Given a set of objects O and a set of tests T, the abstract decision tree problem (DTP) is to construct a tree with minimum height that completely identifies the objects of O, by using the tests of T. No algorithm wit...
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Given a set of objects O and a set of tests T, the abstract decision tree problem (DTP) is to construct a tree with minimum height that completely identifies the objects of O, by using the tests of T. No algorithm with a good approximation ratio is known to solve this problem. We give a theoretical support for this fact by showing that DTP does not admit an o(log n)-approximation algorithm unless P = NP. (C) 2004 Elsevier B.V. All rights reserved.
We study the approximability of three versions of the Steiner tree problem. For the first one where the input graph is only supposed connected, we show that it is not approximable within better than \V\N\(-epsilon) fo...
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We study the approximability of three versions of the Steiner tree problem. For the first one where the input graph is only supposed connected, we show that it is not approximable within better than \V\N\(-epsilon) for any epsilon is an element of (0, 1), where V and N are the vertex-set of the input graph and the set of terminal vertices, respectively. For the second of the Steiner tree versions considered, the one where the input graph is supposed complete and the edge distances are arbitrary, we prove that it can be differentially approximated within 1/2. For the third one defined on complete graphs with edge distances 1 or 2, we show that it is differentially approximable within 0.82. Also, extending the result of Bern and Plassmann [1], we show that the Steiner tree problem with edge lengths 1 and 2 is MaxSNP-complete even in the case where \V\ less than or equal to r\N\, for any r > 0. This allows us to finally show that the Steiner tree problem with edge lengths 1 and 2 cannot by approximated by polynomial time differential approximation schemata. (C) 2003 Elsevier Science Ltd. All rights reserved.
We consider the problem MAX CSP over multi-valued domains with variables ranging over sets of size s(i) less than or equal to s and constraints involving k(j) less than or equal to k variables. We study two algorithms...
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We consider the problem MAX CSP over multi-valued domains with variables ranging over sets of size s(i) less than or equal to s and constraints involving k(j) less than or equal to k variables. We study two algorithms with approximation ratios A and B, respectively, so we obtain a solution with approximation ratio max(A, B). The first algorithm is based on the linear programming algorithm of Serna, Trevisan, and Xhafa [Proc. 15th Annual Symp. on Theoret. Aspects of Comput. Sci., 1998, pp. 488-498] and gives ratio A which is bounded below by s(1-k). For k = 2, our bound in terms of the individual set sizes is the minimum over all constraints involving two variables of (1/2roots(1) + 1/2roots(2))(2), where s(1) and s(2) are the set sizes for the two variables. We then give a simple combinatorial algorithm which has approximation ratio B, with B > A/e. The bound is greater than s(1-k)/e in general, and greater than s(1-k)(1-(s-1)/2(k- 1)) for s < k-1, thus close to the s(1-k) linear programming bound for large k. For k = 2, the bound is 4/9 if s = 2, 1/2(s-1) if s greater than or equal to 3, and in general greater than the minimum of 1/4s(1) + 1/4s(2) over constraints with set sizes s(1) and s(2), thus within a factor of two of the linear programming bound. For the case of k = 2 and s = 2 we prove an integrality gap of 4/9(1 + O(n(-1/2))). This shows that our analysis is tight for any method that uses the linear programming upper bound. (C) 2002 Elsevier Science B.V. All rights reserved.
The problem of Weighted Hypergraph Embedding in a Cycle (WHEC) is to embed the weighted hyperedges of a hypergraph as adjacent paths around a cycle, such that the maximum congestion over,any physical link in the cycle...
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The problem of Weighted Hypergraph Embedding in a Cycle (WHEC) is to embed the weighted hyperedges of a hypergraph as adjacent paths around a cycle, such that the maximum congestion over,any physical link in the cycle is minimized. In this paper, we first show that even when hyperedges contain exactly two vertices, the WHEC problem is NP-complete. Afterwards we formulate the problem as an Integer Linear Program (ILP). Then, a solution with approximation ratio of two is presented by using LP-based rounding algorithm. Finally, to improve the efficiency, we develop a linear-time approximation algorithm to provide an embedding with congestion at most two times the optimum. (C) 2003 Elsevier B.V. All rights reserved.
The GENERALIZED MAXIMUM LINEAR ARRANGEMENT PROBLEM is to compute for a given vector x is an element of R-n and an n x n non-negative symmetric matrix W = (w(i,j)), a permutation pi of {1,..., n) that maximizes Sigma (...
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The GENERALIZED MAXIMUM LINEAR ARRANGEMENT PROBLEM is to compute for a given vector x is an element of R-n and an n x n non-negative symmetric matrix W = (w(i,j)), a permutation pi of {1,..., n) that maximizes Sigma (i,j) w(pii,pij) \x(j) - x(i)\. We present a fast 1/3-approximation algorithm for the problem. We present a randomized approximation algorithm with a better performance guarantee for the special case where x(i) = i, i = 1,..., n. Finally, we introduce a 1/2-approximation algorithm for MAX k-CUT WITH GIVEN SIZES OF PARTS. This matches the bound obtained by Ageev and Sviridenko, but without using linear programming. (C) 2001 Elsevier Science B.V. All rights reserved.
Provides information a study which examined the maximum quadratic assignment problem. Objective of the problem; Details on the special cases and related problems; Algorithm on the expected weight returned by quadratic...
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Provides information a study which examined the maximum quadratic assignment problem. Objective of the problem; Details on the special cases and related problems; Algorithm on the expected weight returned by quadratic assignment.
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