This paper presents a combinatorial polynomial-time algorithm for minimizing submodular functions, answering an open question posed in 1981 by Grötschel, Lovász, and Schrijver. The algorithm employs a scaling scheme that uses a flow in the complete directed graph on the underlying set with each arc capacity equal to the scaled parameter. The resulting algorithm runs in time bounded by a polynomial in the size of the underlying set and the length of the largest absolute function value. The paper also presents a strongly polynomial version in which the number of steps is bounded by a polynomial in the size of the underlying set, independent of the function values.A preliminary version has appeared in

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Flows over time (also called dynamic flows) generalize standard network flows by introducing an element of time. They naturally model problems where travel and transmission are not instantaneous. Traditionally, flows over time are solved in time-expanded networks that contain one copy of the original network for each discrete time step. While this method makes available the whole algorithmic toolbox developed for static flows, its main and often fatal drawback is the enormous size of the time-expanded network. We present several approaches for coping with this difficulty. First, inspired by the work of Ford and Fulkerson on maximal s-t-flows over time (or "maximal dynamic s-t-flows"), we show that static length-bounded flows lead to provably good multicommodity flows over time. Second, we investigate "condensed" time-expanded networks which rely on a rougher discretization of time. We prove that a solution of arbitrary precision can be computed in polynomial time through an appropriate discretization leading to a condensed time-expanded network of polynomial size. In particular, our approach yields fully polynomial-time approximation schemes for the NP-hard quickest min-cost and multicommodity flow problems. For single commodity problems, we show that storage of flow at intermediate nodes is unnecessary, and our approximation schemes do not use any.

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We describe fully polynomial time approximation schemes for various multicommodity flow problems in graphs with m edges and n vertices. We present the first approximation scheme for maximum multicommodity flow that is independent of the number of commodities k, and our algorithm improves upon the runtime of previous algorithms by this factor of k, running in O * (−2 m 2) time. For maximum concurrent flow, and minimum cost concurrent flow, we present algorithms that are faster than the current known algorithms when the graph is sparse or the number of commodities k is large, i.e. k > m/n. Our algorithms build on the framework proposed by Garg and Könemann in FOCS 1998. They are simple, deterministic, and for the versions without costs, they are strongly polynomial. The approximation guarantees are obtained by comparison with dual feasible solutions found by our algorithm. Our maximum multicommodity flow algorithm extends to an approximation scheme for the maximum weighted multicommodity flow, which is faster than those implied by previous algorithms by a factor of k/ log W where W is the maximum weight of a commodity.

The survivable network design problem (SNDP) is the following problem: given an undirected graph and values r ij for each pair of vertices i and j , find a minimum-cost subgraph such that there are at least r ij disjoint paths between vertices i and j . In the edge connected version of this problem (EC-SNDP), these paths must be edge-disjoint. In the vertex connected version of the problem (VC-SNDP), the paths must be vertex disjoint. The element connectivity problem (ELC-SNDP, or ELC) is a problem of intermediate difficulty. In this problem, the set of vertices is partitioned into terminals and nonterminals. The edges and nonterminals of the graph are called elements. The values r ij are only specified for pairs of terminals i, j , and the paths from i to j must be element disjoint. Thus if r ij − 1 elements fail, terminals i and j are still connected by a path in the network.These variants of SNDP are all known to be NP-hard. The best known approximation algorithm for the EC-SNDP has performance guarantee of 2 and iteratively rounds solutions to a linear program-✩ This paper is the union of two previously published extended abstracts. The extended abstract [L. Fleischer, A 2-approximation for minimum cost {0, 1, 2} vertex connectivity, in: 8th International Integer Programming and Combinatorial Optimization Conference, 2001, pp. 115-129] presents negative examples for {0, 1, . . . , k}-vertex connectivity, and the 2-approximation algorithm for {0, 1, 2}-vertex connectivity. This result is generalized to include element connectivity in [L. Fleischer, K. Jain, D.P. Williamson, An iterative rounding 2-approximation algorithm for the element connectivity problem, in: 42nd 839ming relaxation of the problem. ELC has a primal-dual O(log k)-approximation algorithm, where k = max i,j r ij . Since this work first appeared as an extended abstract, it has been shown that it is hard to approximate VC-SNDP to factor 2 log 1− n .In this paper we investigate applying iterative rounding to ELC and VC-SNDP. We show that iterative rounding will not yield a constant factor approximation algorithm for general VC-SNDP. On the other hand, we show how to extend the analysis of iterative rounding applied to EC-SNDP to yield 2-approximation algorithms for both general ELC, and for the case of VC-SNDP when r ij ∈ {0, 1, 2}. The latter result improves on an existing 3-approximation algorithm. The former is the first constant factor approximation algorithm for a general survivable network design problem that allows node failures.