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Abstract:
Let G = (V, E) be a given (directed) graph in which every edge is with a cost and a delay that are nonnegative. The k-disjoint restricted shortest path (kRSP) problem is to compute k (edge) disjoint minimum cost paths between two distinct vertices s, t. is an element of V, such that the total delay of these paths are bounded by a given delay constraint D. R-0(+). This problem is known to be NP-hard, even when k = 1 [4]. Approximation algorithms with bifactor ratio (1+ 1 r, r(1+ 2(log r+ 1) r)(1+ 1/r)) and (1+ 1 r, r(1+ 2(log r+ 1) r)) have been developed for its special case when k = 2 respectively in [11] and [3]. For general k, an approximation algorithm with ratio (1, O(ln n)) has been developed for a weaker version of kRSP, the k bi-constraint path problem of computing k disjoint st-paths to satisfy the given cost constraint and delay constraint simultaneously [7]. n this paper, an approximation algorithm with bifactor ratio (2, 2) is first given for the kRSP problem. Then it is improved such that for any resulted solution, there exists a real number 0 = a = 2 that the delay and the cost of the solution is bounded, respectively, by a times and 2 - alpha times of that of an optimal solution. These two algorithms are both based on rounding a basic optimal solution of a LP formula, which is a relaxation of an integral linear programming (ILP) formula for the kRSP problem. The key observation of the two ratio proofs is to show that, the fractional edges of a basic solution to the LP formula will compose a graph in which the degree of every vertex is exactly 2. To the best of our knowledge, it is the first algorithm with a single factor polylogarithmic ratio for the kRSP problem.
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Source :
FRONTIERS IN ALGORITHMICS, FAW 2014
ISSN: 0302-9743
Year: 2014
Volume: 8497
Page: 94-104
Language: English
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JCR@2005
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