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10
Polynomial time approximation schemes for Euclidean TSP and other geometric problems
 In Proceedings of the 37th IEEE Symposium on Foundations of Computer Science (FOCS’96
, 1996
"... Abstract. We present a polynomial time approximation scheme for Euclidean TSP in fixed dimensions. For every fixed c � 1 and given any n nodes in � 2, a randomized version of the scheme finds a (1 � 1/c)approximation to the optimum traveling salesman tour in O(n(log n) O(c) ) time. When the nodes a ..."
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Cited by 315 (3 self)
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Abstract. We present a polynomial time approximation scheme for Euclidean TSP in fixed dimensions. For every fixed c � 1 and given any n nodes in � 2, a randomized version of the scheme finds a (1 � 1/c)approximation to the optimum traveling salesman tour in O(n(log n) O(c) ) time. When the nodes are in � d, the running time increases to O(n(log n) (O(�dc))d�1). For every fixed c, d the running time is n � poly(log n), that is nearly linear in n. The algorithm can be derandomized, but this increases the running time by a factor O(n d). The previous best approximation algorithm for the problem (due to Christofides) achieves a 3/2approximation in polynomial time. We also give similar approximation schemes for some other NPhard Euclidean problems: Minimum Steiner Tree, kTSP, and kMST. (The running times of the algorithm for kTSP and kMST involve an additional multiplicative factor k.) The previous best approximation algorithms for all these problems achieved a constantfactor approximation. We also give efficient approximation schemes for Euclidean MinCost Matching, a problem that can be solved exactly in polynomial time. All our algorithms also work, with almost no modification, when distance is measured using any geometric norm (such as �p for p � 1 or other Minkowski norms). They also have simple parallel (i.e., NC) implementations.
The Complexity of Pure Nash Equilibria
, 2004
"... We investigate from the computational viewpoint multiplayer games that are guaranteed to have pure Nash equilibria. We focus on congestion games, and show that a pure Nash equilibrium can be computed in polynomial time in the symmetric network case, while the problem is PLScomplete in general. ..."
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Cited by 140 (6 self)
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We investigate from the computational viewpoint multiplayer games that are guaranteed to have pure Nash equilibria. We focus on congestion games, and show that a pure Nash equilibrium can be computed in polynomial time in the symmetric network case, while the problem is PLScomplete in general. We discuss implications to nonatomic congestion games, and we explore the scope of the potential function method for proving existence of pure Nash equilibria.
Network formation games and the potential function method
 Algorithmic Game Theory, chapter 19
, 2007
"... Large computer networks such as the Internet are built, operated, and used by a large number of diverse and competitive entities. In light of these competing forces, it is surprising how efficient these networks are. An exciting challenge in the area of algorithmic game theory is to understand the s ..."
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Cited by 12 (1 self)
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Large computer networks such as the Internet are built, operated, and used by a large number of diverse and competitive entities. In light of these competing forces, it is surprising how efficient these networks are. An exciting challenge in the area of algorithmic game theory is to understand the success of these networks in game theoretic terms: what principles of interaction lead selfish participants to form such efficient networks? In this chapter we present a number of network formation games. We focus on simple games that have been analyzed in terms of the efficiency loss that results from selfishness. We also highlight a fundamental technique used in analyzing inefficiency in many games: the potential function method. The design and operation of many large computer networks, such as the Internet, are carried out by a large number of independent service providers (Autonomous Systems), all of whom seek to selfishly optimize the quality and cost of their own operation. Game theory provides a natural framework for modeling such selfish interests and
Worst case and probabilistic analysis of the 2Opt algorithm for the TSP
 In Proceedings of the 18th ACMSIAM Symposium on Discrete Algorithms (SODA
, 2007
"... 2Opt is probably the most basic local search heuristic for the TSP. This heuristic achieves amazingly good results on “real world ” Euclidean instances both with respect to running time and approximation ratio. There are numerous experimental studies on the performance of 2Opt. However, the theore ..."
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Cited by 10 (5 self)
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2Opt is probably the most basic local search heuristic for the TSP. This heuristic achieves amazingly good results on “real world ” Euclidean instances both with respect to running time and approximation ratio. There are numerous experimental studies on the performance of 2Opt. However, the theoretical knowledge about this heuristic is still very limited. Not even its worst case running time on Euclidean instances was known so far. We clarify this issue by presenting, for every p ∈ N, a family of Lp instances on which 2Opt can take an exponential number of steps. Previous probabilistic analyses were restricted to instances in which n points are placed uniformly at random in the unit square [0, 1] 2, where it was shown that the expected number of steps is bounded by Õ(n10) for Euclidean instances. We consider a more advanced model of probabilistic instances in which the points can be placed according to general distributions on [0, 1] d, for an arbitrary d ≥ 2. In particular, we allow different distributions for different points. We study the expected number of local improvements in terms of the number n of points and the
The npcompleteness column: Finding needles in haystacks
 ACM Transactions on Algorithms
, 2007
"... Abstract. This is the 26th edition of a column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that which M. R. Garey and I used in our book “Computers and Intractability: A Guide to the Theory of NPCompleteness, ” W. H. Freeman & Co., New York, 1979, h ..."
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Cited by 7 (0 self)
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Abstract. This is the 26th edition of a column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that which M. R. Garey and I used in our book “Computers and Intractability: A Guide to the Theory of NPCompleteness, ” W. H. Freeman & Co., New York, 1979, hereinafter referred to as “[G&J]. ” Previous columns, the first 23 of which appeared in J. Algorithms, will be referred to by a combination of their sequence number and year of appearance, e.g., “Column 1 [1981]. ” Full bibliographic details on the previous columns, as well as downloadable unofficial versions of them, can be found at
New Results on the Old kOpt Algorithm for the TSP
 In Proc. 5th ACMSIAM Symposium on Discrete Algorithms
, 1994
"... Local search with kchange neighborhoods is perhaps the oldest and most widely used heuristic method for the traveling salesman problem, yet almost no theoretical performance guarantees for it were previously known. This paper develops several results, some worstcase and some probabilistic, on the ..."
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Cited by 6 (0 self)
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Local search with kchange neighborhoods is perhaps the oldest and most widely used heuristic method for the traveling salesman problem, yet almost no theoretical performance guarantees for it were previously known. This paper develops several results, some worstcase and some probabilistic, on the performance of 2 and kopt local search for the TSP, with respect to both the quality of the solution and the speed with which it is obtained. 1 Introduction Local search with kchange neighborhoods is perhaps the oldest and most widely used heuristic method for the traveling salesman problem [13, 17]. Given a graph G = (V; E) and a tour T of G ("tour" is synonymous with "Hamiltonian cycle"), a tour T 0 is said to be obtained from T by an improving kchange if T 0 is shorter than T , and T 0 is obtained by removing k edges from T and adding k new edges. The kopt algorithm starts with an arbitrary initial tour and incrementally improves on this tour by making successive improving k ...
A Note on Total Functions, Existence Theorems, and Computational Complexity
 Theoretical Computer Science
, 1989
"... . Nondeterministic multivalued functions with values that are polynomially verifiable and guaranteed to exist form an interesting complexity class between P and NP. We show that this class, which we call TFNP, contains a host of important problems, whose membership in P is currently not known. These ..."
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Cited by 5 (0 self)
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. Nondeterministic multivalued functions with values that are polynomially verifiable and guaranteed to exist form an interesting complexity class between P and NP. We show that this class, which we call TFNP, contains a host of important problems, whose membership in P is currently not known. These include, besides factoring, local optimization, Brouwer fixed points, a computational version of Sperner's Lemma, bimatrix equilibria in games, and linear complementarity for P matrices. 1. The class TFNP Let \Sigma be an alphabet with two or more symbols, and suppose that R ` \Sigma \Theta \Sigma is a polynomialtime recognizable relation which is polynomially balanced, that is, (x; y) 2 R implies that jyj p(jxj) for some polynomial p. The relation R defines the following computational problem \Pi R : given an x 2 \Sigma , find any y 2 \Sigma such that (x; y) 2 R, if such a y exists, and reply NO otherwise. The class of all such problems is denoted FNP. The subset of FNP th...
Complexity of Finding Alphabet Indexing
 IEICE Trans. Inf. Sys
, 1995
"... For two finite disjoint sets P and Q of strings over an alphabet 6, an alphabet indexing / for P; Q by an indexing alphabet 0 with j0j ! j6j is a mapping / : 6 ! 0 satisfying ~ /(P ) " ~ /(Q) = ;, where ~ / : 6 3 ! 0 3 is the homomorphism derived from /. We defined this notion through experimen ..."
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Cited by 3 (3 self)
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For two finite disjoint sets P and Q of strings over an alphabet 6, an alphabet indexing / for P; Q by an indexing alphabet 0 with j0j ! j6j is a mapping / : 6 ! 0 satisfying ~ /(P ) " ~ /(Q) = ;, where ~ / : 6 3 ! 0 3 is the homomorphism derived from /. We defined this notion through experiments of knowledge acquisition from amino acid sequences of proteins by learning algorithms. This paper analyzes the complexity of finding an alphabet indexing. We first show that the problem is NPcomplete. Then we give a local search algorithm for this problem and show a result on PLScompleteness. Key words: Algorithm and Computational Complexity, Alphabet Indexing, Local Search Algorithm, NPComplete, PLSComplete 1 Introduction Machine learning methods have been developed in [1, 2] to discover bioinformatical knowledge from amino acid sequences of proteins which are compiled together with their functional information in databases such as PIR [13]. The learning algorithm in [1] uses elemen...
Integer Programming: Optimization and Evaluation are Equivalent
"... Abstract We show that if one can find the optimal value of an integer programming problem min{cx: Ax ≥ b, x ∈ Z n +} in polynomial time, then one can find an optimal solution in polynomial time. We also present a proper generalization to general integer programs and to local search problems of the w ..."
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Abstract We show that if one can find the optimal value of an integer programming problem min{cx: Ax ≥ b, x ∈ Z n +} in polynomial time, then one can find an optimal solution in polynomial time. We also present a proper generalization to general integer programs and to local search problems of the wellknown result that optimization and augmentation are equivalent for 0/1integer programs. Our results imply that, among other things, PLScomplete problems cannot have “nearexact” neighborhoods, unless PLS = P. 1
Random Structure of Error Surfaces: Toward New Stochastic Learning Methods Invited Presentation
"... Learning in neural networks can be formulated as global optimization of a multimodal error function that is ..."
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Learning in neural networks can be formulated as global optimization of a multimodal error function that is