Improved Approximation Algorithms for Computing k Disjoint Paths Subject to Two Constraints

Abstract

For a given graph G with positive integral cost and delay on edges, distinct vertices s and t, cost bound C ∈ Z ⁺ and delay bound D ∈ Z ⁺, the k bi-constraint path (kBCP) problem is to compute k disjoint st-paths subject to C and D. This problem is known NP-hard, even when k = 1 [4]. 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 $(1+\beta,\,\max\{2,\,1+\ln\frac{1}{\beta}\})$ 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 by setting $1+\ln\frac{1}{\beta}=2$ and a factor-(1.567, 1.567) algorithm by setting $1+\beta=1+\ln\frac{1}{\beta}$. Besides, by setting β = 0, an approximation algorithm with ratio (1, O(ln n)), i.e. an algorithm with only a single factor ratio O(ln n) on cost, can be immediately obtained. To the best of our knowledge, this is the first non-trivial approximation algorithm for the kBCP problem that strictly obeys the delay constraint.