ACM Transactions on Algorithms最新文献

{"title":"Flow-augmentation II: Undirected graphs","authors":"Eun Jung Kim, Stefan Kratsch, Marcin Pilipczuk, Magnus Wahlström","doi":"10.1145/3641105","DOIUrl":"https://doi.org/10.1145/3641105","url":null,"abstract":"<p>We present an undirected version of the recently introduced <i>flow-augmentation</i> technique: Given an undirected multigraph <i>G</i> with distinguished vertices <i>s</i>, <i>t</i> ∈ <i>V</i>(<i>G</i>) and an integer <i>k</i>, one can in randomized (k^{mathcal {O}(1)} cdot (|V(G)| + |E(G)|) ) time sample a set (A subseteq binom{V(G)}{2} ) such that the following holds: for every inclusion-wise minimal <i>st</i>-cut <i>Z</i> in <i>G</i> of cardinality at most <i>k</i>, <i>Z</i> becomes a <i>minimum-cardinality</i> cut between <i>s</i> and <i>t</i> in <i>G</i> + <i>A</i> (i.e., in the multigraph <i>G</i> with all edges of <i>A</i> added) with probability (2^{-mathcal {O}(k log k)} ). </p><p>Compared to the version for directed graphs [STOC 2022], the version presented here has improved success probability ((2^{-mathcal {O}(k log k)} ) instead of (2^{-mathcal {O}(k^4 log k)} )), linear dependency on the graph size in the running time bound, and an arguably simpler proof. </p><p>An immediate corollary is that the <span>Bi-objective <i>st</i>-Cut</span> problem can be solved in randomized FPT time (2^{mathcal {O}(k log k)} (|V(G)|+|E(G)|) ) on undirected graphs.</p>","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":"208 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-01-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139510189","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":"Counting list homomorphisms from graphs of bounded treewidth: tight complexity bounds","authors":"Jacob Focke, Dániel Marx, Paweł Rzążewski","doi":"10.1145/3640814","DOIUrl":"https://doi.org/10.1145/3640814","url":null,"abstract":"<p>The goal of this work is to give precise bounds on the counting complexity of a family of generalized coloring problems (list homomorphisms) on bounded-treewidth graphs. Given graphs <i>G</i>, <i>H</i>, and lists <i>L</i>(<i>v</i>)⊆<i>V</i>(<i>H</i>) for every <i>v</i> ∈ <i>V</i>(<i>G</i>), a <i>list homomorphism</i> is a function <i>f</i>: <i>V</i>(<i>G</i>) → <i>V</i>(<i>H</i>) that preserves the edges (i.e., <i>uv</i> ∈ <i>E</i>(<i>G</i>) implies <i>f</i>(<i>u</i>)<i>f</i>(<i>v</i>) ∈ <i>E</i>(<i>H</i>)) and respects the lists (i.e., <i>f</i>(<i>v</i>) ∈ <i>L</i>(<i>v</i>)). Standard techniques show that if <i>G</i> is given with a tree decomposition of width <i>t</i>, then the number of list homomorphisms can be counted in time (|V(H)|^tcdot n^{mathcal {O}(1)} ). Our main result is determining, for every fixed graph <i>H</i>, how much the base |<i>V</i>(<i>H</i>)| in the running time can be improved. For a connected graph <i>H</i> we define (operatorname{irr}(H) ) in the following way: if <i>H</i> has a loop or is nonbipartite, then (operatorname{irr}(H) ) is the maximum size of a set <i>S</i>⊆<i>V</i>(<i>H</i>) where any two vertices have different neighborhoods; if <i>H</i> is bipartite, then (operatorname{irr}(H) ) is the maximum size of such a set that is fully in one of the bipartition classes. For disconnected <i>H</i>, we define (operatorname{irr}(H) ) as the maximum of (operatorname{irr}(C) ) over every connected component <i>C</i> of <i>H</i>. It follows from earlier results that if (operatorname{irr}(H)=1 ), then the problem of counting list homomorphisms to <i>H</i> is polynomial-time solvable, and otherwise it is #P-hard. We show that, for every fixed graph <i>H</i>, the number of list homomorphisms from (<i>G</i>, <i>L</i>) to <i>H</i><p><table border=\"0\" list-type=\"bullet\" width=\"95%\"><tr><td valign=\"top\"><p>•</p></td><td colspan=\"5\" valign=\"top\"><p>can be counted in time (operatorname{irr}(H)^tcdot n^{mathcal {O}(1)} ) if a tree decomposition of <i>G</i> having width at most <i>t</i> is given in the input, and</p></td></tr><tr><td valign=\"top\"><p>•</p></td><td colspan=\"5\" valign=\"top\"><p>given that (operatorname{irr}(H)ge 2 ), cannot be counted in time ((operatorname{irr}(H)-epsilon)^tcdot n^{mathcal {O}(1)} ) for any ϵ &gt; 0, even if a tree decomposition of <i>G</i> having width at most <i>t</i> is given in the input, unless the Counting Strong Exponential-Time Hypothesis (#SETH) fails.</p></td></tr></table></p>\u0000Thereby we give a precise and complete complexity classification featuring matching upper and lower bounds for all target graphs with or without loops.</p>","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":"1 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-01-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139475707","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":"Popular Matchings with One-Sided Bias","authors":"Telikepalli Kavitha","doi":"10.1145/3638764","DOIUrl":"https://doi.org/10.1145/3638764","url":null,"abstract":"<p>Let <i>G</i> = (<i>A</i>∪<i>B</i>, <i>E</i>) be a bipartite graph where the set <i>A</i> consists of agents or main players and the set <i>B</i> consists of jobs or secondary players. Every vertex in <i>A</i>∪<i>B</i> has a strict ranking of its neighbors. A matching <i>M</i> is <i>popular</i> if for any matching <i>N</i>, the number of vertices that prefer <i>M</i> to <i>N</i> is at least the number that prefer <i>N</i> to <i>M</i>. Popular matchings always exist in <i>G</i> since every stable matching is popular. A matching <i>M</i> is <i><i>A</i>-popular</i> if for any matching <i>N</i>, the number of <i>agents</i> (i.e., vertices in <i>A</i>) that prefer <i>M</i> to <i>N</i> is at least the number of agents that prefer <i>N</i> to <i>M</i>. Unlike popular matchings, <i>A</i>-popular matchings need not exist in a given instance <i>G</i> and there is a simple linear time algorithm to decide if <i>G</i> admits an <i>A</i>-popular matching and compute one, if so. </p><p>We consider the problem of deciding if <i>G</i> admits a matching that is both popular and <i>A</i>-popular and finding one, if so. We call such matchings <i>fully popular</i>. A fully popular matching is useful when <i>A</i> is the more important side—so along with overall popularity, we would like to maintain “popularity within the set <i>A</i>”. A fully popular matching is not necessarily a min-size/max-size popular matching and all known polynomial-time algorithms for popular matching problems compute either min-size or max-size popular matchings. Here we show a linear time algorithm for the fully popular matching problem, thus our result shows a new tractable subclass of popular matchings.</p>","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":"462 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-12-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139054629","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":"Optimal Inapproximability with Universal Factor Graphs","authors":"Per Austrin, Jonah Brown-Cohen, Johan Håstad","doi":"10.1145/3631119","DOIUrl":"https://doi.org/10.1145/3631119","url":null,"abstract":"<p>The factor graph of an instance of a constraint satisfaction problem (CSP) is the bipartite graph indicating which variables appear in each constraint. An instance of the CSP is given by the factor graph together with a list of which predicate is applied for each constraint. We establish that many Max-CSPs remain as hard to approximate as in the general case even when the factor graph is fixed (depending only on the size of the instance) and known in advance. </p><p>Examples of results obtained for this restricted setting are: <p><table border=\"0\" list-type=\"ordered\" width=\"95%\"><tr><td valign=\"top\"><p>(1)</p></td><td colspan=\"5\" valign=\"top\"><p>Optimal inapproximability for Max-3-Lin and Max-3-Sat (Håstad, J. ACM 2001).</p></td></tr><tr><td valign=\"top\"><p>(2)</p></td><td colspan=\"5\" valign=\"top\"><p>Approximation resistance for predicates supporting pairwise independent subgroups (Chan, J. ACM 2016).</p></td></tr><tr><td valign=\"top\"><p>(3)</p></td><td colspan=\"5\" valign=\"top\"><p>Hardness of the “(2 + ϵ)-Sat” problem and other Promise CSPs (Austrin et al., SIAM J. Comput. 2017).</p></td></tr></table></p>\u0000The main technical tool used to establish these results is a new way of folding the long code which we call “functional folding”.</p>","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":"1 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138689588","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":"On the External Validity of Average-Case Analyses of Graph Algorithms","authors":"Thomas Bläsius, Philipp Fischbeck","doi":"10.1145/3633778","DOIUrl":"https://doi.org/10.1145/3633778","url":null,"abstract":"<p>The number one criticism of average-case analysis is that we do not actually know the probability distribution of real-world inputs. Thus, analyzing an algorithm on some random model has no implications for practical performance. At its core, this criticism doubts the existence of <i>external validity</i>, i.e., it assumes that algorithmic behavior on the somewhat simple and clean models does not translate beyond the models to practical performance real-world input. </p><p>With this paper, we provide a first step towards studying the question of external validity systematically. To this end, we evaluate the performance of six graph algorithms on a collection of 2740 sparse real-world networks depending on two properties; the heterogeneity (variance in the degree distribution) and locality (tendency of edges to connect vertices that are already close). We compare this with the performance on generated networks with varying locality and heterogeneity. We find that the performance in the idealized setting of network models translates surprisingly well to real-world networks. Moreover, heterogeneity and locality appear to be the core properties impacting the performance of many graph algorithms.</p>","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":"6 20","pages":""},"PeriodicalIF":1.3,"publicationDate":"2023-11-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138494910","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":"Approximating Sparsest Cut in Low-Treewidth Graphs via Combinatorial Diameter","authors":"Parinya Chalermsook, Matthias Kaul, Matthias Mnich, Joachim Spoerhase, Sumedha Uniyal, Daniel Vaz","doi":"10.1145/3632623","DOIUrl":"https://doi.org/10.1145/3632623","url":null,"abstract":"The fundamental Sparsest Cut problem takes as input a graph G together with edge capacities and demands, and seeks a cut that minimizes the ratio between the capacities and demands across the cuts. For n -vertex graphs G of treewidth k , Chlamtáč, Krauthgamer, and Raghavendra (APPROX 2010) presented an algorithm that yields a factor- (2^{2^k} ) approximation in time 2 O ( k ) · n O (1) . Later, Gupta, Talwar and Witmer (STOC 2013) showed how to obtain a 2-approximation algorithm with a blown-up run time of n O ( k ) . An intriguing open question is whether one can simultaneously achieve the best out of the aforementioned results, that is, a factor-2 approximation in time 2 O ( k ) · n O (1) . In this paper, we make significant progress towards this goal, via the following results: (i) A factor- O ( k 2 ) approximation that runs in time 2 O ( k ) · n O (1) , directly improving the work of Chlamtáč et al. while keeping the run time single-exponential in k . (ii) For any ε ∈ (0, 1], a factor- O (1/ε 2 ) approximation whose run time is (2^{O(k^{1+varepsilon }/varepsilon)} cdot n^{O(1)} ) , implying a constant-factor approximation whose run time is nearly single-exponential in k and a factor- O (log 2 k ) approximation in time k O ( k ) · n O (1) . Key to these results is a new measure of a tree decomposition that we call combinatorial diameter , which may be of independent interest.","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":"5 19","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136229841","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":"Shortest Cycles With Monotone Submodular Costs","authors":"Fedor V. Fomin, Petr A. Golovach, Tuukka Korhonen, Daniel Lokshtanov, Giannos Stamoulis","doi":"10.1145/3626824","DOIUrl":"https://doi.org/10.1145/3626824","url":null,"abstract":"We introduce the following submodular generalization of the Shortest Cycle problem. For a nonnegative monotone submodular cost function f defined on the edges (or the vertices) of an undirected graph G , we seek for a cycle C in G of minimum cost 𝖮𝖯𝖳 = f(C) . We give an algorithm that given an n -vertex graph G , parameter ɛ &gt; 0, and the function f represented by an oracle, in time n 𝒪 (log 1/ɛ) finds a cycle C in G with f(C) ≤ (1+ɛ). 𝖮𝖯𝖳. This is in sharp contrast with the non-approximability of the closely related Monotone Submodular Shortest ( s,t -Path problem, which requires exponentially many queries to the oracle for finding an n 2/3-ɛ -approximation Goel et al. [ 7 ], FOCS 2009. We complement our algorithm with a matching lower bound. We show that for every ɛ &gt; 0, obtaining a (1+ɛ)-approximation requires at least n Ω (log 1/ ɛ) queries to the oracle. When the function f is integer-valued, our algorithm yields that a cycle of cost 𝖮𝖯𝖳 can be found in time n 𝒪(log 𝖮𝖯𝖳) . In particular, for 𝖮𝖯𝖳 = n 𝒪(1) this gives a quasipolynomial-time algorithm computing a cycle of minimum submodular cost. Interestingly, while a quasipolynomial-time algorithm often serves as a good indication that a polynomial time complexity could be achieved, we show a lower bound that n 𝒪(log n ) queries are required even when 𝖮𝖯𝖳= 𝒪( n ). We also consider special cases of monotone submodular functions, corresponding to the number of different color classes needed to cover a cycle in an edge-colored multigraph G . For special cases of the corresponding minimization problem, we obtain fixed-parameter tractable algorithms and polynomial-time algorithms, when restricted to certain classes of inputs.","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":"1 9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134992700","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":"Additive Sparsification of CSPs","authors":"Eden Pelleg, Stanislav Živný","doi":"10.1145/3625824","DOIUrl":"https://doi.org/10.1145/3625824","url":null,"abstract":"Multiplicative cut sparsifiers, introduced by Benczúr and Karger [STOC’96], have proved extremely influential and found various applications. Precise characterisations were established for sparsifiability of graphs with other 2-variable predicates on Boolean domains by Filtser and Krauthgamer [SIDMA’17] and non-Boolean domains by Butti and Živný [SIDMA’20]. Bansal, Svensson and Trevisan [FOCS’19] introduced a weaker notion of sparsification termed “additive sparsification”, which does not require weights on the edges of the graph. In particular, Bansal et al. designed algorithms for additive sparsifiers for cuts in graphs and hypergraphs. As our main result, we establish that all Boolean Constraint Satisfaction Problems (CSPs) admit an additive sparsifier; that is, for every Boolean predicate P :{ 0,1} k → { 0,1} of a fixed arity k , we show that CSP( P ) admits an additive sparsifier. Under our newly introduced notion of all-but-one sparsification for non-Boolean predicates, we show that CSP( P ) admits an additive sparsifier for any predicate P : D k → { 0,1} of a fixed arity k on an arbitrary finite domain D .","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":"9 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134993016","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":"Fast and perfect sampling of subgraphs and polymer systems","authors":"Antonio Blanca, Sarah Cannon, Will Perkins","doi":"10.1145/3632294","DOIUrl":"https://doi.org/10.1145/3632294","url":null,"abstract":"We give an efficient perfect sampling algorithm for weighted, connected induced subgraphs (or graphlets ) of rooted, bounded degree graphs. Our algorithm utilizes a vertex-percolation process with a carefully chosen rejection filter and works under a percolation subcriticality condition. We show that this condition is optimal in the sense that the task of (approximately) sampling weighted rooted graphlets becomes impossible in finite expected time for infinite graphs and intractable for finite graphs when the condition does not hold. We apply our sampling algorithm as a subroutine to give near linear-time perfect sampling algorithms for polymer models and weighted non-rooted graphlets in finite graphs, two widely studied yet very different problems. This new perfect sampling algorithm for polymer models gives improved sampling algorithms for spin systems at low temperatures on expander graphs and unbalanced bipartite graphs, among other applications.","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":"116 36","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135137595","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":"Fast sampling via spectral independence beyond bounded-degree graphs","authors":"Ivona Bezáková, Andreas Galanis, Leslie Ann Goldberg, Daniel Štefankovič","doi":"10.1145/3631354","DOIUrl":"https://doi.org/10.1145/3631354","url":null,"abstract":"Spectral independence is a recently-developed framework for obtaining sharp bounds on the convergence time of the classical Glauber dynamics. This new framework has yielded optimal O ( n log n ) sampling algorithms on bounded-degree graphs for a large class of problems throughout the so-called uniqueness regime, including, for example, the problems of sampling independent sets, matchings, and Ising-model configurations. Our main contribution is to relax the bounded-degree assumption that has so far been important in establishing and applying spectral independence. Previous methods for avoiding degree bounds rely on using L p -norms to analyse contraction on graphs with bounded connective constant (Sinclair, Srivastava, Yin; FOCS’13). The non-linearity of L p -norms is an obstacle to applying these results to bound spectral independence. Our solution is to capture the L p -analysis recursively by amortising over the subtrees of the recurrence used to analyse contraction. Our method generalises previous analyses that applied only to bounded-degree graphs. As a main application of our techniques, we consider the random graph G ( n , d / n ), where the previously known algorithms run in time n O (log d ) or applied only to large d . We refine these algorithmic bounds significantly, and develop fast nearly linear algorithms based on Glauber dynamics that apply to all constant d , throughout the uniqueness regime.","PeriodicalId":50922,"journal":{"name":"ACM Transactions on Algorithms","volume":" 83","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135191540","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}