A dual algorithm for submodular flow problems

The submodular flow (SF) problem, introduced by Edmonds and Giles [6], is a general network flow problem, where net flows into subsets of nodes are restricted by submodular set constraints. Suppose that we are given (i) a directed graph G = (V, E) with node set V and arc set E, (ii) a submodular function b" on a crossing family ,~-", (iii) lower and upper capacity functions f a n d g where f : E ~ ( R U ~ ) , g : E (R U ~) and f < g, and (iv) a cost function c :E J R . For a vector x ~ R E, denote px(A) = Y:(x(e): e ~ E , e enters A c V ) and 6x(A)= px(X). Let the difference p x ( A ) ~ ( A ) be denoted by Ax(A). A polyhedron Q(f, g; b") consisting of vectors x ~ RE satisfying f < x < g and Ax(A) < b"(A) for every A c_ V is called a submodular flow polyhedron. The SF problem is to minimize cx subject to x ~ Q( f , g; b"). The generality of submodular functions provides the SF problem with a wide range of applicability. This problem includes, among a host of its special cases, the minimum cost flow (MCF) problem, the independent flow problem, the polymatroidal network flow problem, and the directed cutting packing problem (see [6,10]). As for solution methods, negative circuit [15], simplex [2], primal-dual [4] and out-of-kilter [11] methods have been reported. Each of these algorithms is closely related to the corresponding algorithm for the MCF problem. Dual algorithms for the MCF problem are recognized to be computationally efficient with a certain type of data structures and more suitable for sensitivity analysis of key parameters in various network models [1,3,13]. In this paper, we propose a dual algorithm for the SF problem which uses only the so-called node potentials. By incorporating the three search algorithm by Hassin [13] for the MCF problem, we suggest a systematic way to find the steepest ascent direction of the dual objective function, referred to as the 'best improving set'. This dual-ascent process keeps all arc flows within their capacity bounds, while allowing the set constraints to be violated. The next section defines the SF problem and describes the dual optimality conditions due to Frank and Tardos [9]. Section 3 introduces the concept of improving set. Section 4 describes our