An optimization model for communication networks with partial multiple link failures Michal Pi´ oro †* Dritan Nace ‡ Yoann Fouquet ‡ * Institute of Telecommunications, Warsaw University of Technology, Poland † Department of Electrical and Information Technology, Lund University, Sweden ‡ Laboratoire Heudiasyc UMR CNRS 7253, Universit´ e de Technologie de Compi` egne, France Email: michal.pioro@eit.lth.se, dritan.nace@hds.utc.fr, yoann.fouquet@hds.utc.fr Abstract—This paper studies optimization issues related to a proposed traffic protection strategy referred to as flow-thinning strategy (FTS). FTS is an extension of the well known protection concept, called path diversity, to the case of partial multiple link failures. Path diversity assumes that when a certain subset of links fail, then their capacity is entirely lost and the nominal path-flows that are not affected by the failure are sufficient to realize the required demand. FTS, in turn, is designed to work also when link failures are partial, that is, the failing links in general lose only a fraction of capacity. We consider a link dimensioning problem for FTS, called flow-thinning optimization problem (FTOP), and discuss its properties. It turns out that FTOP, in its general setting, is NP -hard so that its linear programming formulations are unavoidably non-compact and require path generation. As in the general case path generation is also difficult, we propose an integer programming formulation of the pricing problem for the general case of failure scenarios. We also exhibit two special cases when the pricing problem can be solved in polynomial time. Finally, we present a numerical study illustrating cost effectiveness of FTS. The considered partial multi-link failure scenarios are relevant for wireless networks and for the upper layers of fixed communication networks, as for example the MPLS layer with greedy elastic traffic demands. Keywords: survivable network design, multi-link failures, protection strategies, linear and mixed-integer programming, multicommodity flow networks, path generation. I. I NTRODUCTION In this paper we study a network design problem consisting in link dimensioning and routing optimization for a given list of multiple link failure (MPLF) states, under a realistic protection strategy called flow-thinning strategy (FTS). To account for MPLF, we consider, on top of the nominal state, a finite set of failure states S . For each state s ∈S and for each link e ∈E , we specify a fraction 0 ≤ α s e ≤ 1 of the nominal capacity of link e that is assumed to be available in state s. FTS adapts the network path-flows to the varying link availability. More precisely, FTS assumes that the nominal path-flows (i.e., path-flows defined for the nominal state with all links fully available) can adapt to capacity fluctuations only by being thinned, and not increased or restored in any way. On top of this, we assume that the demand volumes to be realized in the failure states are possibly reduced as compared to the nominal demands. For the so specified problem setting, we formulate a non- compact linear programming (LP) problem and describe a way of solving it using path generation. This LP problem, referred to as FTOP (flow-thinning optimization problem), aims at minimizing the cost of link capacities under several constraints: (i) path-flows satisfy traffic demand in the nominal state and all states in S , (ii) path-flows obey the thinning assumption of FTS, and (iii) link loads do not exceed the available link capacities. As the studied problem is NP -hard, the LP formulation is not straightforward to solve and requires path generation to reach optimal solutions. For generating paths, however, we in general need to consider a mixed-integer pricing problem—here the difficulty of FTOP is manifested. The introduced model is original. In fact, not much work has been done in survivable network optimization under the MPLF assumption. To the best of our knowledge, only the so called global rerouting (restoring flows from scratch in surviving capacity) has been studied in this context (see [1] and the references therein)—a strategy quite impractical in telecommunication networks. It happens that multiple failures have so far been considered in survivable network design almost always (i.e., besides for global rerouting) assuming total link failures: α s e ∈{0, 1},s ∈S ,e ∈E . (A set of simultaneously and totally failing links is sometimes referred to as shared-risk link group—SRLG [2].) Although the work done for SRLG is substantial (see [3] and the references therein), this work is, unfortunately, of limited relevance when partial link failures are considered. In fact, SRLG is a special case of our model, and its value for our investigations consists mainly in the fact that the network design problem considered in this paper is NP -hard already for SLRG [4]. Partial capacity failure scenarios are relevant to at least two networking cases of outmost importance. The first case are wireless networks. There, a typical failure state, which in this case should be more appropriately called link availability state, is characterized by a subset of those links that lose a fraction of their nominal capacity (achieved under the nominal state characterized by perfect signal propagation) because of current weather state that directly affects the radio channel condition. The links adapt to the detected channel condition by switching to an adequate modulation and coding scheme (MCS, see [5]) and the current capacity is set by the MCS used. In fact, the optimization model considered in this paper is suitable for wireless broadband networks applying microwave communication [6], [7] or free space optical communication V International Workshop on Reliable Networks Design and Modeling (RNDM 2013) co-located with ICUMT 2013 978-1-4799-1177-6/13/$31.00 ©2013 IEEE 171