Improved LP-rounding Approximations for the k-Disjoint Restricted Shortest Paths Problem

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 ∈ V, such that the total delay of these paths are bounded by a given delay constraint $D\in\mathbb{R}_{0}^{+}$. This problem is known to be NP-hard, even when k = 1 [4]. Approximation algorithms with bifactor ratio $(1+\frac{1}{r},\, r(1+\frac{2(\log r+1)}{r})(1+\epsilon))$ and $(1+\frac{1}{r},\, r(1+\frac{2(\log r+1)}{r}))$ 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].

In 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 ≤ α ≤ 2 that the delay and the cost of the solution is bounded, respectively, by α times and 2 − α 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.

This research was supported by Natural Science Foundation of China (#61300025), Doctoral Funds of Ministry of Education of China for Young Scholars (#20123514120013), Natural Science Foundation of Fujian Province (#2012J05115), and Fuzhou University Development Fund (2012-XQ-26).