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Property Testing and its connection to Learning and Approximation
"... We study the question of determining whether an unknown function has a particular property or is fflfar from any function with that property. A property testing algorithm is given a sample of the value of the function on instances drawn according to some distribution, and possibly may query the fun ..."
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Cited by 498 (68 self)
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the function on instances of its choice. First, we establish some connections between property testing and problems in learning theory. Next, we focus on testing graph properties, and devise algorithms to test whether a graph has properties such as being kcolorable or having a aeclique (clique of density ae
Parameterized Complexity
, 1998
"... the rapidly developing systematic connections between FPT and useful heuristic algorithms  a new and exciting bridge between the theory of computing and computing in practice. The organizers of the seminar strongly believe that knowledge of parameterized complexity techniques and results belongs ..."
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Cited by 1218 (75 self)
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the rapidly developing systematic connections between FPT and useful heuristic algorithms  a new and exciting bridge between the theory of computing and computing in practice. The organizers of the seminar strongly believe that knowledge of parameterized complexity techniques and results belongs into the toolkit of every algorithm designer. The purpose of the seminar was to bring together leading experts from all over the world, and from the diverse areas of computer science that have been attracted to this new framework. The seminar was intended as the rst larger international meeting with a specic focus on parameterized complexity, and it hopefully serves as a driving force in the development of the eld. 1 We had 49 participants from Australia, Canada, India, Israel, New Zealand, USA, and various European countries. During the workshop 25 lectures were given. Moreover, one night session was devoted to open problems and Thursday was basically used for problem discussion
A PTAS for
"... 4> + \Delta: Since L is the average load of a machine, there must be another machine with a load less than the average. Assume without loss of generality, that this is machine 2, i.e., M 2 ! L. Therefore, we have M 1 \Gamma M 2 ! \Delta. Consider a schedule obtained by reassigning all the jobs, ..."
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4> + \Delta: Since L is the average load of a machine, there must be another machine with a load less than the average. Assume without loss of generality, that this is machine 2, i.e., M 2 ! L. Therefore, we have M 1 \Gamma M 2 ! \Delta. Consider a schedule obtained by reassigning all the jobs, assigned to machine 1, except the job j, to machine 2. The contribution of machines 1 and 2 to the objective function in the new schedule is, then, (M 1 \Gamma \Delta) 2 + (M 2 + \Delta) 2 = M 2 1 +M 2 2 + 2\Delta(\Delta \Gamma (M 1 \Gamma M 2 )) M 2 1 +M 2 2 : Therefore, the new schedule has
A PTAS for the Multiple Knapsack Problem
, 1993
"... The Multiple Knapsackproblem (MKP) is a natural and well known generalization of the single knapsack problem and is defined as follows. We are given a set of n items and m bins (knapsacks) such that each item i has a profit p(i) and a size s(i), and each bin j has a capacity c(j). The goal is to fin ..."
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Cited by 112 (2 self)
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The Multiple Knapsackproblem (MKP) is a natural and well known generalization of the single knapsack problem and is defined as follows. We are given a set of n items and m bins (knapsacks) such that each item i has a profit p(i) and a size s(i), and each bin j has a capacity c(j). The goal is to find a subset of items of maximum profit such that they have a feasible packing in the bins. MKP is a special case of the Generalized Assignment problem (GAP) where the profit and the size of an item can vary based on the specific bin that it is assigned to. GAP is APXhard and a 2approximation for it is implicit in the work of Shmoys and Tardos [26], and thus far, this was also the best known approximation for MKP. The main result of this paper is a polynomial time approximation scheme for MKP. Apart from its inherent theoretical interest as a common generalization of the wellstudied knapsack and bin packing problems, it appears to be the strongest special case of GAP that is not APXhard. We substantiate this by showing that slight generalizations of MKP that are very restricted versions of GAP are APXhard. Thus our results help demarcate the boundary at which instances of GAP becomeAPXhard. An interesting and novel aspect of our approach is an approximation preserving reduction from an arbitrary instance of MKP to an instance with O(log n) distinct sizes and profits.
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 399 (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.
PTAS for Minimax Approval Voting
"... Abstract. We consider Approval Voting systems where each voter decides on a subset of candidates he/she approves. We focus on the optimization problem of finding the committee of fixed size k, minimizing the maximal Hamming distance from a vote. In this paper we give a PTAS for this problem and he ..."
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Abstract. We consider Approval Voting systems where each voter decides on a subset of candidates he/she approves. We focus on the optimization problem of finding the committee of fixed size k, minimizing the maximal Hamming distance from a vote. In this paper we give a PTAS for this problem
Towards a Syntactic Characterization of PTAS
 In Proceedings of the 28th ACM Symposium on Theory of Computing
, 1996
"... The class PTAS is defined to consist of all NP optimization problems that permit polynomialtime approximation schemes. This paper explores the possibility that a core of PTAS may be characterized through syntactic classes endowed with restrictions on the structure of the input instances. Recent wor ..."
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Cited by 37 (7 self)
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The class PTAS is defined to consist of all NP optimization problems that permit polynomialtime approximation schemes. This paper explores the possibility that a core of PTAS may be characterized through syntactic classes endowed with restrictions on the structure of the input instances. Recent
How to Rank with Fewer Errors A PTAS for Feedback Arc Set in Tournaments
"... We present the first polynomial time approximation scheme (PTAS) for the minimum feedback arc set problem in tournaments. A weighted generalization gives the first PTAS for Kemeny rank aggregation. The runtime is singly exponential in 1/ǫ, improving on the conference version of this work, which was ..."
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We present the first polynomial time approximation scheme (PTAS) for the minimum feedback arc set problem in tournaments. A weighted generalization gives the first PTAS for Kemeny rank aggregation. The runtime is singly exponential in 1/ǫ, improving on the conference version of this work, which
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