We give a strongly polynomial-time algorithm minimizing a submodular function f given by a value-giving oracle. The algorithm does not use the ellipsoid method or any other linear programming method. No bound on the complexity of the values of f is needed to be known a priori. The number of oracle calls is bounded by a polynomial in the size of the underlying set. 1.

Recently, the first combinatorial strongly polynomial algorithms for submodular function minimization have been devised independently by Iwata, Fleischer, and Fujishige and by Schrijver. In this paper, we improve the running time of Schrijver's algorithm by designing a push-relabel framework for submodular function minimization (SFM). We also extend this algorithm to carry out parametric minimization for a strong map sequence of submodular functions in the same asymptotic running time as a single SFM. Applications include an eicient algorithm for finding a lexicographically optimal base.

Abstract. In this survey we have attempted to bring together most of the results and papers that deal with toughness related to cycle structure. We begin with a brief introduction and a section on terminology and notation, and then try to organize the work into a few self explanatory categories. These categories are circumference, the disproof of the 2-tough conjecture, factors, special graph classes, computational complexity, and miscellaneous results as they relate to toughness. We complete the survey with some tough open problems!

We now know that not every 2-tough graph is hamiltonian. In fact for every ɛ>0, there exists a (9/4−ɛ)- tough nontraceable graph. We continue our quadrennial survey of results that relate the toughness of a graph to its cycle structure.

The Whitehead minimization problem consists in finding a minimum size element in the automorphic orbit of a word, a cyclic word or a finitely generated subgroup in a finite rank free group. We give the first fully polynomial algorithm to solve this problem, that is, an algorithm that is polynomial both in the length of the input word and in the rank of the free group. Earlier algorithms had an exponential dependency in the rank of the free group. It follows that the primitivity problem – to decide whether a word is an element of some basis of the free group – and the free factor problem can also be solved in polynomial time. ∗Part of this work was produced while the second author was a Visiting Professor at

This is the first in a series of papers that explores a class of polyhedra we call 2-lattice polyhedra. 2-Lattice polyhedra are a special class of lattice polyhedra that include network flow polyhedra, fractional matching polyhedra, matroid intersection polyhedra, the intersection of two polymatroids, etc. In this paper we show that the maximum sum of components of a vector in a 2-lattice polyhderon is equal to the minimum capacity of a cover for the polyhedron. For special classes of 2-lattice polyhedra, called matching 2-lattice polyhedra, that include all of the mentioned special cases except the intersection of two polymatroids, we characterize the largest member in the family of minimum covers in terms of the maximum "cardinality" vectors in the polyhedron. In fact, we show that this same characterization arises from considering only the extreme maximum cardinality vectors. This characterization is at the heart of our extreme point algorithm [3] for finding a maximum cardinality vector in a matching 2-lattice polyhedron. 1 1

In a recent paper [13], Y. Boykov et al. propose an approach for computing curve and surface evolution using a variational approach and the geo-cuts method of Boykov and Kolmogorov [11]. We recall in this paper how this is related to well-known approaches for mean curvature motion, introduced by F. Almgren et al. [3] and S. Luckhaus and T. Sturzenhecker [44], and show how the corresponding problems can be solved with sub-pixel accuracy using Parametric Maximum Flow techniques. This provides interesting algorithms for computing crystalline curvature motion, possibly with a forcing term.