Dec 3, 2010 - Another extension of the Local-Ratio theorem to randomized ..... fairness criteria, adapted from the literature: envy-freeness and order ...
Discrete Algorithms Seminar List of Presentations Fall 2010
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August 27, 2010 Title: “One for the Price of Two: a Unified Approach for Approximating Covering Problems” Authors: Reuven Bar-Yehuda Journal or Conference: Algorithmica 27(2): 131–144 (2000) Abstract: We present a simple and unified approach for developing and analyzing approximation algorithms for covering problems. We illustrate this on approximation algorithms for the following problems: Vertex Cover, Set Cover, Feedback Vertex Set, Generalized Steiner Forest and related problems. The main idea can be phrased as follows: iteratively, pay two dollars (at most) to reduce the total optimum by one dollar (at least), so the rate of payment is no more than twice the rate of the optimum reduction. This implies a total payment (i.e., approximation cost) = twice the optimum cost. Our main contribution is based on a formal definition for covering problems, which includes all the above fundamental problems and others. We further extend the Bafna, Berman and Fujito Local-Ratio theorem. This extension eventually yields a short generic rapproximation algorithm which can generate most known approximation algorithms for most covering problems. Another extension of the Local-Ratio theorem to randomized algorithms gives a simple proof of Pitt’s randomized approximation for Vertex Cover. Using this approach, we develop a modified greedy algorithm, which for Vertex Cover, gives an expected performance ratio ≤ 2. Internet: http://www.cs.technion.ac.il/~reuven/PDF/Bar98a.pdf Speaker: Reuven Bar-Yehuda (Technion, Israel)
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September 03, 2010 Title: “A Unified Approach to Approximating Resource Allocation and Scheduling” Authors: Amotz Bar-Noy, Reuven Bar-Yehuda, Ari Freund, Joseph (Seffi) Naor, and Baruch Schieber Journal or Conference: JACM 48(5): 1069–1090 (2001) Abstract: We present a general framework for solving resource allocation and scheduling problems. Given a resource of fixed size, we present algorithms that approximate the maximum throughput or the minimum loss by a constant factor. Our approximation factors apply to many problems, among which are: (i) real-time scheduling of jobs on parallel machines, (ii) bandwidth allocation for sessions between two endpoints, (iii) general caching, (iv) dynamic storage allocation, and (v) bandwidth allocation on optical line and ring topologies. For some of these problems we provide the first constant factor approximation algorithm. Our algorithms are simple and efficient and are based on the local-ratio technique. We note that they can equivalently be interpreted within the primal-dual schema. Internet: http://www.cs.technion.ac.il/~reuven/PDF/BarBar.pdf Speaker: Reuven Bar-Yehuda (Technion, Israel)
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September 24, 2010 Title: “Labeling Schemes with Queries” Authors: Amos Korman and Shay Kutten Journal or Conference: SIROCCO 2007: 109–123 Abstract: Recently, quite a few papers studied methods for representing network properties by assigning informative labels to the vertices of a network. Consulting the labels given to any two vertices u and v for some function f (e.g. ”distance(u, v)”) one can compute the function (e.g. the graph distance between u and v). Some very involved lower bounds for the sizes of the labels were proven. In this paper, we demonstrate that such lower bounds are very sensitive to the number of vertices consulted. That is, we show several almost trivial constructions of such labeling schemes that beat the lower bounds by large margins. The catch is that one needs to consult the labels of three vertices instead of two. We term our generalized model labeling schemes with queries. Additional contributions are several extensions. In particular, we show that it is easy to extend our schemes for tree to work also in the the dynamic scenario. We also demonstrate that the study of the queries model can help in designing a scheme for the traditional model too. Finally, we demonstrate extensions to the non-distributed environment. In particular, we show that one can preprocess a general weighted graph using almost linear space so that flow queries can be answered in almost constant time. Internet: http://iew3.technion.ac.il/~pandit/query.pdf Speaker: Valia Mitsou
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October 01, 2010 Title I: “Known Algorithms on Graphs of Bounded Treewidth are Probably Optimal” Authors I: Daniel Lokshtanov, D´aniel Marx, and Saket Saurabh Journal or Conference I: SODA 2011 Abstract I: We obtain a number of lower bounds on the running time of algorithms solving problems on graphs of bounded treewidth. We prove the results under the Strong Exponential Time Hypothesis of Impagliazzo and Paturi. In particular, assuming that SAT cannot be solved in (2 − )n mO(1) time, we show that for any e > 0; Independent Set cannot be solved in (2−e)tw(G) |V (G)|O(1) time, Dominating Set cannot be solved in (3−e)tw(G) |V (G)|O(1) time, Max Cut cannot be solved in (2−e)tw(G) |V (G)|O(1) time, Odd Cycle Transversal cannot be solved in (3 − e)tw(G) |V (G)|O(1) time, for any q ≥ 3, q-Coloring cannot be solved in (q − e)tw(G) |V (G)|O(1) time, Partition Into Triangles cannot be solved in (2 − e)tw(G) |V (G)|O(1) time. Our lower bounds match the running times for the best known algorithms for the problems, up to the e in the base. Internet I: http://arxiv.org/PS_cache/arxiv/pdf/1007/1007.5450v1.pdf Title II: “Slightly Superexponential Parameterized Problems” Authors II: Daniel Lokshtanov, D´aniel Marx, and Saket Saurabh Journal or Conference II: SODA 2011 Abstract II: A central problem in parameterized algorithms is to obtain algorithms with running time f (k) · nO (1) such that f is as slow growing function of the parameter k as possible. In particular, the first natural goal is to make f (k) single-exponential, that is, ck for some constant c. This has led to the development of parameterized algorithms for various problems where f (k) appearing in their running time is of form 2O(k) . However there are still plenty of problems where the “slightly superexponential” f (k) appearing in the best known running time has remained non single-exponential even after a lot of attempts to bring it down. A natural question to ask is whether the f (k) appearing in the running time of the best-known algorithms is optimal for any of these problems. In this paper, we examine parameterized problems where f (k) is k O(k) = 2O(k log k) in the best known running time and for a number of such problems, we show that the dependence on k in the running time cannot be improved to single exponential. More precisely we prove following tight lower bounds, for three natural problems, arising from three different domains: The pattern matching problem CLOSEST STRING is known to be solvable in time 2O(d log d) · nO(1) and 2O(d log |Σ|) · nO(1) . We show that there is no 2o(d log d) · nO(1) and 2o(d log |Sigma|) · nO(1) time algorithm, unless Exponential Time Hypothesis (ETH) fails. The graph embedding problem DISTORTION, that is, deciding whether a graph G has a metric embedding into the integers with distortion at most d can be done in time 2O(d log d) · nO(1) . We show that there is no 2o(d log d) · nO(1) time algorithm, unless ETH 5
fails. The DISJOINT PATHS problem can be solved in time in time 2O(w log w) · nO(1) on graphs of treewidth at most w. We show that there is no 2o(w log w) · nO(1) time algorithm, unless ETH fails. To obtain our result we first prove the lower bound for variants of basic problems: finding cliques, independent sets, and hitting sets. These artificially constrained variants form a good starting point for proving lower bounds on natural problems without any technical restrictions and could be of independent interest. We believe that many further results of this form can be obtained by using the framework of the current paper. Internet II: http://www.ii.uib.no/~daniello/papers/slightlySuper.pdf Speaker: Michael Lampis
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October 15, 2010 Title: “Fixing a tournament” Authors: Virginia Vassilevska Williams Journal or Conference: AAAI 2010: Abstract: We consider a very natural problem concerned with game manipulation. Let G be a directed graph where the nodes represent players of a game, and an edge from u to v means that u can beat v in the game. (If an edge (u, v) is not present, one cannot match u and v.) Given G and a favorite node A, is it possible to set up the bracket of a balanced single-elimination tournament so that A is guaranteed to win, if matches occur as predicted by G? We show that the problem is NP-complete for general graphs. For the case when G is a tournament graph we give several interesting conditions on the desired winner A for which there exists a balanced single-elimination tournament which A wins, and it can be found in polynomial time. Internet: http://www.eecs.berkeley.edu/~virgi/tournament3.pdf Speaker: Simon Shamoun
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October 22, 2010 Title: “A Continuous, Local Strategy for Constructing a Short Chain of Mobile Robots” Authors: Bastian Degener, Barbara Kempkes, Peter Kling, and Friedhelm Meyer auf der Heide Journal or Conference: SIROCCO 2010: Abstract: We are given an arbitrarily shaped chain of n robots with fixed end points in the plane. We assume that each robot can only see its two neighbors in the chain, which have to be within its viewing range. The goal is to move the robots to the straight line between the end points. Each robot has to base the decision where to move on the relative positions of its neighbors only. Such local strategies considered until now are based on discrete rounds, where a round consists of a movement of each robot. In this paper, we initiate the study of continuous local strategies: The robots may perpetually observe the relative positions of their neighbors, and may perpetually adjust their speed and direction in response to these observations. We assume a speed limit for the robots, that we normalize to one, which corresponds to the viewing range. Our contribution is a continuous, local strategy that needs time O(min n, (OP T + d) log(n)). Here d is the distance between the two stationary end points, and OP T is the time needed by an optimal global strategy. Our strategy has the property that the robot which reaches its destination last always moves with maximum speed. Thus, the same bound as above also holds for the distance travelled. Internet: http://www.springerlink.com/content/1h438276v6332832/ Speaker: Peter Terlecky
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October 29, 2010 Title: “Knapsack Auctions” Authors: Gagan Aggarwal and Jason D. Hartline Journal or Conference: SODA 2006: 1083–1092 Abstract: We consider a game theoretic knapsack problem that has application to auctions for selling advertisements on Internet search engines. Consider n agents each wishing to place an object in the knapsack. Each agent has a private valuation for having their object in the knapsack and each object has a publicly known size. For this setting, we consider the design of auctions in which agents have an incentive to truthfully reveal their private valuations. Following the framework of Goldberg et al. [10], we look to design an auction that obtains a constant fraction of the profit obtainable by a natural optimal pricing algorithm that knows the agents’ valuations and object sizes.We give an auction that obtains a constant factor approximation in the non-trivial special case where the knapsack has unlimited capacity. We then reduce the limited capacity version of the problem to the unlimited capacity version via an approximately efficient auction (i.e., one that maximizes the social welfare). This reduction follows from generalizable principles. Internet: http://portal.acm.org/citation.cfm?id=1109677 Speaker: George Rabanca
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November 05, 2010 Title: “Escaping Offline Searchers and Isoperimetric Theorems” Authors: Peter Brass, Kyue D. Kim, Hyeon-Suk Nab, and Chan-Su Shin Journal or Conference: Computational Geometry Theory and Applications 2008: 119–126 Abstract: Given a set of searchers in the grid, whose search paths are known in advance, can a target that moves at the same speed as the searchers escape detection indefinitely? We study the number of searchers against which the target can still escape. This number is less than n in an n × n grid, since a row of searchers can sweep the allowed region. In an alternating-move-model where at each time searchers first move and then the target n strategy moves, we show that a target can always escape 2 searchers and there is a √ for n2 + 1 searchers to catch the target. This improves a recent bound Ω( n) [A. Dumitrescu, I. Suzuki, P. Zylinski, Offline variants of the “lion and man” problem, in: SoCG 2007, Proc. 23rd Annual Symposium on Computational Geometry, ACM Press, 2007, pp. 102.111] in the simultaneous-move-model where at each time searchers and target moves simultaneously. We also prove similar bounds for the continuous analogue, as well as for searchers and targets moving with different speeds. In the proof, we use new isoperimetric theorems for subsets of the n × n grid and the n × n square, which is of independent interest. Internet: http://linkinghub.elsevier.com/retrieve/pii/S0925772108000722 Speaker: Ivo Vigan
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November 12, 2010 Title: “Manipulating Tournaments in Cup and Round Robin Competitions” Authors: Tyrel Russell and Toby Walsh Journal or Conference: ADT 2009: 26–37 Abstract: In sports competitions, teams can manipulate the result by, for instance, throwing games. We show that we can decide how to manipulate round robin and cup competitions, two of the most popular types of sporting competitions in polynomial time. In addition, we show that finding the minimal number of games that need to be thrown to manipulate the result can also be determined in polynomial time. Finally, we show that there are several different variations of standard cup competitions where manipulation remains polynomial. Internet: http://www.cs.uwaterloo.ca/~tcrussel/documents/adt2009.pdf Speaker: Brian Phelan
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November 19, 2010 Title: “Maximizing Data Gathering Capacity of Wireless Sensor Networks using Mobile Relays” Authors: Fatme El-Moukaddem, Eric Torng, and Guoliang Xing Journal or Conference: MASS 2010 Abstract: Recently, the availability of numerous low-cost robotic units (eg, Packbot, Robomote, and Khepera) has made it possible to massively deploy mobile sensors in a network and use them in a disposable manner. It has been shown that the controlled mobility offered by sensors can be exploited to improve the energy efficiency of a network. In this paper, we study a new problem called max-data mobile relay configuration (MMRC) that finds the positions of a set of mobile sensors, referred to as relays, such that the total amount of data that can be transmitted during the lifetime of a network topology is maximized. Different from previous controlled mobility approaches, we account for several characteristics of existing practical mobile sensing platforms including limited mobility and the high energy consumption of locomotion. The MMRC problem is shown to be surprisingly complex even for a trivial network topology due to the joint consideration of the energy consumption of both wireless communication and mechanical locomotion. We present optimal MMRC algorithms for several important network topologies and a practical distributed implementation. Our extensive simulations based on realistic energy models of existing mobile sensing platforms show that our approach can increase the data gathering capacity by a factor of at least 2. Moreover, our distributed algorithm converges quickly and incurs low messaging overhead. Internet: Speaker: Yosef Alayev
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December 03, 2010 Title: “Decomposition of Multiple Coverings into More Parts” Authors: Greg Aloupis, Jean Cardinal, Sebastien Collette, Stefan Langerman, David Orden, and Pedro Ramos Journal or Conference: SODA 2009: 302–310 Abstract: We prove that for every centrally symmetric convex polygon Q, there exists a constant α such that any αk-fold covering of the plane by translates of Q can be decomposed into k coverings. This improves on a quadratic upper bound proved by Pach and T´oth (SoCG’07). The question is motivated by a sensor network problem, in which a region has to be monitored by sensors with limited battery life. Internet: http://www.siam.org/proceedings/soda/2009/SODA09_034_aloupisg.pdf Speaker: Ben Baumer
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December 10, 2010 Title: “An Approximation Algorithm for the Traveling Tournament Problem” Authors: Ryuhei Miyashiro, Tomomi Matsui, and Shinji Imahori Journal or Conference: Abstract: This paper describes the traveling tournament problem, a well-known benchmark problem in the field of tournament timetabling. We propose a new lower bound for the traveling tournament problem, and construct a randomized approximation algorithm yielding a feasible solution whose approximation ratio is less than 2 + (9/4)/(n − 1), where n is the number of teams. Additionally, we propose a deterministic approximation algorithm with the same approximation ratio using a derandomization technique. For the traveling tournament problem, the proposed algorithms are the first approximation algorithms with a constant approximation ratio, which is less than 2 + 3/4. Internet: http://www.keisu.t.u-tokyo.ac.jp/research/techrep/data/2008/METR08-42.pdf Speaker: Yi Chucai
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December 17, 2010 Title: “Fair Seeding in Knockout Tournaments” Authors: Thuc Vu and Yoav Shoham Journal or Conference: AAMAS 2010: 1579–1580 Abstract: Most of the past work on the seeding of a knockout tournament has focused on maximizing the winning probability of the strongest player (so-called “predictive power”). In contrast, we focus on finding a fair seeding. We consider two alternative fairness criteria, adapted from the literature: envy-freeness and order preservation. For the first criterion we provide a solution for unconstrained tournaments, and provide an impossibility result for balanced tournaments. For the second criterion we have a similar result for unconstrained tournaments, but not for the balanced case. We provide instead a heuristic algorithm which we show through experiments to be efficient and effective. Surprisingly, the criterion becomes provably impossible to achieve when we add a weak condition guarding against the phenomenon of tournament dropout. Internet: http://www.stanford.edu/~thucvu/papers/fairness-journal.pdf Speaker: Ali Assarpour
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