Finding disjoint paths with different path‐costs: Complexity and algorithms

Abstract

Consider a network C = ( V , E ) with distinguished vertices s and t , and with k different costs on every edge. We consider the problem of finding k disjoint paths from s to t such that the total cost of the paths is minimized, where thejth edge-cost is associated with thejth path. The problem has several variants: The paths may be vertex-disjoint or arcdisjoint and the network may be directed or undirected. We show that all four versions of the problem are strongly NP-complete even for k = 2. We describe polynomial time heuristics for the problem and a polynomial time algorithm for the acyclic directed case.Q 1992 John Wiley & Sons, Inc.

Finding Two Disjoint Paths in a Network with Normalized α + -MIN-SUM Objective Function

“…The objective of this problem is to find e disjoint paths such that the total length of the paths is minimized, where the f t h edge-length is associated with the f t h path. They showed that all four versions of the G-MIN-SUM e -path problem are strongly NP-complete even for e g h W [10]. In [12], we considered the problem of finding two disjoint paths such that the length of the shorter path is minimized (named the MIN-MIN 2-path problem).…”

On finding disjoint paths in single and dual link cost networks

“…In [8], it was shown that in a network where the relationship between C a and C b on each link is arbitrary, the Min-Sum problem is NPC. A more restricted version of the problem was also studied, where for each and every link, C b ≤ C a .…”

“…The Min-Sum problem in an arbitrary dual-cost network has been proved to be NPC [8]. But the network used in [8] for its NPC proof is not for MSOD (with ordered dual-cost links).…”

“…But the network used in [8] for its NPC proof is not for MSOD (with ordered dual-cost links). In this section, we use a well-known NPC problem, called 3-Satisfiability (3SAT) to show that Min-Min problem in a single cost network is NPC and the MSOD problem is also NPC in an undirected network by extending the graph construction in [8], [21].…”

“…More specifically, denote by C b the additional units of backup bandwidth needed on a link, we have 0 ≤ C b ≤ w because of the possible sharing of the backup bandwidth among this BP and other existing BPs using the same link. In fact, despite of extensive research [6]- [8], one of the open questions is whether the Min-Sum problem is NPComplete (NPC) in undirected networks where each link has two ordered "costs", and the cost used by a BP is always a fraction of the cost used by an AP.…”