Improved approximation algorithms for computing $k$ disjoint paths subject to two constraints

Abstract

For a given graph $G$ with distinct vertices $s$ and $t$, nonnegative integral cost and delay on edges, and positive integral bound $C$ and $D$ on cost and delay respectively, the $k$ bi-constraint path ($k$BCP) problem is to compute $k$ disjoint $st$-paths subject to $C$ and $D$. This problem is known to be NP-hard, even when $k=1$ (Garey and Johnson, Computers and Intractability, 1979). This paper first gives a simple approximation algorithm with factor-$(2,2)$, i.e. the algorithm computes a solution with delay and cost bounded by $2*D$ and $2*C$ respectively. Later, a novel improved approximation algorithm with ratio $(1+\beta ,\,\max \{2,\,1+\ln (1/\beta )\})$ is developed by constructing interesting auxiliary graphs and employing the cycle cancellation method. As a consequence, we can obtain a factor-$(1.369,\,2)$ approximation algorithm immediately and a factor-$(1.567,\,1.567)$ algorithm by slightly modifying the algorithm. Besides, when $\beta =0$, the algorithm is shown to be with ratio $(1,\, O(\ln n))$, i.e. it is an algorithm with only a single factor ratio $O(\ln n)$ on cost. To the best of our knowledge, this is the first non-trivial approximation algorithm that strictly obeys the delay constraint for the $k$BCP problem.