SIAM Journal on Computing最新文献

{"title":"Tree-Depth and the Formula Complexity of Subgraph Isomorphism","authors":"Deepanshu Kush, Benjamin Rossman","doi":"10.1137/20m1372925","DOIUrl":"https://doi.org/10.1137/20m1372925","url":null,"abstract":"For a fixed “pattern” graph , the colored -subgraph isomorphism problem (denoted by ) asks, given an -vertex graph and a coloring , whether contains a properly colored copy of . The complexity of this problem is tied to parameterized versions of and , among other questions. An overarching goal is to understand the complexity of , under different computational models, in terms of natural invariants of the pattern graph . In this paper, we establish a close relationship between the formula complexity of and an invariant known as tree-depth (denoted by). is known to be solvable by monotone formulas of size . Our main result is an lower bound for formulas that are monotone or have sublogarithmic depth. This complements a lower bound of Li, Razborov, and Rossman [SIAM J. Comput., 46 (2017), pp. 936–971] relating tree-width and circuit size. As a corollary, it implies a stronger homomorphism preservation theorem for first-order logic on finite structures [B. Rossman, An improved homomorphism preservation theorem from lower bounds in circuit complexity, in 8th Innovations in Theoretical Computer Science Conference, LIPIcs. Leibniz Int. Proc. Inform. 67, Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, Germany, 2017, 27]. The technical core of this result is an lower bound in the special case where is a complete binary tree of height , which we establish using the pathset framework introduced in B. Rossman [SIAM J. Comput., 47 (2018), pp. 1986–2028]. (The lower bound for general patterns follows via a recent excluded-minor characterization of tree-depth [W. Czerwiński, W. Nadara, and M. Pilipczuk, SIAM J. Discrete Math., 35 (2021), pp. 934–947; K. Kawarabayashi and B. Rossman, A polynomial excluded-minor approximation of treedepth, in Proceedings of the 2018 Annual ACM-SIAM Symposium on Discrete Algorithms, 2018, pp. 234–246]. Additional results of this paper extend the pathset framework and improve upon both the best known upper and lower bounds on the average-case formula size of when is a path.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135533409","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":"Improved Truthful Mechanisms for Combinatorial Auctions with Submodular Bidders","authors":"Sepehr Assadi, Sahil Singla","doi":"10.1137/20m1316068","DOIUrl":"https://doi.org/10.1137/20m1316068","url":null,"abstract":"A longstanding open problem in algorithmic mechanism design is to design truthful mechanisms that are computationally efficient and (approximately) maximize welfare in combinatorial auctions with submodular bidders. The first such mechanism was obtained by Dobzinski, Nisan, and Schapira [Proceedings of the 37th Annual ACM Symposium on Theory of Computing, Baltimore, MD, ACM, New York, 2005, pp. 610–618] who gave an -approximation, where is the number of items. This problem has been studied extensively since, culminating in an -approximation mechanism by Dobzinski [Proceedings of the 48th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2016, Cambridge, MA, ACM, New York, 2016, pp. 940–948]. We present a computationally-efficient truthful mechanism with an approximation ratio that improves upon the state-of-the-art by an exponential factor. In particular, our mechanism achieves an -approximation in expectation, uses only demand queries, and has universal truthfulness guarantee. This settles an open question of Dobzinski on whether is the best approximation ratio in this setting in the negative.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"356 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135727871","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 Complexity of Equilibrium Computation in First-Price Auctions","authors":"Aris Filos-Ratsikas, Yiannis Giannakopoulos, Alexandros Hollender, Philip Lazos, Diogo Poças","doi":"10.1137/21m1435823","DOIUrl":"https://doi.org/10.1137/21m1435823","url":null,"abstract":"We consider the problem of computing a (pure) Bayes-Nash equilibrium in the first-price auction with continuous value distributions and discrete bidding space. We prove that when bidders have independent subjective prior beliefs about the value distributions of the other bidders, computing an $varepsilon$-equilibrium of the auction is PPAD-complete, and computing an exact equilibrium is FIXP-complete. We also provide an efficient algorithm for solving a special case of the problem, for a fixed number of bidders and available bids.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"88 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135727872","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":"Hop-Constrained Oblivious Routing","authors":"Mohsen Ghaffari, Bernhard Haeupler, Goran Zuzic","doi":"10.1137/21m1443467","DOIUrl":"https://doi.org/10.1137/21m1443467","url":null,"abstract":"We prove the existence of an oblivious routing scheme that is -competitive in terms of , thus resolving a well-known question in oblivious routing. Concretely, consider an undirected network and a set of packets each with its own source and destination. The objective is to choose a path for each packet, from its source to its destination, so as to minimize , defined as follows: The dilation is the maximum path hop length, and the congestion is the maximum number of paths that include any single edge. The routing scheme obliviously and randomly selects a path for each packet independent of (the existence of) the other packets. Despite this obliviousness, the selected paths have within a factor of the best possible value. More precisely, for any integer hop constraint , this oblivious routing scheme selects paths of length at most and is -competitive in terms of congestion in comparison to the best possible congestion achievable via paths of length at most hops. These paths can be sampled in polynomial time. This result can be viewed as an analogue of the celebrated oblivious routing results of Räcke [Proceedings of the 43rd Annual IEEE Symposium on Foundations of Computer Science, 2002; Proceedings of the 40th Annual ACM Symposium on Theory of Computing, 2008], which are -competitive in terms of congestion but are not competitive in terms of dilation.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136252248","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":"Competitively Chasing Convex Bodies","authors":"Sébastien Bubeck, Yin Tat Lee, Yuanzhi Li, Mark Sellke","doi":"10.1137/20m1312332","DOIUrl":"https://doi.org/10.1137/20m1312332","url":null,"abstract":"Let be a family of sets in some metric space. In the -chasing problem, an online algorithm observes a request sequence of sets in and responds (online) by giving a sequence of points in these sets. The movement cost is the distance between consecutive such points. The competitive ratio is the worst case ratio (over request sequences) between the total movement of the online algorithm and the smallest movement one could have achieved by knowing in advance the request sequence. The family is said to be chaseable if there exists an online algorithm with finite competitive ratio. In 1991, Linial and Friedman conjectured that the family of convex sets in Euclidean space is chaseable. We prove this conjecture.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136180936","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 Lower Bound on the Number of Rounds for Parallel Submodular Function Minimization and Matroid Intersection","authors":"Deeparnab Chakrabarty, Yu Chen, Sanjeev Khanna","doi":"10.1137/22m147685x","DOIUrl":"https://doi.org/10.1137/22m147685x","url":null,"abstract":"Submodular function minimization (SFM) and matroid intersection are fundamental discrete optimization problems with applications in many fields. It is well known that both of these can be solved making queries to a relevant oracle (evaluation oracle for SFM and rank oracle for matroid intersection), where denotes the universe size. However, all known polynomial query algorithms are highly adaptive, requiring at least rounds of querying the oracle. A natural question is whether these can be efficiently solved in a highly parallel manner, namely, with queries using only polylogarithmic rounds of adaptivity. An important step towards understanding the adaptivity needed for efficient parallel SFM was taken recently in the work of Balkanski and Singer who showed that any SFM algorithm making queries necessarily requires rounds. This left open the possibility of efficient SFM algorithms in polylogarithmic rounds. For matroid intersection, even the possibility of a constant round, query algorithm was not hitherto ruled out. In this work, we prove that any, possibly randomized, algorithm for submodular function minimization or matroid intersection making queries requires (Throughout the paper, we use the usual convention of using to denote and using to denote for some unspecified constant ) rounds of adaptivity. In fact, we show a polynomial lower bound on the number of rounds of adaptivity even for algorithms that make at most queries for any constant . Therefore, even though SFM and matroid intersection are efficiently solvable, they are not highly parallelizable in the oracle model.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136252251","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":"Quantum Speedup for Graph Sparsification, Cut Approximation, and Laplacian Solving","authors":"Simon Apers, Ronald de Wolf","doi":"10.1137/21m1391018","DOIUrl":"https://doi.org/10.1137/21m1391018","url":null,"abstract":"SIAM Journal on Computing, Volume 51, Issue 6, Page 1703-1742, December 2022. <br/> Abstract. Graph sparsification underlies a large number of algorithms, ranging from approximation algorithms for cut problems to solvers for linear systems in the graph Laplacian. In its strongest form, “spectral sparsification” reduces the number of edges to near-linear in the number of nodes, while approximately preserving the cut and spectral structure of the graph. In this work we demonstrate a polynomial quantum speedup for spectral sparsification and many of its applications. In particular, we give a quantum algorithm that, given a weighted graph with [math] nodes and [math] edges, outputs a classical description of an [math]-spectral sparsifier in sublinear time [math]. This contrasts with the optimal classical complexity [math]. We also prove that our quantum algorithm is optimal up to polylog-factors. The algorithm builds on a string of existing results on sparsification, graph spanners, quantum algorithms for shortest paths, and efficient constructions for [math]-wise independent random strings. Our algorithm implies a quantum speedup for solving Laplacian systems and for approximating a range of cut problems such as min cut and sparsest cut.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"22 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2022-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138520759","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":"Subexponential Parameterized Algorithms for Planar and Apex-Minor-Free Graphs via Low Treewidth Pattern Covering","authors":"Fedor V. Fomin, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, Saket Saurabh","doi":"10.1137/19m1262504","DOIUrl":"https://doi.org/10.1137/19m1262504","url":null,"abstract":"SIAM Journal on Computing, Volume 51, Issue 6, Page 1866-1930, December 2022. <br/> Abstract. We prove the following theorem. Given a planar graph [math] and an integer [math], it is possible in polynomial time to randomly sample a subset [math] of vertices of [math] with the following properties: [math] induces a subgraph of [math] of treewidth [math], and for every connected subgraph [math] of [math] on at most [math] vertices, the probability that [math] covers the whole vertex set of [math] is at least [math], where [math] is the number of vertices of [math]. Together with standard dynamic programming techniques for graphs of bounded treewidth, this result gives a versatile technique for obtaining (randomized) subexponential-time parameterized algorithms for problems on planar graphs, usually with running time bound [math]. The technique can be applied to problems expressible as searching for a small, connected pattern with a prescribed property in a large host graph; examples of such problems include Directed [math]-Path, Weighted [math]-Path, Vertex Cover Local Search, and Subgraph Isomorphism, among others. Up to this point, it was open whether these problems could be solved in subexponential parameterized time on planar graphs, because they are not amenable to the classic technique of bidimensionality. Furthermore, all our results hold in fact on any class of graphs that exclude a fixed apex graph as a minor, in particular on graphs embeddable in any fixed surface.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"18 4","pages":""},"PeriodicalIF":1.6,"publicationDate":"2022-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138520764","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":"One-Way Functions and (Im)perfect Obfuscation","authors":"Ilan Komargodski, Tal Moran, Moni Naor, Rafael Pass, Alon Rosen, Eylon Yogev","doi":"10.1137/15m1048549","DOIUrl":"https://doi.org/10.1137/15m1048549","url":null,"abstract":"SIAM Journal on Computing, Volume 51, Issue 6, Page 1769-1795, December 2022. <br/> Abstract. A program obfuscator takes a program and outputs a “scrambled” version of it, where the goal is that the obfuscated program will not reveal much about its structure beyond what is apparent from executing it. There are several ways of formalizing this goal. Specifically, in indistinguishability obfuscation, first defined by Barak et al. [Advances in Cryptology - CRYPTO, 2001, Lect. Notes Comput. Sci. 2139, Springer, Berlin, Heidelberg, pp. 1–18], the requirement is that the results of obfuscating any two functionally equivalent programs (circuits) will be computationally indistinguishable. In 2013, a fascinating candidate construction for indistinguishability obfuscation was proposed by Garg et al. [Proceedings of the Symposium on Theory of Computing Conference, STOC, ACM, 2013, pp. 467–476]. This has led to a flurry of discovery of intriguing constructions of primitives and protocols whose existence was not previously known (for instance, fully deniable encryption by Sahai and Waters [Proceedings of the Symposium on Theory of Computing, 2014, STOC, pp. 475–484]). Most of them explicitly rely on additional hardness assumptions, such as one-way functions. Our goal is to get rid of this extra assumption. We cannot argue that indistinguishability obfuscation of all polynomial-time circuits implies the existence of one-way functions, since if [math], then program obfuscation (under the indistinguishability notion) is possible. Instead, the ultimate goal is to argue that if [math] and program obfuscation is possible, then one-way functions exist. Our main result is that if [math] and there is an efficient (even imperfect) indistinguishability obfuscator, then there are one-way functions. In addition, we show that the existence of an indistinguishability obfuscator implies (unconditionally) the existence of SZK-arguments for [math]. This, in turn, provides an alternative version of our main result, based on the assumption of hard-on-the-average [math] problems. To get some of our results we need obfuscators for simple programs such as [math] circuits.","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"8 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2022-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138520758","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":"Relaxed Locally Correctable Codes with Nearly-Linear Block Length and Constant Query Complexity","authors":"Alessandro Chiesa, Tom Gur, Igor Shinkar","doi":"10.1137/20m135515x","DOIUrl":"https://doi.org/10.1137/20m135515x","url":null,"abstract":"SIAM Journal on Computing, Volume 51, Issue 6, Page 1839-1865, December 2022. <br/> Abstract. Locally correctable codes (LCCs) are error correcting codes [math] which admit local algorithms that correct any individual symbol of a corrupted codeword via a minuscule number of queries. For systematic codes, this notion is stronger than that of locally decodable codes (LDCs), where the goal is to only recover individual symbols of the message. One of the central problems in algorithmic coding theory is to construct [math]-query LCCs and LDCs with minimal block length. Alas, state-of-the-art of such codes requires super-polynomial block length to admit [math]-query algorithms for local correction and decoding, despite much attention during the last two decades. The study of relaxed LCCs and LDCs, which allow the correction algorithm to abort (but not err) on a small fraction of the locations, provides a way to circumvent this barrier. This relaxation turned out to allow constant-query correcting and decoding algorithms for codes with polynomial block length. Focusing on local correction, Gur, Ramnarayan, and Rothblum [Proceedings of the 9th Innovations in Theoretical Computer Science Conference, ITCS’18, 2018, pp. 1–27] showed that there exist [math]-query relaxed LCCs that achieve nearly-quartic block length [math], for an arbitrarily small constant [math]. We construct an [math]-query relaxed LCC with nearly-linear block length [math], for an arbitrarily small constant [math]. This significantly narrows the gap between the lower bound which states that there are no [math]-query relaxed LCCs with block length [math]. In particular, our construction matches the parameters achieved by Ben-Sasson et al. [SIAM J. Comput., 36 (2006), pp. 889–974], who constructed relaxed LDCs with the same parameters. This resolves an open problem raised by Gur, Ramnarayan, and Rothblum [Proceedings of the 9th Innovations in Theoretical Computer Science Conference, ITCS’18, 2018, pp. 1–27].","PeriodicalId":49532,"journal":{"name":"SIAM Journal on Computing","volume":"11 1","pages":""},"PeriodicalIF":1.6,"publicationDate":"2022-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138520739","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}