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 boundC and D on cost and delay respectively, the k bi-constraint path (kBCP) 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 + β, max{2, 1 + ln(1/β)}) 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 β = 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 kBCP problem.