Submodular Flows Research Articles

Erratum to: Arc Connectivity and Submodular Flows in Digraphs

Erratum to: Arc Connectivity and Submodular Flows in Digraphs

Arc Connectivity and Submodular Flows in Digraphs

Let D=(V,A)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$D=(V,A)$$\\end{document} be a digraph. For an integer k≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k\\ge 1$$\\end{document}, a k-arc-connected flip is an arc subset of D such that after reversing the arcs in it the digraph becomes (strongly) k-arc-connected. The first main result of this paper introduces a sufficient condition for the existence of a k-arc-connected flip that is also a submodular flow for a crossing submodular function. More specifically, given some integer τ≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au \\ge 1$$\\end{document}, suppose dA+(U)+(τk-1)dA-(U)≥τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d_A^+(U)+(\\frac{\ au }{k}-1)d_A^-(U)\\ge \ au $$\\end{document} for all U⊊V,U≠∅\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U\\subsetneq V, U\ e \\emptyset $$\\end{document}, where dA+(U)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d_A^+(U)$$\\end{document} and dA-(U)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d_A^-(U)$$\\end{document} denote the number of arcs in A leaving and entering U, respectively. Let C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {C}}$$\\end{document} be a crossing family over ground set V, and let f:C→Z\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:{\\mathcal {C}}\\rightarrow {\\mathbb {Z}}$$\\end{document} be a crossing submodular function such that f(U)≥kτ(dA+(U)-dA-(U))\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(U)\\ge \\frac{k}{\ au }(d_A^+(U)-d_A^-(U))$$\\end{document} for all U∈C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U\\in {\\mathcal {C}}$$\\end{document}. Then D has a k-arc-connected flip J such that f(U)≥dJ+(U)-dJ-(U)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f(U)\\ge d_J^+(U)-d_J^-(U)$$\\end{document} for all U∈C\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$U\\in {\\mathcal {C}}$$\\end{document}. The result has several applications to Graph Orientations and Combinatorial Optimization. In particular, it strengthens Nash-Williams’ so-called weak orientation theorem, and proves a weaker variant of Woodall’s conjecture on digraphs whose underlying undirected graph is τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document}-edge-connected. The second main result of this paper is even more general. It introduces a sufficient condition for the existence of capacitated integral solutions to the intersection of two submodular flow systems. This sufficient condition implies the classic result of Edmonds and Giles on the box-total dual integrality of a submodular flow system. It also has the consequence that in a weakly connected digraph, the intersection of two submodular flow systems is totally dual integral.

Fair integral submodular flows

Integer-valued elements of an integral submodular flow polyhedron Q are investigated which are decreasingly minimal (dec-min) in the sense that their largest component is as small as possible, within this, the second largest component is as small as possible, and so on. As a main result, we prove that the set of dec-min integral elements of Q is the set of integral elements of another integral submodular flow polyhedron arising from Q by intersecting a face of Q with a box. Based on this description, we develop a strongly polynomial algorithm for computing not only a dec-min integer-valued submodular flow but even a cheapest one with respect to a linear cost-function. A special case is the problem of finding a strongly connected (or k -edge-connected) orientation of a mixed graph whose in-degree vector is decreasingly minimal.

A Discrete Convex Min-Max Formula for Box-TDI Polyhedra

A min-max formula is proved for the minimum of an integer-valued separable discrete convex function in which the minimum is taken over the set of integral elements of a box total dual integral polyhedron. One variant of the theorem uses the notion of conjugate function (a fundamental concept in nonlinear optimization), but we also provide another version that avoids conjugates, and its spirit is conceptually closer to the standard form of classic min-max theorems in combinatorial optimization. The presented framework provides a unified background for separable convex minimization over the set of integral elements of the intersection of two integral base-polyhedra, submodular flows, L-convex sets, and polyhedra defined by totally unimodular matrices. As an unexpected application, we show how a wide class of inverse combinatorial optimization problems can be covered by this new framework.

Cooperative Regenerating Codes

One of the design objectives in distributed storage system is the minimization of the data traffic during the repair of failed storage nodes. By repairing multiple failures simultaneously and cooperatively rather than successively and independently, further reduction of repair traffic is made possible. A closed-form expression of the optimal tradeoff between the repair traffic and the amount of storage in each node for cooperative repair is given. We show that the points on the tradeoff curve can be achieved by linear cooperative regenerating codes, with an explicit bound on the required finite-field size. The proof relies on a max-flow-min-cut-type theorem from combinatorial optimization for submodular flows. Two families of explicit constructions are given.

Degree bounded matroids and submodular flows

We consider two related problems, the Minimum Bounded Degree Matroid Basis problem and the Minimum Bounded Degree Submodular Flow problem. The first problem is a generalization of the Minimum Bounded Degree Spanning Tree problem: We are given a matroid and a hypergraph on its ground set with lower and upper bounds f(e)≤g(e) for each hyperedge e. The task is to find a minimum cost basis which contains at least f(e) and at most g(e) elements from each hyperedge e. In the second problem we have a submodular flow problem, a lower bound f(v) and an upper bound g(v) for each node v, and the task is to find a minimum cost 0–1 submodular flow with the additional constraint that the sum of the incoming and outgoing flow at each node v is between f(v) and g(v). Both of these problems are NP-hard (even the feasibility problems are NP-complete), but we show that they can be approximated in the following sense. Let opt be the value of the optimal solution. For the first problem we give an algorithm that finds a basis B of cost no more than opt such that f(e)−2Δ+1≤|B∩e|≤g(e)+2Δ−1 for every hyperedge e, where Δ is the maximum degree of the hypergraph. If there are only upper bounds (or only lower bounds), then the violation can be decreased to Δ−1. For the second problem we can find a 0–1 submodular flow of cost at most opt where the sum of the incoming and outgoing flow at each node v is between f(v)−1 and g(v)+1. These results can be applied to obtain approximation algorithms for several combinatorial optimization problems with degree constraints, including the Minimum Crossing Spanning Tree problem, the Minimum Bounded Degree Spanning Tree Union problem, the Minimum Bounded Degree Directed Cut Cover problem, and the Minimum Bounded Degree Graph Orientation problem.

Simple push-relabel algorithms for matroids and submodular flows

We derive simple push-relabel algorithms for the matroid partitioning, matroid membership, and submodular flow feasibility problems. It turns out that, in order to have a strongly polynomial algorithm, the lexicographic rule used in all previous algorithms for the two latter problems can be avoided. Its proper role is that it helps speeding up the algorithm in the last problem.

Lattice polyhedra and submodular flows

Lattice polyhedra, as introduced by Gröflin and Hoffman, form a common framework for various discrete optimization problems. They are specified by a lattice structure on the underlying matrix satisfying certain sub- and supermodularity constraints. Lattice polyhedra provide one of the most general frameworks of total dual integral systems. So far no combinatorial algorithm has been found for the corresponding linear optimization problem. We show that the important class of lattice polyhedra in which the underlying lattice is of modular characteristic can be reduced to the Edmonds–Giles polyhedra. Thus, submodular flow algorithms can be applied to this class of lattice polyhedra. In contrast to a previous result of Schrijver, we do not explicitly require that the lattice is distributive. Moreover, our reduction is very simple in that it only uses an arbitrary maximal chain in the lattice.

Minimizing a sum of submodular functions

We consider the problem of minimizing a function represented as a sum of submodular terms. We assume each term allows an efficient computation of exchange capacities. This holds, for example, for terms depending on a small number of variables, or for certain cardinality-dependent terms.A naive application of submodular minimization algorithms would not exploit the existence of specialized exchange capacity subroutines for individual terms. To overcome this, we cast the problem as a submodular flow (SF) problem in an auxiliary graph in such a way that applying most existing SF algorithms would rely only on these subroutines.We then explore in more detail Iwata’s capacity scaling approach for submodular flows (Iwata 1997 [19]). In particular, we show how to improve its complexity in the case when the function contains cardinality-dependent terms.

An FPTAS for minimizing the product of two non-negative linear cost functions

We consider a quadratic programming (QP) problem (Π) of the form min x T C x subject to Ax ≥ b, x ≥ 0 where $${C\in {\mathbb R}^{n \times n}_+, {\rm rank}(C)=1}$$ and $${A\in {\mathbb R}^{m \times n}, b\in {\mathbb R}^m}$$ . We present an fully polynomial time approximation scheme (FPTAS) for this problem by reformulating the QP (Π) as a parameterized LP and “rounding” the optimal solution. Furthermore, our algorithm returns an extreme point solution of the polytope. Therefore, our results apply directly to 0–1 problems for which the convex hull of feasible integer solutions is known such as spanning tree, matchings and sub-modular flows. They also apply to problems for which the convex hull of the dominant of the feasible integer solutions is known such as s, t-shortest paths and s, t-min-cuts. For the above discrete problems, the quadratic program Π models the problem of obtaining an integer solution that minimizes the product of two linear non-negative cost functions.

Rooted [formula omitted]-connections in digraphs

The problem of computing a minimum cost subgraph D ′ = ( V , A ′ ) of a directed graph D = ( V , A ) so as to contain k edge-disjoint paths from a specified root r 0 ∈ V to every other node in V was solved by Edmonds [J. Edmonds, Submodular functions, matroids, and certain polyhedra, in: R. Guy, H. Hanani, N. Sauer, J. Schönheim (Eds.), Combinatorial Structures and their Applications, Gordon and Breach, New York, 1970, pp. 69–87] by an elegant reduction to weighted matroid intersection. A corresponding problem when openly disjoint paths are requested rather than edge-disjoint ones was solved in [A. Frank, É. Tardos, An application of submodular flows, Linear Algebra Appl. 114–115 (1989) 329–348] with the help of submodular flows. Here we show that the use of submodular flows is actually avoidable and even a common generalization of the two rooted k -connection problems reduces to matroid intersection. The approach is based on a new matroid construction extending what Whiteley [W. Whiteley, Some matroids from discrete applied geometry, in: J.E. Bonin, J.G. Oxley, B. Servatius (Eds.), Matroid Theory, in: Contemp. Math., vol. 197, Amer. Math. Soc, Providence, RI, 1996, pp. 171–311] calls count matroids. We also provide a polyhedral description using supermodular functions on bi-sets and this approach enables us to handle more general rooted k -connection problems. For example, with the help of a submodular flow algorithm the following restricted version of the generalized Steiner-network problem is solvable in polynomial time: given a digraph D = ( V , A ) with a root-node r 0 , a terminal set T , and a cost function c : A → R + so that each edge of positive cost has its head in T , find a subgraph D ′ = ( V , A ′ ) of D of minimum cost so that there are k openly disjoint paths in D ′ from r 0 to every node in T .

A polynomial cycle canceling algorithm for submodular flows

Submodular flow problems, introduced by Edmonds and Giles [2], generalize network flow problems. Many algorithms for solving network flow problems have been generalized to submodular flow problems (cf. references in Fujishige [4]), e.g. the cycle canceling method of Klein [9]. For network flow problems, the choice of minimum-mean cycles in Goldberg and Tarjan [6], and the choice of minimum-ratio cycles in Wallacher [12] lead to polynomial cycle canceling methods. For submodular flow problems, Cui and Fujishige [1] show finiteness for the minimum-mean cycle method while Zimmermann [16] develops a pseudo-polynomial minimum ratio cycle method. Here, we prove pseudo-polynomiality of a larger class of the minimum-ratio variants and, by combining both methods, we develop a polynomial cycle canceling algorithm for submodular flow problems.

Increasing the rooted-connectivity of a digraph by one

In order to unify these approaches, here we describe a two-phase greedy algorithm working on a slight extension of lattice polyhedra. This framework includes the rooted node-connectivity augmentation problem, as well, and hence the resulting algorithm, when appropriately specialized, finds a minimum cost of new edges whose addition to a digraph increases its rooted connectivity by one. The only known algorithm for this problem used submodular flows. Actually, the specialized algorithm solves an extension of the rooted edge-connectivity and node-connectivity augmentation problem.

A cost-scaling algorithm for 0–1 submodular flows

This paper presents a cost-scaling algorithm for minimum cost 0–1 submodular flows. The algorithm works by splitting the arc costs approximately and maintaining an optimal submodular pseudoflow with respect to the split costs obtained by a greedy algorithm. Each scaling phase of the algorithm is a hybrid version of an auction-like method with cost-splitting and a successive-shortest-path method, as a generalization of Orlin and Ahuja's algorithm for the assignment problem.

A capacity scaling algorithm for convex cost submodular flows

This paper presents a scaling scheme for submodular functions. A small but strictly submodular function is added before scaling so that the resulting functions should be submodular. This scaling scheme leads to a weakly polynomial algorithm to solve minimum cost integral submodular flow problems with separable convex cost functions, provided that an oracle for exchange capacities is available.

A Push/Relabel framework for submodular flows and its definement for 0-1 submodular flows

We consider the submodular flow problem of Edmonds and Giles. A submodular flow is a flow in a network satisfying capacity constraints and flow-boundary constraints given in terms of the base polyhedron of a submodular system. A cost scaling framework is constructed by using e-optimality concept associated with dual variables of a flow, originally due to Tardos and Bertsekas. The framework is a generalization of Goldberg and Tarjan's push/relabel algorithm for minimum-cost flows and also a generalization of Fujishige and Zhang's algorithm for the submodular intersection problem. Each phase of the cost scaling, called procedure Refine, improves a 2∊-optimal submodular flow to an ∊,-optimal submodular flow. Furthermore, we devise a faster hybrid algorithm of procedure Refine for the 0-1 submodular flow problem which is a natural generalization of Fujishige and Zhang's algorithm for the independent assignment problem. For a network with n vertices, m arcs and integer arc costs bounded by Г, an optimal 0-1 su...

Centroids, Representations, and Submodular Flows

The notion of centroid of a tree is generalized to apply to an arbitrary intersecting family of sets. Centroids are used to construct a compact representation for any intersecting family of sets, as well as any crossing family. The size of the representation for a family on n elements is O(n2), compared to size O(n3) for previous representations. Efficient algorithms to construct the representation are given. For example on a network of n vertices and m edges, the representation of all minimum cuts uses O(m log(n2/m)) space; it is constructed in O(nm log(n2/m)) time (this is the best-known time for finding one minimum cut). The representation is used to improve several submodular flow algorithms. For example a minimum-cost dijoin is found in time O(n2m); as a result a minimum-cost planar feedback are set is found in time O(n3). The previous best-known time bounds for these two problems are both a factor n larger.

Negative circuits for flows and submodular flows

For solving minimum cost flow problems Goldberg and Tarjan [7] prove strongly polynomial bounds on the negative circuit method of Klein [9] which previously was not even known to be finite. Following the proposal of Goldberg and Tarjan, Cui and Fujishige [1] discuss the use of minimum mean circuits for solving the much more general minimum cost submodular flow problem and prove finiteness where the minimum mean circuit is chosen using a secondary criterium. We introduce certain additional positive weights on negative circuits and propose selecting a negative circuit with minimum ration of cost and weight. The resulting method for solving minimum cost submodular flow problems is pseudopolynomial. In fact, it terminates after at most m· U minimum ratio computations where m denotes the number of arcs and U the maximum capacity of an arc.

An application of submodular flows

Extending theorems of Rado and Lovász, we introduce a new framework for problems concerning supermodular functions and graphs. Among the application is an optimization problem for finding a minimum-cost subgraph H of a digraph G=( V, E) such that H contains k disjoint paths from a fixed paths from a node of G to any other node. Another consequence is a characterization for graphs having a branching that meets all directed cuts. A theorem of Vidyasankar on optimal covering by arborescences and a matroid intersection theorem of Gröflin and Hoffman are also shown to be special cases.

Generalized polymatroids and submodular flows

Polyhedra related to matroids and sub- or supermodular functions play a central role in combinatorial optimization. The purpose of this paper is to present a unified treatment of the subject. The structure of generalized polymatroids and submodular flow systems is discussed in detail along with their close interrelation. In addition to providing several applications, we summarize many known results within this general framework.