Complexity of strict robust integer minimum cost flow problems: An overview and further results

Highlights

• Robustness minimum cost flow label setting integral flows.

Abstract

We consider a robust version of the integer minimum cost flow problem (IMCF) with uncertainty in the cost function, which is represented by a discrete scenario set. It is known that this problem is NP-hard in general. Besides general graphs, we analyze the problem complexities for acyclic and series-parallel graphs. For both classes we are able to present pseudo-polynomial algorithms when the flow value F and the size of the uncertainty set is fixed. On series-parallel graphs, this was even possible for unbounded F by using the recursive structure to develop a dynamic programming algorithm. For acyclic networks, we transfer the robust flow problem to a robust shortest path problem on a new graph. This graph can be deduced from the original network and the flow value F. Beside the theoretical studies, we also test practical ideas to improve the efficiency of the label setting algorithm, which we use to solve the robust shortest path problem. We improve the running time by using an upper bound and precomputed lower bounds in each vertex, similar to the A*-search.

Introduction

The underlying structure of many real world problems forms a graph, for example supply chain optimization, designing IT networks or evacuation planning. In these problems different commodities, information or persons need to be moved across the graph. These movements are usually summarized as flows. Many of these real world problems can be modeled and solved as a minimum cost flow (MCF) problem. The goal of the MCF problem is to find a flow through a graph with minimal costs. The classic version of the minimum cost flow problem is well studied and can be solved very efficiently (Ahuja, Magnanti, Orlin, 1993, Goldberg, 1990). However, for some problems it is necessary to add additional constraints to the model to make it applicable for real world problems. An important requirement is to restrict the flow to be integral since the moved commodities may be indivisible, for example, for evacuation planning if buses have to be moved through the network (Cova, Johnson, 2003, Hamacher, Tjandra, Hoppe, Tardos, 1994).

The interest in robust optimization has been constantly growing during the last two decades. The seminal papers of Ben-Tal and A. (2000); Ben-Tal, Nemirovski, 1998, Ben-Tal, Nemirovski, 1999 and Bertsimas, Sim, 2003, Bertsimas, Sim, 2004 have built the foundation for the research of robust optimization. The goal of robust optimization is to find mathematical models which are applicable even if the input data is affected by uncertainty. Since uncertainty can affect the problem in many different ways, different concepts have been developed to handle the circumstances in question (Ben-Tal, Bertsimas, Brown, 2010, Ben-Tal, Goryashko, Guslitzer, Nemirovski, 2004, Liebchen, Lübbecke, Möhring, Stiller, 2009). In almost all concepts, one defines a so called uncertainty set which resembles the parameter realizations which have to be taken into account by the robust model. In the classic version of robust optimization, it is required that the solution is feasible for all realizations of the uncertainty set. Further, the value of a solution is defined as the worst case performance guarantee among all realizations from the uncertainty set.

Robust optimization has already been applied in evacuation planning. Different concepts and uncertainty sets have been considered to approach the modeling of evacuation. Do Chung et al. (2011) and Yao et al. (2009) discuss demand uncertainty where the number of people is not known exactly. Demand uncertainty is also assumed in Goerigk and Grün (2014). In this paper, the authors do not consider individual evacuation but an evacuation by public transportation by using buses. Besides demand, uncertainty can also occur in other parameters like travel time or capacity. In Boginski et al. (2009) they consider arc failure in a network, i.e., when the capacity of an arc becomes zero. Uncertain travel times are dealt with in Goerigk et al. (2015) again with focus on bus evacuation. In general, evacuation planning is modeled by the usage of dynamic network flows. It is well known that for even for dynamic network flows, static network flow problem are useful, in particular the MCF (see Ford Jr. and Fulkerson (1958)).

Since MCF problems can be used to define various real world problems and most of these problems are affected by uncertainty, it is not surprising that different robust versions of the MCF problem have already been studied. In Bertsimas et al. (2013), the problem is studied in a variant in which arcs may fail. It is shown that the maximum flow problem, which is closely related to the MCF, is still tractable in this setting. However, if one searches for an adaptive solution, the problem becomes $\mathsf{NP}$-complete. Furthermore, in Bertsimas and Sim (2003), the MCF problem is studied for the case the cost vector is affected by uncertainty. The authors use the bounded uncertainty sets introduced in Bertsimas and Sim (2004) and define a combinatorial, polynomial algorithm for the resulting problem.

In our paper, we study a similar problem. Instead of the bounded uncertainty set, we assume to have a finite uncertainty set. Especially in the context of evacuation planning considering a finite uncertainty set does make sense. For instance, different and finitely many flood scenarios lead to various conditions on the streets and thus, on the underlying mathematical network. Those scenarios, will be use in the following. Moreover, we require the flow to be integral which makes the problem far more challenging from a computational point of view. As mentioned before, especially for evacuation planning, the assumption of integral flows is important. In general, people, cars and buses cannot be separated and must be considered as a unit. Whereas for the continuous minimum cost flow problem (Bertsimas and Sim, 2003) have shown that this problem can be solved in polynomial time, the integer version is known to be NP-hard (see Yu and Yang (1998) and Kouvelis and Yu (1997)). In this paper we focus in the complexity results of the min-max IMCF. We distinguish further between strongly and weakly NP-hardness for different problem variants. Some of those results are known from literature and other we contribute. In particular, we present two pseudopolynomial algorithms for the min-max IMCF on acyclic and series-parallel graphs, respectively. However, there are still some open problems in this field as we will see in the following sections.

The paper is structured as follows. The definitions and notations used throughout the paper are defined in Section 2. In Section 3, we give an overview on already existing complexity results concerning the min-max integer minimum cost flow problem. Moreover, we extend those results by discussing further special cases on the underlying graph structure. Here, a focus is placed on acyclic graphs and how to solve robust integer MCFs on those kind of networks with bounded number of scenarios and a bounded flow value which has to be sent over the network. For series parallel graphs with unbounded number of scenarios, we show that this problem is still strongly NP-hard. Moreover, we develop a pseudopolynomial procedure also for a bounded scenario number but this time we allow the flow value to be unbounded. A combinatorial algorithm for acyclic graphs, whose idea is already discussed in Section 3, is continued and further improved in Section 4. To verify those improvements, we present computational experiments in Section 5. We summarize the paper in Section 6 and pose further research questions.