Complexity of a classical flow restoration problem

Abstract

In this article, we revisit a classical optimization problem occurring in designing survivable multicommodity flow networks. The problem, referred to as FR, assumes flow restoration that takes advantage of the so-called stub release. As no compact linear programming (LP) formulation of FR is known and at the same time all known noncompact LP formulations of FR exhibit N Phard dual separation, the problem itself is believed to be N P-hard, although without a proof. In this article, we study a restriction of FR (RFR) that assumes only elementary (cycle-free) admissible paths-an important case virtually not considered in the literature. The two problems have the same noncompact LP formulations as they differ only in the definition of admissible paths: all paths (also those including cycles) are allowed in FR, while only elementary paths are allowed in RFR. Because of that, RFR is in general computationally more complex than FR. The purpose of this article, is three-fold. First, the article reveals an interesting special case of RFR-the case with only one failing link-for which a natural noncompact LP formulation obtained by reducing the general RFR formulation still exhibits N P-hard dual separation, but nevertheless this special case of RFR is polynomial. The constructed example of a polynomial multicommodity flow problem with difficult dual separation is of interest since, to our knowledge, no example of this kind has been known. In this article, we also examine a second special case of RFR, this time assuming two failing links instead of one, which turns out to be N P-hard. This implies that problem RFR is N P-hard in general (more precisely, for two or more failure states). This new result is the second contribution of the article. Finally, we discuss the complexity of FR in the light of our new findings, emphasizing the differences between RFR and FR.

Generalized elastic flow rerouting scheme

“…For instance let us consider RR: taking a = 0 implies that flow thinning is not allowed, while b → ∞ means that new rerouting paths can be created in practice; this is because the flow on some paths can be enlarged at any finite value starting from practically insignificant flow values. All these special cases have different levels of complexity for the single link failure case: GR and PD fall into the polynomial time complexity class [2], while RR is shown to be N P-hard for both the directed [9,18] and undirected [20] cases. The observation suggests that both problems will exhibit the same N P complexity.…”

“…Not surprisingly, we will use similar arguments to show the N P-hardness in question. The proof given for EFR is inspired by the RR N P-hardness proof presented in [9]. The proof is based on a specific network constructed to show that finding an RR solution is equivalent to solving the elementary path problem EL-PATH, which consists in answering the question whether there exists an elementary path going through a fixed link in a directed graph.…”

“…We will show that in this case the cost C 0 of an optimal solution of EFR must be greater than or equal to 7 4 ξ . Indeed, it has been shown in [9] that any solution for RR in the considered network is necessarily greater than or equal to 7 4 ξ , which is greater than 61 36 ξ . This holds also for EFR as for a = 0 all EFR solutions are necessarily solutions of RR, so EFR cannot do better than RR.…”

“…Let this problem be denoted by EL-PATH. As discussed in [9], EL-PATH (the decision problem consisting in answering whether there exists an elementary path going through a given link in a directed graph) is N P-hard.…”

An optimization framework for traffic restoration in optical wireless networks with partial link failures

“…Table 1: End-to-end flow restoration/protection strategies. name mechanism assumed failures Unrestricted Restoration [22,20] restoration multiple total and partial Global Rerouting Restricted Restoration [22,17] restoration multiple total End-to-End Recovery with Stub Release [20] Elastic Flow Routing [6] restoration single total Path Diversity [20] protection multiple total Demand-Wise Shared Protection [11] Flow Thinning [23] protection multiple partial A commonly studied strategy is end-to-end flow rerouting utilizing stub release called Restricted Restoration (RR) [22,20,17]. The strategy restores the affected (failed) flows for the duration of the failure, while the unaffected flows are kept unchanged.…”