{"title":"Between O(nm) and O(nalpha)","authors":"D. Kratsch, J. Spinrad","doi":"10.1137/S0097539704441435","DOIUrl":"https://doi.org/10.1137/S0097539704441435","url":null,"abstract":"This paper uses periodic matrix multiplication to improve the time complexities for a number of graph problems. The time for finding an asteroidal triple is reduced from O(nm) to O(n2.82), and the time for finding a star cutset, a two-pair, and a dominating pair is reduced from O(nm) to O(n2.79). It is also shown that each of these problems is at least as hard as one of three basic graph problems for which the best known algorithms run in times O(nm) and O(nalpha). We note that the fast matrix multiplication algorithms do not seem to be practical because of the enormous constants needed to achieve the asymptotic time bounds. These results are important theoretically for breaking the n3 barrier rather than giving efficient algorithms for a user.","PeriodicalId":160227,"journal":{"name":"SIAM journal on computing (Print)","volume":"74 2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116860035","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Edge Disjoint Paths in Moderately Connected Graphs","authors":"Satish Rao, Shuheng Zhou","doi":"10.1137/080715093","DOIUrl":"https://doi.org/10.1137/080715093","url":null,"abstract":"We study the Edge Disjoint Paths (EDP) problem in undirected graphs: Given a graph G with n nodes and a set ${mathcal T}$ of pairs of terminals, connect as many terminal pairs as possible using paths that are mutually edge disjoint. This leads to a variety of classic NP-complete problems, for which approximability is not well understood. We show a polylogarithmic approximation algorithm for the undirected EDP problem in general graphs with a moderate restriction on graph connectivity; we require the global minimum cut of G to be Ω(log5n). Previously, constant or polylogarithmic approximation algorithms were known for trees with parallel edges, expanders, grids and grid-like graphs, and most recently, even-degree planar graphs. These graphs either have special structure (e.g., they exclude minors) or there are large numbers of short disjoint paths. Our algorithm extends previous techniques in that it applies to graphs with high diameters and asymptotically large minors.","PeriodicalId":160227,"journal":{"name":"SIAM journal on computing (Print)","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128494720","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
S. Aland, Dominic Dumrauf, M. Gairing, B. Monien, F. Schoppmann
{"title":"Exact Price of Anarchy for Polynomial Congestion Games","authors":"S. Aland, Dominic Dumrauf, M. Gairing, B. Monien, F. Schoppmann","doi":"10.1137/090748986","DOIUrl":"https://doi.org/10.1137/090748986","url":null,"abstract":"We show exact values for the price of anarchy of weighted and unweighted congestion games with polynomial latency functions. The given values also hold for weighted and unweighted network congestion games.","PeriodicalId":160227,"journal":{"name":"SIAM journal on computing (Print)","volume":"47 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124945188","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Convergence of Autonomous Mobile Robots with Inaccurate Sensors and Movements","authors":"Reuven Cohen, D. Peleg","doi":"10.1137/060665257","DOIUrl":"https://doi.org/10.1137/060665257","url":null,"abstract":"The common theoretical model adopted in recent studies on algorithms for systems of autonomous mobile robots assumes that the positional input of the robots is obtained by perfectly accurate visual sensors, that robot movements are accurate, and that internal calculations performed by the robots on (real) coordinates are perfectly accurate as well. The current paper concentrates on the effect of weakening this rather strong set of assumptions, and replacing it with the more realistic assumption that the robot sensors, movement and internal calculations may have slight inaccuracies. Specifically, the paper concentrates on the ability of robot systems with inaccurate sensors, movements and calculations to carry out the task of convergence. The paper presents several impossibility results, limiting the inaccuracy allowing convergence. The main positive result is an algorithm for convergence under bounded measurement, movement and calculation errors.","PeriodicalId":160227,"journal":{"name":"SIAM journal on computing (Print)","volume":"52 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2006-02-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131121531","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Parametric Duality and Kernelization: Lower Bounds and Upper Bounds on Kernel Size","authors":"Jianer Chen, H. Fernau, Iyad A. Kanj, Ge Xia","doi":"10.1137/050646354","DOIUrl":"https://doi.org/10.1137/050646354","url":null,"abstract":"We develop new techniques to derive lower bounds on the kernel size for certain parameterized problems. For example, we show that unless $mathcal{P}$=$mathcal{NP}$, planar vertex cover does not have a problem kernel of size smaller than 4k/3, and planar independent set and planar dominating set do not have kernels of size smaller than 2k. We derive an upper bound of 67k on the problem kernel for planar dominating set improving the previous 335k upper bound by Alber et al.","PeriodicalId":160227,"journal":{"name":"SIAM journal on computing (Print)","volume":"112 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2005-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128101118","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On the Complexity of Network Synchronization","authors":"Darin Goldstein, Kojiro Kobayashi","doi":"10.1137/S0097539705447086","DOIUrl":"https://doi.org/10.1137/S0097539705447086","url":null,"abstract":"We show that if a minimal-time solution to a fundamental distributed computation primitive, synchronizing a network path of finite-state processors, exists on the three-dimensional, undirected grid, then we can conclude the purely complexity-theoretic result P=NP. \u0000 \u0000Every previous result on network synchronization for various network topologies either demonstrates the existence of fast synchronization solutions or proves that a synchronization solution cannot exist at all To date, it is unknown whether there is a network topology for which there exists a synchronization solution but for which no minimal-time synchronization solution exists Under the assumption that P ≠ NP, this paper solves this longstanding open problem in the affirmative.","PeriodicalId":160227,"journal":{"name":"SIAM journal on computing (Print)","volume":"37 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130268482","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"O(sqrt(log(n)) Approximation to SPARSEST CUT in Õ(n2) Time","authors":"Sanjeev Arora, Elad Hazan, Satyen Kale","doi":"10.1137/080731049","DOIUrl":"https://doi.org/10.1137/080731049","url":null,"abstract":"We show how to compute 0(sqrt {log n)}-approximations to SPARSEST CUT and BALANCED SEPARATOR problems in ?(n²) time, thus improving upon the recent algorithm of Arora, Rao and Vazirani (2004). Their algorithm uses semidefinite programming and required ?(n^{4.5} ) time. Our algorithm relies on efficiently finding expander flows in the graph and does not solve semidefinite programs. The existence of expander flows was also established by Arora, Rao, and Vazirani.","PeriodicalId":160227,"journal":{"name":"SIAM journal on computing (Print)","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131212533","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Cryptography in NC0","authors":"B. Applebaum, Y. Ishai, E. Kushilevitz","doi":"10.1137/S0097539705446950","DOIUrl":"https://doi.org/10.1137/S0097539705446950","url":null,"abstract":"We study the parallel time-complexity of basic cryptographic primitives such as one-way functions (OWFs) and pseudorandom generators (PRGs). Specifically, we study the possibility of computing instances of these primitives by NC/sup 0/ circuits, in which each output bit depends on a constant number of input bits. Despite previous efforts in this direction, there has been no significant theoretical evidence supporting this possibility, which was posed as an open question in several previous works. We essentially settle this question by providing overwhelming positive evidence for the possibility of cryptography in NC/sup 0/. Our main result is that every \"moderately easy\" OWF (resp., PRG), say computable in NC/sup 1/, can be compiled into a corresponding OWF (resp., low-stretch PRG) in NC/sub 4//sup 0/, i.e. whose output bits each depend on at most 4 input bits. The existence of OWF and PRG in NC/sup 1/ is a relatively mild assumption, implied by most number-theoretic or algebraic intractability assumptions commonly used in cryptography. Hence, the existence of OWF and PRG in NC/sup 0/ follows from a variety of standard assumptions. A similar compiler can also be obtained for other cryptographic primitives such as one-way permutations, encryption, commitment, and collision-resistant flashing. The above results leave a small gap between the possibility of cryptography in NC/sub 4//sup 0/, and the known impossibility of implementing even OWF in NC/sub 2//sup 0/. We partially close this gap by providing evidence for the existence of OWF in NC/sub 3//sup 0/. Finally, our techniques can also be applied to obtain unconditionally provable constructions of non-cryptographic PRGs. In particular, we obtain e-biased generators in NC/sub 3//sup 0/, resolving an open question posed by Mossel et al. (2003), as well as a PRG for logspace in NC/sup 0/. Our results make use of the machinery of randomizing polynomials which was originally motivated by questions in the domain of information-theoretic secure multiparty computation.","PeriodicalId":160227,"journal":{"name":"SIAM journal on computing (Print)","volume":"108 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127966314","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Online Scheduling of Equal-Length Jobs: Randomization and Restarts Help","authors":"M. Chrobak, Wojciech Jawor, J. Sgall, T. Tichý","doi":"10.1137/S0097539704446608","DOIUrl":"https://doi.org/10.1137/S0097539704446608","url":null,"abstract":"The input of the studied scheduling problem is a set of jobs with equal processing times, where each job is specified by its release time and deadline. The goal is to determine a single-processor, non-preemptive schedule that maximizes the number of completed jobs. In the online version, each job arrives at its release time.","PeriodicalId":160227,"journal":{"name":"SIAM journal on computing (Print)","volume":"38 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-07-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127236716","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Complexity of Self-Assembled Shapes","authors":"D. Soloveichik, E. Winfree","doi":"10.1137/S0097539704446712","DOIUrl":"https://doi.org/10.1137/S0097539704446712","url":null,"abstract":"The connection between self-assembly and computation suggests that a shape can be considered the output of a self-assembly “program,” a set of tiles that fit together to create a shape. It seems plausible that the size of the smallest self-assembly program that builds a shape and the shape's descriptional (Kolmogorov) complexity should be related. We show that under the notion of a shape that is independent of scale this is indeed so: in the Tile Assembly Model, the minimal number of distinct tile types necessary to self-assemble an arbitrarily scaled shape can be bounded both above and below in terms of the shape's Kolmogorov complexity. As part of the proof of the main result, we sketch a general method for converting a program outputting a shape as a list of locations into a set of tile types that self-assembles into a scaled up version of that shape. Our result implies, somewhat counter-intuitively, that self-assembly of a scaled up version of a shape often requires fewer tile types, and suggests that the independence of scale in self-assembly theory plays the same crucial role as the independence of running time in the theory of computability.","PeriodicalId":160227,"journal":{"name":"SIAM journal on computing (Print)","volume":"356 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2004-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123109310","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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