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 + β, max{2, 1 + ln 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 1 β = 2 and a factor-(1.567, 1.567) algorithm by setting 1 + β = 1 + ln 1 β . 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.
A closely related problem, the k disjoint bi-constrained path problem (kBCP), targets k disjoint st-paths that satisfy the given cost constraint ) or a single factor ratio of O(ln n) (i.e. bifactor approximation ratio (1, O(ln n))) have been developed for general k in REF , where β > 0 is any fixed positive real number.
Improved Approximation Algorithms for Computing k Disjoint Paths Subject to Two Constraints
ArXiv
ArXiv
Mathematics Computer Science