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

الأصل · EN

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 garey1979computers. 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,{2,1+1β}) 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+1β=2 and a factor-(1.567,1.567) algorithm by setting 1+β=1+1β. Besides, by setting β=0, an approximation algorithm with ratio (1, O(n)), i.e. an algorithm with only a single factor ratio O(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.