# 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 (*k*RSP) 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 *k*RSP, 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 *k*RSP 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 *k*RSP 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 *k*RSP problem.

## Keywords

LP rounding flow theory*k*-disjoint restricted shortest path problem bifactor approximation algorithm

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