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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 322 (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 141 (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.
Approximating Geometrical Graphs Via Spanners and Banyans
, 1998
"... The main result of this paper is an improvement of Arora's method to find (1+ ffl) approximations for geometric NPhard problems including the Euclidean Traveling Salesman Problem and the Euclidean Steiner Minimum Tree problems. For fixed dimension d and ffl, our algorithms run in O(N log N) t ..."
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Cited by 61 (0 self)
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The main result of this paper is an improvement of Arora's method to find (1+ ffl) approximations for geometric NPhard problems including the Euclidean Traveling Salesman Problem and the Euclidean Steiner Minimum Tree problems. For fixed dimension d and ffl, our algorithms run in O(N log N) time. An interesting byproduct of our work is the definition and construction of banyans, a generalization of graph spanners. A (1 + ffl)banyan for a set of points A is a set of points A 0 and line segments S with endpoints in A [ A 0 such that a 1 + ffl optimal Steiner Minimum Tree for any subset of A is contained in S. We give a construction for banyans such that the total length of the line segments in S is within a constant factor of the length of the minimum spanning tree of A, and jA 0 j = O(jAj), when ffl and d are fixed. In this abbreviated paper, we only provide proofs of these results in two dimensions. The full paper on WDS's web page (http://www.neci.nj.nec.com/homepages/wds, c...
Approximation schemes for NPhard geometric optimization problems: A survey
 Mathematical Programming
, 2003
"... NPhard geometric optimization problems arise in many disciplines. Perhaps the most famous one is the traveling salesman problem (TSP): given n nodes in ℜ 2 (more generally, in ℜ d), find the minimum length path that visits each node exactly once. If distance is computed using the Euclidean norm (di ..."
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Cited by 38 (2 self)
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NPhard geometric optimization problems arise in many disciplines. Perhaps the most famous one is the traveling salesman problem (TSP): given n nodes in ℜ 2 (more generally, in ℜ d), find the minimum length path that visits each node exactly once. If distance is computed using the Euclidean norm (distance between nodes (x1, y1) and (x2, y2) is ((x1−x2) 2 +(y1−y2) 2) 1/2) then the problem is called Euclidean TSP. More generally the distance could be defined using other norms, such as ℓp norms for any p> 1. All these are subcases of the more general notion of a geometric norm or Minkowski norm. We will refer to the version of the problem with a general geometric norm as geometric TSP. Some other NPhard geometric optimization problems are Minimum Steiner Tree (“Given n points, find the smallest network connecting them,”), kTSP(“Given n points and a number k, find the shortest salesman tour that visits k points”), kMST (“Given n points and a number k, find the shortest tree that contains k points”), vehicle routing, degree restricted minimum
Improved Approximation Schemes for Geometrical Graphs Via Spanners and Banyans
 In 30th ACM Symposium on Theory of Computing (STOC'98
, 1998
"... We give deterministic and randomized algorithms to find a Euclidean traveling salesman tour (TST) of length within (1 + 1=s) times optimal. They run in O(N log N) time and O(N) space for constant dimension and s. These time and space bounds are optimal in an algebraic computation tree model. We can ..."
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Cited by 17 (2 self)
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We give deterministic and randomized algorithms to find a Euclidean traveling salesman tour (TST) of length within (1 + 1=s) times optimal. They run in O(N log N) time and O(N) space for constant dimension and s. These time and space bounds are optimal in an algebraic computation tree model. We can also find a (1 + 1=s) times optimal length 2matching (M2M), edge cover (EC), minimum spanning tree (MST), Steiner minimal tree (SMT), rectilinear ditto (RSMT), and related graphs in the same time bound. This improves recent algorithms of Arora, which had used N(log N) O(s d\Gamma1 ) time in fixed dimension d to produce a (1 + 1=s) times optimal TST (or SMT, RSMT) with success probability 1=2. To verify success, however, Arora could only use a deterministic version of his algorithm that took a factor of N d more time. The increase in running time for our deterministic version depends only on s. Arora's approach can also be extended to produce other (1 + ffl)approximate geometrical grap...
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
Tabu Search on the Geometric Traveling Salesman Problem
, 1994
"... The Traveling Salesman Problem (TSP) is probably the most wellknown problem of its genre: Combinatorial Optimization. New heuristics, built on the concept of local search, i.e. continuously to make transitions from one solution to another in the search for better solutions have shown to be successf ..."
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Cited by 9 (0 self)
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The Traveling Salesman Problem (TSP) is probably the most wellknown problem of its genre: Combinatorial Optimization. New heuristics, built on the concept of local search, i.e. continuously to make transitions from one solution to another in the search for better solutions have shown to be successful, especially for medium to largescale problems (+1000 cities). This thesis presents a new classification of TSPtransitions and evaluates a new tabu search implementation for the geometric TSP. The use of complex TSP transitions in a tabu search context is investigated; among these transitions are the classical LinKernighan transition and a new transition, called the Flower transition. The neighbourhood of the complex transitions is reduced strategically by using computational geometry forming a socalled variable candidate set of neighbouring solutions; the average quality of candidate set solutions is controlled by parameter. A new diversification method based on a notion of solutiond...
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, 197 ..."
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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
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 6 (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...
Property Analysis of Symmetric Travelling Salesman Problem Instances Acquired Through Evolution
 Proceedings of the Fifth Conference on Evolutionary Computation in Combinatorial Optimization, volume 3448 of LNCS
, 2005
"... Abstract. We show how an evolutionary algorithm can successfully be used to evolve a set of difficult to solve symmetric travelling salesman problem instances for two variants of the LinKernighan algorithm. Then we analyse the instances in those sets to guide us towards deferring general knowledge ..."
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Cited by 4 (1 self)
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Abstract. We show how an evolutionary algorithm can successfully be used to evolve a set of difficult to solve symmetric travelling salesman problem instances for two variants of the LinKernighan algorithm. Then we analyse the instances in those sets to guide us towards deferring general knowledge about the efficiency of the two variants in relation to structural properties of the symmetric travelling salesman problem. 1