A new model for selfish routing

In this work, we introduce and study a new, potentially rich model for selfish routing over non-cooperative networks, as an interesting hybridization of the two prevailing such models, namely the KPmodel [E. Koutsoupias, C.H. Papadimitriou, Worst-case equilibria, in: G. Meinel, S. Tison (Eds.), Proceedings of the 16th Annual Symposium on Theoretical Aspects of Computer Science, in: Lecture Notes in Computer Science, vol. 1563, Springer-Verlag, 1999, pp. 404–413] and the Wmodel [J.G. Wardrop, Some theoretical aspects of road traffic research, Proceedings of the of the Institute of Civil Engineers 1 (Pt. II) (1952) 325–378].     

Walrasian Equilibrium: Hardness,Approximations and Tractable Instances

We study the complexity issues for Walrasian equilibrium in a special case of combinatorial auction, called single-minded auction, in which every participant is interested in only one subset of commodities. Chen et al. (J. Comput. Syst. Sci. 69(4): 675–687, 2004) showed that it is NP-hard to decide the existence of a Walrasian equilibrium for a single-minded auction and proposed a notion of approximate Walrasian equilibrium called relaxed Walrasian equilibrium. We show that every single-minded auction has a relaxed Walrasian equilibrium that satisfies at least two-thirds of the participants, proving a conjecture posed in Chen et al. (J. Comput. Syst. Sci. 69(4): 675–687, 2004). Motivated by practical considerations, we introduce another concept of approximate Walrasian equilibrium called weak Walrasian equilibrium. We show NP-completeness and hardness of approximation results for weak Walrasian equilibria. In search of positive results, we restrict our attention to the tollbooth problem (Guruswami et al. in Proceedings of the Symposium on Discrete Algorithms (SODA), pp. 1164–1173, 2005), where every participant is interested in a single path in some underlying graph. We give a polynomial time algorithm to determine the existence of a Walrasian equilibrium and compute one (if it exists), when the graph is a tree. However, the problem is still NP-hard for general graphs.     

Isomorphic Implication

We study the isomorphic implication problem for Boolean constraints. We show that this is a natural analog of the subgraph isomorphism problem. We prove that, depending on the set of constraints, this problem is in P, or is NP-complete, or is NP-hard, coNP-hard, and in P_‖^NP. We show how to extend the NP-hardness and coNP-hardness to P_‖^NP-hardness for some cases, and conjecture that this can be done in all cases. Supported in part by grants NSF-CCR-0311021 and DFG VO 630/5-1 and VO 630/5-2. An extended abstract of this paper appears in Proceedings of the 30th International Symposium on Mathematical Foundations of Computer Science (MFCS 2005), pp. 119–130, Springer-Verlag Lecture Notes in Computer Science #3618, August 2005. Work of M. Bauland done in part while visiting CASCI’s Laboratory for Complexity at Rochester Institute of Technology. Work of E. Hemaspaandra done in part while on sabbatical at the University of Rochester.     

Online rectangle filling

We study an online geometric problem arising in channel-aware scheduling of wireless networks, which we call the online rectangle filling. We present an online algorithm (with one-lookahead) for this problem with a competitive ratio of 1.848, improving the previously best-known 8/3-competitive algorithm in Arora et al. (2006) [4]. We also prove a lower bound of 1.6358 on the competitive ratio of the problem, improving the previous 1.6 lower bound in [4]. In addition, we give an O(n²)-time optimal algorithm for the offline version of the problem, where n is the size of the input, which improves the O(n³)-time solution in Arora and Choi (2006) [3], Arora et al. (2006) [5]. Our techniques are based on interesting techniques and new observations of the combinatorial structures in the problem.     

Lower Bounds for Kernelizations and Other Preprocessing Procedures

We first present a method to rule out the existence of parameter non-increasing polynomial kernelizations of parameterized problems under the hypothesis P≠NP. This method is applicable, for example, to the problem Sat parameterized by the number of variables of the input formula. Then we obtain further improvements of corresponding results in (Bodlaender et al. in Lecture Notes in Computer Science, vol. 5125, pp. 563–574, Springer, Berlin, 2008; Fortnow and Santhanam in Proceedings of the 40th ACM Symposium on the Theory of Computing (STOC’08), ACM, New York, pp. 133–142, 2008) by refining the central lemma of their proof method, a lemma due to Fortnow and Santhanam. In particular, assuming that the polynomial hierarchy does not collapse to its third level, we show that every parameterized problem with a “linear OR” and with NP-hard underlying classical problem does not have polynomial self-reductions that assign to every instance x with parameter k an instance y with |y|=k ^O(1)⋅|x|^1−ε (here ε is any given real number greater than zero). We give various applications of these results. On the structural side we prove several results clarifying the relationship between the different notions of preprocessing procedures, namely the various notions of kernelizations, self-reductions and compressions.     

Approximation algorithm for coloring of dotted interval graphs

Dotted interval graphs were introduced by Aumann et al. [Y. Aumann, M. Lewenstein, O. Melamud, R. Pinter, Z. Yakhini, Dotted interval graphs and high throughput genotyping, in: ACM-SIAM Symposium on Discrete Algorithms, SODA 2005, pp. 339-348] as a generalization of interval graphs. The problem of coloring these graphs found application in high-throughput genotyping. Jiang [M. Jiang, Approximating minimum coloring and maximum independent set in dotted interval graphs, Information Processing Letters 98 (2006) 29-33] improves the approximation ratio of Aumann et al. [Y. Aumann, M. Lewenstein, O. Melamud, R. Pinter, Z. Yakhini, Dotted interval graphs and high throughput genotyping, in: ACM-SIAM Symposium on Discrete Algorithms, SODA 2005, pp. 339-348]. In this work we improve the approximation ratio of Jiang [M. Jiang, Approximating minimum coloring and maximum independent set in dotted interval graphs, Information Processing Letters 98 (2006) 29-33] and Aumann et al. [Y. Aumann, M. Lewenstein, O. Melamud, R. Pinter, Z. Yakhini, Dotted interval graphs and high throughput genotyping, in: ACM-SIAM Symposium on Discrete Algorithms, SODA 2005, pp. 339-348]. In the exposition we develop a generalization of the problem of finding the maximum number of non-attacking queens on a triangle.     

An improved FPTAS for Restricted Shortest Path

Given a graph with a cost and a delay on each edge, Restricted Shortest Path (RSP) aims to find a min-cost s-t path subject to an end-to-end delay constraint. The problem is NP-hard. In this note we present an FPTAS with an improved running time of O(mn/ε) for acyclic graphs, where m and n denote the number of edges and nodes in the graph. Our algorithm uses a scaling and rounding technique similar to that of Hassin [Math. Oper. Res. 17 (1) (1992) 36-42]. The novelty of our algorithm lies in its “adaptivity”. During each iteration of our algorithm the approximation parameters are fine-tuned according to the quality of the current solution so that the running time is kept low while progress is guaranteed at each iteration. Our result improves those of Hassin [Math. Oper. Res. 17 (1) (1992) 36-42], Phillips [Proc. 25th Annual ACM Symposium on the Theory of Computing, 1993, pp. 776-785], and Raz and Lorenz [Technical Report, 1999].     

Approximately dominating representatives

We propose and investigate from the algorithmic standpoint a novel form of fuzzy query called approximately dominating representatives or ADRs. The ADRs of a multidimensional point set consist of a few points guaranteed to contain an approximate optimum of any monotone Lipschitz continuous combining function of the dimensions. ADRs can be computed by appropriately post-processing Pareto, or “skyline”, queries [Kian-Lee Tan, Pin-Kwang Eng, Beng Chin Ooi, Efficient progressive skyline computation, in: VLDB, 2001, pp. 301–310; Wolf-Tilo Balke, Ulrich Güntzer, Jason Xin Zheng, Efficient distributed skylining for web information systems, in: EDBT, 2004.  [14]]. We show that the problem of minimizing the number of points returned, for a user-specified desired approximation, can be solved in polynomial time in two dimensions; for three and more it is NP-hard but has a polynomial-time logarithmic approximation. Finally, we present a polynomial-time, constant factor approximation algorithm for three dimensions.     

A linear time algorithm for monadic querying of indefinite data over linearly ordered domains

This paper demonstrates the generation of a linear-time query-answering algorithm based on the constructive proof of Higman’s lemma by Murthy and Russell [Proceedings of the 5th IEEE Symposium on Logic in Computer Science, 1990, p. 257–267]. The target problem is linear-time evaluation of a fixed disjunctive monadic query on an indefinite database over linearly ordered domains, first posed by van der Meyden [Proceedings of the 11th ACM SIGACT-SIGMOD-SIGART Symposium on Principles of Database Systems, 1992, p. 331–345]. Van der Meyden showed the existence of a linear-time algorithm, but an explicit construction has, until now, not been reported.     

Improved survivability analysis for SONET SHRs

Recently, a special type of Markov model called parametric state reward Markov model (SRMM/p) [H.C. Cankaya, V.S.S. Nair, in: IEEE Proceedings of GLOBECOM 97, vol. 1, IEEE Computer Soc. Press, Silver Spring, MD, 1997, pp. 252–256] and a set of survivability metrics comprising reliability, availability, and restorability have been proposed for the evaluation of self-healing SONET mesh networks [H.C. Cankaya, V.S.S, Nair, in: Proceedings of ISCIS 'XII, vol. 1, Bogazici University, Bogazici University Press, 1997, pp. 269–276]. The SRMM/p accommodates multiple consecutive link failures and uses topology-free approximation in order to calculate the average performance loss due to a failure. The SRMM/p is equally applicable to the analysis of self-healing SONET rings by considering a ring as a special case of a mesh topology [H.C. Cankaya, V.S.S. Nair, in: IEEE Proceedings of GLOBECOM '98, vol. 4, IEEE Computer Soc. Press, Silver Spring, MD, 1998, pp. 2276–2281]. Further, the topological uniformity and simplicity of rings allow one to include more detailed features of the network in the model so that the analysis will be more accurate. For this purpose, we propose an improved approach to the survivability analysis of self-healing SONET rings which employs a probability-tree based evaluation of the probability of various system states resulting from all possible combinations of node and link failures. The corresponding tree-construction and traversal algorithms are presented. Survivability of rings with distinctive demand patterns are studied with the improved analysis and compared experimentally. One limitation to the model is the high run-time complexity caused mainly by the disparity between transition rates amongst various states in the SRMM/p [H.C. Cankaya, V.S.S. Nair, ACM Comput. Commun. Rev. 28 (4) (1998) 268–277]. In this paper, we also present an approach to circumvent this problem by state aggregation method and compare the results in terms of run-time complexity and accuracy by conducting an experimental study.     

Improved Approximations for Guarding 1.5-Dimensional Terrains

We present a 4-approximation algorithm for the problem of placing the fewest guards on a 1.5D terrain so that every point of the terrain is seen by at least one guard. This improves on the previous best approximation factor of 5 (see King in Proceedings of the 13th Latin American Symposium on Theoretical Informatics, pp. 629–640, 2006). Unlike most of the previous techniques, our method is based on rounding the linear programming relaxation of the corresponding covering problem. Besides the simplicity of the analysis, which mainly relies on decomposing the constraint matrix of the LP into totally balanced matrices, our algorithm, unlike previous work, generalizes to the weighted and partial versions of the basic problem.     

Analysis of Approximation Algorithms for <Emphasis Type="Italic">k</Emphasis>-Set Cover Using Factor-Revealing Linear Programs

We present new combinatorial approximation algorithms for the k-set cover problem. Previous approaches are based on extending the greedy algorithm by efficiently handling small sets. The new algorithms further extend these approaches by utilizing the natural idea of computing large packings of elements into sets of large size. Our results improve the previously best approximation bounds for the k-set cover problem for all values of k≥6. The analysis technique used could be of independent interest; the upper bound on the approximation factor is obtained by bounding the objective value of a factor-revealing linear program. A preliminary version of the results in this paper appeared in Proceedings of the 16th International Symposium on Fundamentals of Computation Theory (FCT ’07), LNCS 4639, Springer, pp. 52–63, 2007. This work was partially supported by the European Union under IST FET Integrated Project 015964 AEOLUS and by the General Secretariat for Research and Technology of the Greek Ministry of Development under programme PENED 2003.     

Edge Pricing of Multicommodity Networks for Selfish Users with Elastic Demands

We examine how to induce selfish heterogeneous users in a multicommodity network to reach an equilibrium that minimizes the social cost. In the absence of centralized coordination, we use the classical method of imposing appropriate taxes (tolls) on the edges of the network. We significantly generalize previous work (Yang and Huang in Transp. Res. Part B 38:1–15, [2004]; Karakostas and Kolliopoulos in Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, pp. 268–276, [2004]; Fleischer et al. in Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, pp. 277–285, [2004]) by allowing user demands to be elastic. In this setting the demand of a user is not fixed a priori but it is a function of the routing cost experienced, a most natural assumption in traffic and data networks. Research supported by MITACS and a NSERC Discovery grant.     

欧氏空间货郎担问题的一个多项式时间近似方案的改进与实现

货郎担问题的实例是给定n个结点和任意一对结点{i,j}之间的距离di,j,要求找出一条封闭的回路,该回路经过每个结点一次且仅一次,并且费用最小,这里的费用是指回路上相邻结点间的距离和.货郎担问题是NP难的组合优化问题,是计算机算法研究的热点之一.在过去几十年中,这一经典问题成为许多重要算法思想的测试平台,并促使一些研究领域的出现,如多面体理论和复杂性理论.欧氏空间上的货郎担问题,结点限制在欧氏空间,距离定义为欧氏距离.即使是这样,欧氏空间上的货郎担问题仍然是NP难的.1996年,Arora提出欧氏空间上货郎担问题的第1个多项式时间近似方案.对其中货郎担问题的算法进行了改进：提出一种新的构造方法,使应用于该算法的“补丁引理”结论由常数6改进到常数3,从而使算法的时间复杂度大幅减少;同时,编程实现了该算法,并对实验结果进行了分析.     

Approximation algorithms for partitioning a rectangle with interior points

LetR be a rectangle and letP be a set of points located insideR. Our problem consists of introducing a set of line segments of least total length to partition the interior ofR into rectangles. Each rectangle in a valid partition must not contain points fromP as interior points. Since this partitioning problem is computationally intractable (NP-hard), we present efficient approximation algorithms for its solution. The solutions generated by our algorithms are guaranteed to be within three times the optimal solution value. Our algorithm also generates solutions within four times the optimal solution value whenR is a rectilinear polygon. Our algorithm can be generalized to generate good approximation solutions for the case whenR is a rectilinear polygon, there are rectilinear polygonal holes, and the sum of the length of the boundaries is not more than the sum of the length of the edges in an optimal solution.A preliminary version of this paper appeared in theProceedings of the Symposium on Computational Geometry, June 1985, pp. 281–287. This research was supported in part by the National Science Foundation under Grant DCR-8503163.     

Improved multi-processor scheduling for?flow time and?energy

Energy usage has been an important concern in recent research on online scheduling. In this paper, we study the tradeoff between flow time and energy (Albers and Fujiwara in ACM Trans. Algorithms 3(4), 2007; Bansal et al. in Proceedings of ACM-SIAM Symposium on Discrete Algorithms, pp. 805–813, 2007b, Bansal et al. in Proceedings of International Colloquium on Automata, Languages and Programming, pp. 409–420, 2008; Lam et al. in Proceedings of European Symposium on Algorithms, pp. 647–659, 2008b) in the multi-processor setting. Our main result is an enhanced analysis of a simple non-migratory online algorithm called CRR (classified round robin) on m≥2 processors, showing that its flow time plus energy is within O(1) times of the optimal non-migratory offline algorithm, when the maximum allowable speed is slightly relaxed. The result still holds even if the comparison is made against the optimal migratory offline algorithm. This improves previous analysis that CRR is O(log P)-competitive where P is the ratio of the maximum job size to the minimum job size.     

An LP rounding algorithm for approximating uncapacitated facility location problem with penalties

In this paper, we consider an interesting variant of the facility location problem called uncapacitated facility location problem with penalties (UFLWP, for short) in which each client can be either assigned to some opened facility or rejected by paying a penalty. Existing approaches [M. Charikar, S. Khuller, D. Mount, G. Narasimhan, Algorithms for facility location problems with outliers, in: Proc. Symposium on Discrete Algorithms, 2001, p. 642] and [K. Jain, M. Mahdian, E. Markakis, A. Saberi, V. Vazirani, Greedy facility location algorithms analyzed using dual fitting with factor-revealing LP, J. ACM 50 (2003) 795] for this variant of facility location problem are all based on primal-dual method. In this paper, we present an efficient linear programming (LP) rounding based approach to show that LP rounding techniques are equally capable of solving this variant of facility location problem. Our algorithm uses a two-phase filtering technique (generalized from Lin and Vitter's [?-approximation with minimum packing constraint violation, in: Proc. 24th Annual ACM Symp. on Theory of Computing, 1992, p. 771]) to identify those to-be-rejected clients and open facilities for the remaining ones. Our approach achieves an approximation ratio of 2+2/e (≈2.736) which is worse than the best approximation ratio of 2 achieved by the much more sophisticated dual fitting technique [K. Jain, M. Mahdian, E. Markakis, A. Saberi, V. Vazirani, Greedy facility location algorithms analyzed using dual fitting with factor-revealing LP, J. ACM 50 (2003) 795], but better than the approximation ratio of 3 achieved by the primal-dual method [M. Charikar, S. Khuller, D. Mount, G. Narasimhan, Algorithms for facility location problems with outliers, in: Proc. Symposium on Discrete Algorithms, 2001, p. 642]. Our algorithm is simple, natural, and can be easily integrated into existing LP rounding based algorithms for facility location problem to deal with outliers.     

Fingerprint Clustering with Bounded Number of Missing Values

The problem of clustering fingerprint vectors with missing values is an interesting problem in Computational Biology that has been proposed in Figueroa et al. (J. Comput. Biol. 11(5):887–901, 2004). In this paper we show some improvements in closing the gaps between the known lower bounds and upper bounds on the approximability of variants of the biological problem. Moreover, we have studied two additional variants of the original problem proposed in Figueroa et al. (Proc. 11th Computing: The Australasian Theory Symposium (CATS), CRPIT, vol. 41, pp. 57–60, 2005). We prove that all such problems are APX-hard even when each fingerprint contains only two unknown positions and we present a greedy algorithm that has constant approximation factors for these variants. Despite the hardness of these restricted versions of the problem, we show that the general clustering problem on an unbounded number of missing values such that they occur for every fixed position of an input vector in at most one fingerprint is polynomial time solvable.     

Hierarchical Sampling from Sketches: Estimating Functions over Data Streams

We present a randomized procedure named Hierarchical Sampling from Sketches (Hss) that can be used for estimating a class of functions over the frequency vector f of update streams of the form . We illustrate this by applying the Hss technique to design nearly space-optimal algorithms for estimating the pth moment of the frequency vector, for real p≥2 and for estimating the entropy of a data stream. Preliminary version of this paper appeared as the following conference publications. “Simpler algorithm for estimating frequency moments of data streams,” Lakshminath Bhuvanagiri, Sumit Ganguly, Deepanjan Kesh and Chandan Saha, Proceedings of the ACM Symposium on Discrete Algorithms, 2006, pp. 708–713 and “Estimating entropy over data streams,” Lakshminath Bhuvanagiri and Sumit Ganguly, Proceedings of the European Symposium on Algorithms, LNCS, vol. 4168, pp. 148–159, Springer, 2006.     

On the complexity of working set selection

The decomposition method is currently one of the major methods for solving the convex quadratic optimization problems being associated with Support Vector Machines (SVM-optimization). A key issue in this approach is the policy for working set selection. We would like to find policies that realize (as well as possible) three goals simultaneously: “(fast) convergence to an optimal solution”, “efficient procedures for working set selection”, and “a high degree of generality” (including typical variants of SVM-optimization as special cases). In this paper, we study a general policy for working set selection that has been proposed in [Nikolas List, Hans Ulrich Simon, A general convergence theorem for the decomposition method, in: Proceedings of the 17th Annual Conference on Computational Learning Theory, 2004, pp. 363–377] and further analyzed in [Nikolas List, Hans Ulrich Simon, General polynomial time decomposition algorithms, in: Proceedings of the 17th Annual Conference on Computational Learning Theory, 2005, pp. 308–322]. It is known that it efficiently approaches feasible solutions with minimum cost for any convex quadratic optimization problem. Here, we investigate its computational complexity when it is used for SVM-optimization. It turns out that, for a variable size of the working set, the general policy poses an NP-hard working set selection problem. But a slight variation of it (sharing the convergence properties with the original policy) can be solved in polynomial time. For working sets of fixed size 2, the situation is even better. In this case, the general policy coincides with the “rate certifying pair approach” (introduced by Hush and Scovel). We show that maximum rate certifying pairs can be found in linear time, which leads to a quite efficient decomposition method with a polynomial convergence rate for SVM-optimization.