ACM Transactions on Algorithms最新文献

{"title":"Tiling with Squares and Packing Dominos in Polynomial Time","authors":"Anders Aamand, Mikkel Abrahamsen, Peter M. R. Rasmussen, Thomas D. Ahle","doi":"https://dl.acm.org/doi/10.1145/3597932","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3597932","url":null,"abstract":"<p>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 <i>P</i>. We give polynomial-time algorithms for deciding if <i>P</i> can be tiled with <i>k</i> × <i>k</i> squares for any fixed <i>k</i> which can be part of the input (that is, deciding if <i>P</i> is the union of a set of non-overlapping <i>k × k</i> squares) and for packing <i>P</i> with a maximum number of non-overlapping and axis-parallel <i>2 × 1</i> dominos, allowing rotations by 90°. As packing is more general than tiling, the latter algorithm can also be used to decide if <i>P</i> can be tiled by 2 × 1 dominos.</p><p>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 <i>area</i> or <i>perimeter</i> of <i>P</i>. 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 <i>O(n</i> log <i>n</i>)-time algorithm for tiling with squares, where <i>n</i> is the number of corners of <i>P</i>. 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 <i>O</i>(<i>n</i>³) 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.</p>","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":"7 17","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138494900","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}

{"title":"A Polynomial-Time Algorithm for 1/3-Approximate Nash Equilibria in Bimatrix Games","authors":"Argyrios Deligkas, Michail Fasoulakis, Evangelos Markakis","doi":"https://dl.acm.org/doi/10.1145/3606697","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3606697","url":null,"abstract":"<p>Since the celebrated PPAD-completeness result for Nash equilibria in bimatrix games, a long line of research has focused on polynomial-time algorithms that compute ε-approximate Nash equilibria. Finding the best possible approximation guarantee that we can have in polynomial time has been a fundamental and non-trivial pursuit on settling the complexity of approximate equilibria. Despite a significant amount of effort, the algorithm of Tsaknakis and Spirakis [38], with an approximation guarantee of (0.3393 + <i>δ</i>), remains the state of the art over the last 15 years. In this paper, we propose a new refinement of the Tsaknakis-Spirakis algorithm, resulting in a polynomial-time algorithm that computes a ((frac{1}{3}+delta) )-Nash equilibrium, for any constant <i>δ</i> &gt; 0. The main idea of our approach is to go beyond the use of convex combinations of primal and dual strategies, as defined in the optimization framework of [38], and enrich the pool of strategies from which we build the strategy profiles that we output in certain bottleneck cases of the algorithm.</p>","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":"7 18","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138494899","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}

{"title":"Efficient and Near-optimal Algorithms for Sampling Small Connected Subgraphs","authors":"Marco Bressan","doi":"https://dl.acm.org/doi/10.1145/3596495","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3596495","url":null,"abstract":"<p>We study the following problem: Given an integer <i>k</i> ≥ 3 and a simple graph <i>G</i>, sample a connected induced <i>k</i>-vertex subgraph of <i>G</i> uniformly at random. This is a fundamental graph mining primitive with applications in social network analysis, bioinformatics, and more. Surprisingly, no efficient algorithm is known for uniform sampling; the only somewhat efficient algorithms available yield samples that are only approximately uniform, with running times that are unclear or suboptimal. In this work, we provide: (i) a near-optimal mixing time bound for a well-known random walk technique, (ii) the first efficient algorithm for truly uniform graphlet sampling, and (iii) the first sublinear-time algorithm for ε-uniform graphlet sampling.</p>","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":"7 20","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138494897","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}

{"title":"Towards Optimal Moment Estimation in Streaming and Distributed Models","authors":"Rajesh Jayaram, David P. Woodruff","doi":"https://dl.acm.org/doi/10.1145/3596494","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3596494","url":null,"abstract":"<p>One of the oldest problems in the data stream model is to approximate the <i>p</i>th moment (Vert mathbf {X}Vert _p^p = sum _{i=1}^n mathbf {X}_i^p) of an underlying non-negative vector (mathbf {X}in mathbb {R}^n), which is presented as a sequence of (mathrm{poly}(n)) updates to its coordinates. Of particular interest is when (p in (0,2]). Although a tight space bound of (Theta (epsilon ^{-2} log n)) bits is known for this problem when both positive and negative updates are allowed, surprisingly, there is still a gap in the space complexity of this problem when all updates are positive. Specifically, the upper bound is (O(epsilon ^{-2} log n)) bits, while the lower bound is only (Omega (epsilon ^{-2} + log n)) bits. Recently, an upper bound of (tilde{O}(epsilon ^{-2} + log n)) bits was obtained under the assumption that the updates arrive in a <i>random order</i>. </p><p>We show that for (p in (0, 1]), the random order assumption is not needed. Namely, we give an upper bound for worst-case streams of (tilde{O}(epsilon ^{-2} + log n)) bits for estimating (Vert mathbf {X}Vert _p^p). Our techniques also give new upper bounds for estimating the empirical entropy in a stream. However, we show that for (p in (1,2]), in the natural coordinator and blackboard distributed communication topologies, there is an (tilde{O}(epsilon ^{-2})) bit max-communication upper bound based on a randomized rounding scheme. Our protocols also give rise to protocols for heavy hitters and approximate matrix product. We generalize our results to arbitrary communication topologies <i>G</i>, obtaining an (tilde{O}(epsilon ^{2} log d)) max-communication upper bound, where <i>d</i> is the diameter of <i>G</i>. Interestingly, our upper bound rules out natural communication complexity-based approaches for proving an (Omega (epsilon ^{-2} log n)) bit lower bound for (p in (1,2]) for streaming algorithms. In particular, any such lower bound must come from a topology with large diameter.</p>","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":"7 21","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138494896","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}

{"title":"Competitive Online Search Trees on Trees","authors":"Prosenjit Bose, Jean Cardinal, John Iacono, Grigorios Koumoutsos, Stefan Langerman","doi":"https://dl.acm.org/doi/10.1145/3595180","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3595180","url":null,"abstract":"<p>We consider the design of adaptive data structures for searching elements of a tree-structured space. We use a natural generalization of the rotation-based online binary search tree model in which the underlying search space is the set of vertices of a tree. This model is based on a simple structure for decomposing graphs, previously known under several names including elimination trees, vertex rankings, and tubings. The model is equivalent to the classical binary search tree model exactly when the underlying tree is a path. We describe an online <i>O</i>(log log <i>n</i>)-competitive search tree data structure in this model, where <i>n</i> is the number of vertices. This matches the best-known competitive ratio of binary search trees. Our method is inspired by Tango trees, an online binary search tree algorithm, but critically needs several new notions including one that we call Steiner-closed search trees, which may be of independent interest. Moreover, our technique is based on a novel use of two levels of decomposition, first from search space to a set of Steiner-closed trees and, second, from these trees into paths.</p>","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":"7 19","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138494898","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}

{"title":"Collapsing the Tower - On the Complexity of Multistage Stochastic IPs","authors":"Kim-Manuel Klein, Janina Reuter","doi":"https://dl.acm.org/doi/10.1145/3604554","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3604554","url":null,"abstract":"<p>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 {<i>c</i>^⊺<i>x</i>∣<i>Ax</i> = <i>b</i>, <i>x</i> ≥ <b>0</b>, <i>x</i> integral} where the constraint matrix <i>A</i> 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 <i>n</i>-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 <i>t</i> exponentials, where <i>t</i> 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 <i>t</i> 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 <i>d</i> is the sum of columns in the connecting blocks and <i>rn</i> 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 <i>t</i>. 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. </p><p>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.</p>","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":"7 22","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138494895","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}

{"title":"Matching on the Line Admits no (o(sqrt {log n})) -Competitive Algorithm","authors":"E. Peserico, Michele Scquizzato","doi":"10.1145/3594873","DOIUrl":"https://doi.org/10.1145/3594873","url":null,"abstract":"We present a simple proof that no randomized online matching algorithm for the line can be ((sqrt {log _2(n+1)}/15)) -competitive against an oblivious adversary for any n = 2i - 1 : i ∈ ℕ. This is the first super-constant lower bound for the problem, and disproves as a corollary a recent conjecture on the topology-parametrized competitiveness achievable on generic spaces.","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":"19 1","pages":"1 - 4"},"PeriodicalIF":1.3,"publicationDate":"2023-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43653341","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}

{"title":"Almost-Optimal Deterministic Treasure Hunt in Unweighted Graphs","authors":"Sébastien Bouchard, Yoann Dieudonné, Arnaud Labourel, Andrzej Pelc","doi":"https://dl.acm.org/doi/10.1145/3588437","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3588437","url":null,"abstract":"<p>A mobile agent navigating along edges of a simple connected unweighted graph, either finite or countably infinite, has to find an inert target (treasure) hidden in one of the nodes. This task is known as treasure hunt. The agent has no <i>a priori</i> knowledge of the graph, of the location of the treasure, or of the initial distance to it. The cost of a treasure hunt algorithm is the worst-case number of edge traversals performed by the agent until finding the treasure. Awerbuch et al. [3] considered graph exploration and treasure hunt for finite graphs in a restricted model where the agent has a fuel tank that can be replenished only at the starting node <i>s</i>. The size of the tank is <i>B = 2 (1+α) r</i>, for some positive real constant α, where <i>r</i>, called the radius of the graph, is the maximum distance from <i>s</i> to any other node. The tank of size <i>B</i> allows the agent to make at most ⌊ B ⌋ edge traversals between two consecutive visits at node <i>s</i>.</p><p>Let <i>e(d)</i> be the number of edges whose at least one endpoint is at distance less than <i>d</i> from <i>s</i>. Awerbuch et al. [3] conjectured that it is impossible to find a treasure hidden in a node at distance at most <i>d</i> at cost nearly linear in <i>e(d)</i>. We first design a deterministic treasure hunt algorithm working in the model without any restrictions on the moves of the agent at cost <i>𝒪(e(d)</i> log <i>d</i>) and then show how to modify this algorithm to work in the model from Awerbuch et al. [3] with the same complexity. Thus, we refute the preceding 20-year-old conjecture. We observe that no treasure hunt algorithm can beat cost Θ (<i>e(d)</i>) for all graphs, and thus our algorithms are also almost optimal.</p>","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":"11 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138516999","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}

{"title":"Load Thresholds for Cuckoo Hashing with Overlapping Blocks","authors":"Stefan Walzer","doi":"https://dl.acm.org/doi/10.1145/3589558","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3589558","url":null,"abstract":"<p>We consider a natural variation of cuckoo hashing proposed by Lehman and Panigrahy (2009). Each of <i>cn</i> objects is assigned <i>k</i> = 2 intervals of size ℓ in a linear hash table of size <i>n</i> and both starting points are chosen independently and uniformly at random. Each object must be placed into a table cell within its intervals, but each cell can only hold one object. Experiments suggested that this scheme outperforms the variant with <i>blocks</i> in which intervals are aligned at multiples of ℓ. In particular, the <i>load threshold</i> is higher, i.e., the load <i>c</i> that can be achieved with high probability. For instance, Lehman and Panigrahy (2009) empirically observed the threshold for ℓ = 2 to be around 96.5% as compared to roughly 89.7% using blocks. They pinned down the asymptotics of the thresholds for large ℓ, but the precise values resisted rigorous analysis. </p><p>We establish a method to determine these load thresholds for all ℓ ≥ 2, and, in fact, for general <i>k</i> ≥ 2. For instance, for <i>k</i> = ℓ = 2, we get ≈ 96.4995%. We employ a theorem due to Leconte, Lelarge, and Massoulié (2013), which adapts methods from statistical physics to the world of hypergraph orientability. In effect, the orientability thresholds for our graph families are determined by belief propagation equations for certain graph limits. As a side note, we provide experimental evidence suggesting that placements can be constructed in linear time using an adapted version of an algorithm by Khosla (2013).</p>","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":"7 24","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138494893","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}

{"title":"Hitting Topological Minor Models in Planar Graphs is Fixed Parameter Tractable","authors":"Petr A. Golovach, Giannos Stamoulis, Dimitrios M. Thilikos","doi":"https://dl.acm.org/doi/10.1145/3583688","DOIUrl":"https://doi.org/https://dl.acm.org/doi/10.1145/3583688","url":null,"abstract":"<p>For a finite collection of graphs ℱ, the ℱ-<span>TM-Deletion</span> problem has as input an <i>n</i>-vertex graph <i>G</i> and an integer <i>k</i> and asks whether there exists a set <i>S ⊆ V(G)</i> with <i>|S| ≤ k</i> such that <i>G S</i> does not contain any of the graphs in ℱ as a topological minor. We prove that for every such ℱ, ℱ -<span>TM-Deletion</span> is fixed parameter tractable on planar graphs. Our algorithm runs in a 2^{𝒪(<i>k</i>2)} ⋅ <i>n</i>² time, or, alternatively, in 2^{𝒪(<i>k</i>)} ⋅ <i>n</i>⁴ time. Our techniques can easily be extended to graphs that are embeddable on any fixed surface.</p>","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":"7 23","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-05-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138494894","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}