected graph G has a k-edge-connected orientation if and only if G is 2k-edge connected. As another consequence of the theorem, we prove that the edge-flip graph of k-edge-connected orientations of an undirected graph G is connected if G is (2k+2)-edge connected. This has been known to be true only when k=1.","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":"19 1","pages":"1 - 22"},"PeriodicalIF":1.3,"publicationDate":"2021-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45223403","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The greedy spanner in a low dimensional Euclidean space is a fundamental geometric construction that has been extensively studied over three decades as it possesses the two most basic properties of a good spanner: constant maximum degree and constant lightness. Recently, Eppstein and Khodabandeh [28] showed that the greedy spanner in (mathbb {R}^2 ) admits a sublinear separator in a strong sense: any subgraph of k vertices of the greedy spanner in (mathbb {R}^2 ) has a separator of size (O(sqrt {k}) ) . Their technique is inherently planar and is not extensible to higher dimensions. They left showing the existence of a small separator for the greedy spanner in (mathbb {R}^d ) for any constant d ≥ 3 as an open problem. In this paper, we resolve the problem of Eppstein and Khodabandeh [28] by showing that any subgraph of k vertices of the greedy spanner in (mathbb {R}^d ) has a separator of size O(k1 − 1/d). We introduce a new technique that gives a simple criterion for any geometric graph to have a sublinear separator that we dub τ-lanky: a geometric graph is τ-lanky if any ball of radius r cuts at most τ edges of length at least r in the graph. We show that any τ-lanky geometric graph of n vertices in (mathbb {R}^d ) has a separator of size O(τn1 − 1/d). We then derive our main result by showing that the greedy spanner is O(1)-lanky. We indeed obtain a more general result that applies to unit ball graphs and point sets of low fractal dimensions in (mathbb {R}^d ) . Our technique naturally extends to doubling metrics. We use the τ-lanky criterion to show that there exists a (1 + ϵ)-spanner for doubling metrics of dimension d with a constant maximum degree and a separator of size (O(n^{1-frac{1}{d}}) ) ; this result resolves an open problem posed by Abam and Har-Peled [1] a decade ago. We then introduce another simple criterion for a graph in doubling metrics of dimension d to have a sublinear separator. We use the new criterion to show that the greedy spanner of an n-point metric space of doubling dimension d has a separator of size (O((n^{1-frac{1}{d}}) + log Delta) ) where Δ is the spread of the metric; the factor log (Δ) is tightly connected to the fact that, unlike its Euclidean counterpart, the greedy spanner in doubling metrics has unbounded maximum degree. Finally, we discuss algorithmic implications of our results.
{"title":"Greedy Spanners in Euclidean Spaces Admit Sublinear Separators","authors":"Hung Le, Cuong V. Than","doi":"10.1145/3590771","DOIUrl":"https://doi.org/10.1145/3590771","url":null,"abstract":"The greedy spanner in a low dimensional Euclidean space is a fundamental geometric construction that has been extensively studied over three decades as it possesses the two most basic properties of a good spanner: constant maximum degree and constant lightness. Recently, Eppstein and Khodabandeh [28] showed that the greedy spanner in (mathbb {R}^2 ) admits a sublinear separator in a strong sense: any subgraph of k vertices of the greedy spanner in (mathbb {R}^2 ) has a separator of size (O(sqrt {k}) ) . Their technique is inherently planar and is not extensible to higher dimensions. They left showing the existence of a small separator for the greedy spanner in (mathbb {R}^d ) for any constant d ≥ 3 as an open problem. In this paper, we resolve the problem of Eppstein and Khodabandeh [28] by showing that any subgraph of k vertices of the greedy spanner in (mathbb {R}^d ) has a separator of size O(k1 − 1/d). We introduce a new technique that gives a simple criterion for any geometric graph to have a sublinear separator that we dub τ-lanky: a geometric graph is τ-lanky if any ball of radius r cuts at most τ edges of length at least r in the graph. We show that any τ-lanky geometric graph of n vertices in (mathbb {R}^d ) has a separator of size O(τn1 − 1/d). We then derive our main result by showing that the greedy spanner is O(1)-lanky. We indeed obtain a more general result that applies to unit ball graphs and point sets of low fractal dimensions in (mathbb {R}^d ) . Our technique naturally extends to doubling metrics. We use the τ-lanky criterion to show that there exists a (1 + ϵ)-spanner for doubling metrics of dimension d with a constant maximum degree and a separator of size (O(n^{1-frac{1}{d}}) ) ; this result resolves an open problem posed by Abam and Har-Peled [1] a decade ago. We then introduce another simple criterion for a graph in doubling metrics of dimension d to have a sublinear separator. We use the new criterion to show that the greedy spanner of an n-point metric space of doubling dimension d has a separator of size (O((n^{1-frac{1}{d}}) + log Delta) ) where Δ is the spread of the metric; the factor log (Δ) is tightly connected to the fact that, unlike its Euclidean counterpart, the greedy spanner in doubling metrics has unbounded maximum degree. Finally, we discuss algorithmic implications of our results.","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":" ","pages":""},"PeriodicalIF":1.3,"publicationDate":"2021-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46689132","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we present Approximation Schemes for Capacitated Vehicle Routing Problem (CVRP) on several classes of graphs. In CVRP, introduced by Dantzig and Ramser in 1959 [14], we are given a graph G=(V,E) with metric edges costs, a depot r ∈ V, and a vehicle of bounded capacity Q. The goal is to find a minimum cost collection of tours for the vehicle that returns to the depot, each visiting at most Q nodes, such that they cover all the nodes. This generalizes classic TSP and has been studied extensively. In the more general setting, each node v has a demand dv and the total demand of each tour must be no more than Q. Either the demand of each node must be served by one tour (unsplittable) or can be served by multiple tours (splittable). The best-known approximation algorithm for general graphs has ratio α +2(1-ε) (for the unsplittable) and α +1-ε (for the splittable) for some fixed (ε gt frac{1}{3000}) , where α is the best approximation for TSP. Even for the case of trees, the best approximation ratio is 4/3 [5] and it has been an open question if there is an approximation scheme for this simple class of graphs. Das and Mathieu [15] presented an approximation scheme with time nlogO(1/ε)n for Euclidean plane ℝ2. No other approximation scheme is known for any other class of metrics (without further restrictions on Q). In this article, we make significant progress on this classic problem by presenting Quasi-Polynomial Time Approximation Schemes (QPTAS) for graphs of bounded treewidth, graphs of bounded highway dimensions, and graphs of bounded doubling dimensions. For comparison, our result implies an approximation scheme for the Euclidean plane with run time nO(log6n/ε5).
{"title":"Approximation Schemes for Capacitated Vehicle Routing on Graphs of Bounded Treewidth, Bounded Doubling, or Highway Dimension","authors":"Aditya Jayaprakash, M. Salavatipour","doi":"10.1145/3582500","DOIUrl":"https://doi.org/10.1145/3582500","url":null,"abstract":"In this article, we present Approximation Schemes for Capacitated Vehicle Routing Problem (CVRP) on several classes of graphs. In CVRP, introduced by Dantzig and Ramser in 1959 [14], we are given a graph G=(V,E) with metric edges costs, a depot r ∈ V, and a vehicle of bounded capacity Q. The goal is to find a minimum cost collection of tours for the vehicle that returns to the depot, each visiting at most Q nodes, such that they cover all the nodes. This generalizes classic TSP and has been studied extensively. In the more general setting, each node v has a demand dv and the total demand of each tour must be no more than Q. Either the demand of each node must be served by one tour (unsplittable) or can be served by multiple tours (splittable). The best-known approximation algorithm for general graphs has ratio α +2(1-ε) (for the unsplittable) and α +1-ε (for the splittable) for some fixed (ε gt frac{1}{3000}) , where α is the best approximation for TSP. Even for the case of trees, the best approximation ratio is 4/3 [5] and it has been an open question if there is an approximation scheme for this simple class of graphs. Das and Mathieu [15] presented an approximation scheme with time nlogO(1/ε)n for Euclidean plane ℝ2. No other approximation scheme is known for any other class of metrics (without further restrictions on Q). In this article, we make significant progress on this classic problem by presenting Quasi-Polynomial Time Approximation Schemes (QPTAS) for graphs of bounded treewidth, graphs of bounded highway dimensions, and graphs of bounded doubling dimensions. For comparison, our result implies an approximation scheme for the Euclidean plane with run time nO(log6n/ε5).","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":"19 1","pages":"1 - 36"},"PeriodicalIF":1.3,"publicationDate":"2021-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49221488","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider the classic problem of computing the Longest Common Subsequence (LCS) of two strings of length n. The 40-year-old quadratic-time dynamic programming algorithm has recently been shown to be near-optimal by Abboud, Backurs, and Vassilevska Williams [FOCS’15] and Bringmann and Künnemann [FOCS’15] assuming the Strong Exponential Time Hypothesis. This has led the community to look for subquadratic approximation algorithms for the problem. Yet, unlike the edit distance problem for which a constant-factor approximation in almost-linear time is known, very little progress has been made on LCS, making it a notoriously difficult problem also in the realm of approximation. For the general setting, only a naive O(nɛ/2-approximation algorithm with running time OŠ(n2-ɛ has been known, for any constant 0 < ɛ ≤ 1. Recently, a breakthrough result by Hajiaghayi, Seddighin, Seddighin, and Sun [SODA’19] provided a linear-time algorithm that yields a O(n0.497956-approximation in expectation; improving upon the naive (O(sqrt {n})) -approximation for the first time. In this paper, we provide an algorithm that in time O(n2-ɛ) computes an OŠ(n2ɛ/5-approximation with high probability, for any 0 < ɛ ≤ 1. Our result (1) gives an OŠ(n0.4-approximation in linear time, improving upon the bound of Hajiaghayi, Seddighin, Seddighin, and Sun, (2) provides an algorithm whose approximation scales with any subquadratic running time O(n2-ɛ), improving upon the naive bound of O(nɛ/2) for any ɛ, and (3) instead of only in expectation, succeeds with high probability.
{"title":"A Linear-Time n0.4-Approximation for Longest Common Subsequence","authors":"K. Bringmann, Vincent Cohen-Addad, Debarati Das","doi":"10.1145/3568398","DOIUrl":"https://doi.org/10.1145/3568398","url":null,"abstract":"We consider the classic problem of computing the Longest Common Subsequence (LCS) of two strings of length n. The 40-year-old quadratic-time dynamic programming algorithm has recently been shown to be near-optimal by Abboud, Backurs, and Vassilevska Williams [FOCS’15] and Bringmann and Künnemann [FOCS’15] assuming the Strong Exponential Time Hypothesis. This has led the community to look for subquadratic approximation algorithms for the problem. Yet, unlike the edit distance problem for which a constant-factor approximation in almost-linear time is known, very little progress has been made on LCS, making it a notoriously difficult problem also in the realm of approximation. For the general setting, only a naive O(nɛ/2-approximation algorithm with running time OŠ(n2-ɛ has been known, for any constant 0 < ɛ ≤ 1. Recently, a breakthrough result by Hajiaghayi, Seddighin, Seddighin, and Sun [SODA’19] provided a linear-time algorithm that yields a O(n0.497956-approximation in expectation; improving upon the naive (O(sqrt {n})) -approximation for the first time. In this paper, we provide an algorithm that in time O(n2-ɛ) computes an OŠ(n2ɛ/5-approximation with high probability, for any 0 < ɛ ≤ 1. Our result (1) gives an OŠ(n0.4-approximation in linear time, improving upon the bound of Hajiaghayi, Seddighin, Seddighin, and Sun, (2) provides an algorithm whose approximation scales with any subquadratic running time O(n2-ɛ), improving upon the naive bound of O(nɛ/2) for any ɛ, and (3) instead of only in expectation, succeeds with high probability.","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":"19 1","pages":"1 - 24"},"PeriodicalIF":1.3,"publicationDate":"2021-06-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46708293","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This article presents universal algorithms for clustering problems, including the widely studied k-median, k-means, and k-center objectives. The input is a metric space containing all potential client locations. The algorithm must select k cluster centers such that they are a good solution for any subset of clients that actually realize. Specifically, we aim for low regret, defined as the maximum over all subsets of the difference between the cost of the algorithm’s solution and that of an optimal solution. A universal algorithm’s solution Sol for a clustering problem is said to be an α , β-approximation if for all subsets of clients C′, it satisfies sol (C′) ≤ α ċ opt (C′) + β ċ mr, where opt (C′ is the cost of the optimal solution for clients (C′) and mr is the minimum regret achievable by any solution. Our main results are universal algorithms for the standard clustering objectives of k-median, k-means, and k-center that achieve (O(1), O(1))-approximations. These results are obtained via a novel framework for universal algorithms using linear programming (LP) relaxations. These results generalize to other ℓp-objectives and the setting where some subset of the clients are fixed. We also give hardness results showing that (α, β)-approximation is NP-hard if α or β is at most a certain constant, even for the widely studied special case of Euclidean metric spaces. This shows that in some sense, (O(1), O(1))-approximation is the strongest type of guarantee obtainable for universal clustering.
{"title":"Universal Algorithms for Clustering Problems","authors":"Arun Ganesh, B. Maggs, Debmalya Panigrahi","doi":"10.1145/3572840","DOIUrl":"https://doi.org/10.1145/3572840","url":null,"abstract":"This article presents universal algorithms for clustering problems, including the widely studied k-median, k-means, and k-center objectives. The input is a metric space containing all potential client locations. The algorithm must select k cluster centers such that they are a good solution for any subset of clients that actually realize. Specifically, we aim for low regret, defined as the maximum over all subsets of the difference between the cost of the algorithm’s solution and that of an optimal solution. A universal algorithm’s solution Sol for a clustering problem is said to be an α , β-approximation if for all subsets of clients C′, it satisfies sol (C′) ≤ α ċ opt (C′) + β ċ mr, where opt (C′ is the cost of the optimal solution for clients (C′) and mr is the minimum regret achievable by any solution. Our main results are universal algorithms for the standard clustering objectives of k-median, k-means, and k-center that achieve (O(1), O(1))-approximations. These results are obtained via a novel framework for universal algorithms using linear programming (LP) relaxations. These results generalize to other ℓp-objectives and the setting where some subset of the clients are fixed. We also give hardness results showing that (α, β)-approximation is NP-hard if α or β is at most a certain constant, even for the widely studied special case of Euclidean metric spaces. This shows that in some sense, (O(1), O(1))-approximation is the strongest type of guarantee obtainable for universal clustering.","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":"19 1","pages":"1 - 46"},"PeriodicalIF":1.3,"publicationDate":"2021-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47512912","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Anders Aamand, Mikkel Abrahamsen, P. M. R. Rasmussen, Thomas D. Ahle
A polyomino is a polygonal region with axis-parallel edges and corners of integral coordinates, which may have holes. In this paper, we consider planar tiling and packing problems with polyomino pieces and a polyomino container P. We give polynomial-time algorithms for deciding if P can be tiled with k × k squares for any fixed k which can be part of the input (that is, deciding if P is the union of a set of non-overlapping k × k squares) and for packing P with a maximum number of non-overlapping and axis-parallel 2 × 1 dominos, allowing rotations by 90°. As packing is more general than tiling, the latter algorithm can also be used to decide if P can be tiled by 2 × 1 dominos. These are classical problems with important applications in VLSI design, and the related problem of finding a maximum packing of 2 × 2 squares is known to be NP-hard [6]. For our three problems there are known pseudo-polynomial-time algorithms, that is, algorithms with running times polynomial in the area or perimeter of P. However, the standard, compact way to represent a polygon is by listing the coordinates of the corners in binary. We use this representation, and thus present the first polynomial-time algorithms for the problems. Concretely, we give a simple O(n log n)-time algorithm for tiling with squares, where n is the number of corners of P. We then give a more involved algorithm that reduces the problems of packing and tiling with dominos to finding a maximum and perfect matching in a graph with O(n3) vertices. This leads to algorithms with running times (O(n^3 frac{log ^3 n}{log ^2log n})) and (O(n^3 frac{log ^2 n}{log log n})) , respectively.
{"title":"Tiling with Squares and Packing Dominos in Polynomial Time","authors":"Anders Aamand, Mikkel Abrahamsen, P. M. R. Rasmussen, Thomas D. Ahle","doi":"10.1145/3597932","DOIUrl":"https://doi.org/10.1145/3597932","url":null,"abstract":"A polyomino is a polygonal region with axis-parallel edges and corners of integral coordinates, which may have holes. In this paper, we consider planar tiling and packing problems with polyomino pieces and a polyomino container P. We give polynomial-time algorithms for deciding if P can be tiled with k × k squares for any fixed k which can be part of the input (that is, deciding if P is the union of a set of non-overlapping k × k squares) and for packing P with a maximum number of non-overlapping and axis-parallel 2 × 1 dominos, allowing rotations by 90°. As packing is more general than tiling, the latter algorithm can also be used to decide if P can be tiled by 2 × 1 dominos. These are classical problems with important applications in VLSI design, and the related problem of finding a maximum packing of 2 × 2 squares is known to be NP-hard [6]. For our three problems there are known pseudo-polynomial-time algorithms, that is, algorithms with running times polynomial in the area or perimeter of P. However, the standard, compact way to represent a polygon is by listing the coordinates of the corners in binary. We use this representation, and thus present the first polynomial-time algorithms for the problems. Concretely, we give a simple O(n log n)-time algorithm for tiling with squares, where n is the number of corners of P. We then give a more involved algorithm that reduces the problems of packing and tiling with dominos to finding a maximum and perfect matching in a graph with O(n3) vertices. This leads to algorithms with running times (O(n^3 frac{log ^3 n}{log ^2log n})) and (O(n^3 frac{log ^2 n}{log log n})) , respectively.","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":"19 1","pages":"1 - 28"},"PeriodicalIF":1.3,"publicationDate":"2020-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44913968","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Lior Gishboliner, Yevgeny Levanzov, A. Shapira, R. Yuster
Since counting subgraphs in general graphs is, by and large, a computationally demanding problem, it is natural to try and design fast algorithms for restricted families of graphs. One such family that has been extensively studied is that of graphs of bounded degeneracy (e.g., planar graphs). This line of work, which started in the early 80’s, culminated in a recent work of Gishboliner et al., which highlighted the importance of the task of counting homomorphic copies of cycles (i.e., cyclic walks) in graphs of bounded degeneracy. Our main result in this paper is a surprisingly tight relation between the above task and the well-studied problem of detecting (standard) copies of directed cycles in general directed graphs. More precisely, we prove the following: One can compute the number of homomorphic copies of C2k and C2k+1 in n-vertex graphs of bounded degeneracy in time Õ(ndk), where the fastest known algorithm for detecting directed copies of Ck in general m-edge digraphs runs in time Õ(mdk). Conversely, one can transform any O(nbk) algorithm for computing the number of homomorphic copies of C2k or of C2k+1 in n-vertex graphs of bounded degeneracy, into an Õ(mbk) time algorithm for detecting directed copies of Ck in general m-edge digraphs. We emphasize that our first result does not use a black-box reduction (as opposed to the second result which does). Instead, we design an algorithm for computing the number of Ck-homomorphisms in degenerate graphs and show that one part of its analysis can be reduced to the analysis of the fastest known algorithm for detecting directed cycles in general digraphs, which was carried out in a recent breakthrough of Dalirrooyfard, Vuong and Vassilevska Williams. As a by-product of our algorithm, we obtain a new algorithm for detecting k-cycles in directed and undirected graphs of bounded degeneracy that is faster than all previously known algorithms for 7 ≤ k ≤ 11, and faster for all k ≥ 7 if the matrix multiplication exponent is 2.
{"title":"Counting Homomorphic Cycles in Degenerate Graphs","authors":"Lior Gishboliner, Yevgeny Levanzov, A. Shapira, R. Yuster","doi":"10.1145/3560820","DOIUrl":"https://doi.org/10.1145/3560820","url":null,"abstract":"Since counting subgraphs in general graphs is, by and large, a computationally demanding problem, it is natural to try and design fast algorithms for restricted families of graphs. One such family that has been extensively studied is that of graphs of bounded degeneracy (e.g., planar graphs). This line of work, which started in the early 80’s, culminated in a recent work of Gishboliner et al., which highlighted the importance of the task of counting homomorphic copies of cycles (i.e., cyclic walks) in graphs of bounded degeneracy. Our main result in this paper is a surprisingly tight relation between the above task and the well-studied problem of detecting (standard) copies of directed cycles in general directed graphs. More precisely, we prove the following: One can compute the number of homomorphic copies of C2k and C2k+1 in n-vertex graphs of bounded degeneracy in time Õ(ndk), where the fastest known algorithm for detecting directed copies of Ck in general m-edge digraphs runs in time Õ(mdk). Conversely, one can transform any O(nbk) algorithm for computing the number of homomorphic copies of C2k or of C2k+1 in n-vertex graphs of bounded degeneracy, into an Õ(mbk) time algorithm for detecting directed copies of Ck in general m-edge digraphs. We emphasize that our first result does not use a black-box reduction (as opposed to the second result which does). Instead, we design an algorithm for computing the number of Ck-homomorphisms in degenerate graphs and show that one part of its analysis can be reduced to the analysis of the fastest known algorithm for detecting directed cycles in general digraphs, which was carried out in a recent breakthrough of Dalirrooyfard, Vuong and Vassilevska Williams. As a by-product of our algorithm, we obtain a new algorithm for detecting k-cycles in directed and undirected graphs of bounded degeneracy that is faster than all previously known algorithms for 7 ≤ k ≤ 11, and faster for all k ≥ 7 if the matrix multiplication exponent is 2.","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":"19 1","pages":"1 - 22"},"PeriodicalIF":1.3,"publicationDate":"2020-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46151666","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The complexity of (unbounded-arity) Max-CSPs under structural restrictions is poorly understood. The two most general hypergraph properties known to ensure tractability of Max-CSPs, β-acyclicity an...
{"title":"Point-Width and Max-CSPs","authors":"CarbonnelClément, RomeroMiguel, ŽivnýStanislav","doi":"10.1145/3409447","DOIUrl":"https://doi.org/10.1145/3409447","url":null,"abstract":"The complexity of (unbounded-arity) Max-CSPs under structural restrictions is poorly understood. The two most general hypergraph properties known to ensure tractability of Max-CSPs, β-acyclicity an...","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":"16 1","pages":"1-28"},"PeriodicalIF":1.3,"publicationDate":"2020-09-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1145/3409447","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44013762","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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