Matroid intersection

Abstract: L. G. Khachiyan recently published a polynomial algorithm to check feasibility of a system of linear inequalities. The method is an adaptation of an algorithm proposed by Shor for non-linear optimization problems. In this paper we show that the method also yields interesting results in combinatorial optimization. Thus it yields polynomial algorithms for vertex packing in perfect graphs; for the matching and matroid intersection problems; for optimum covering of directed cuts of a digraph; for the minimum value of a submodular set function; and for other important combinatorial problems. On the negative side, it yields a proof that weighted fractional chromatic number is NP-hard.

Abstract: The viewpoint of the subject of matroids, and related areas of lattice theory, has always been, in one way or another, abstraction of algebraic dependence or, equivalently, abstraction of the incidence relations in geometric representations of algebra. Often one of the main derived facts is that all bases have the same cardinality. (See Van der Waerden, Section 33.)

Abstract: Publisher Summary This chapter describes a min-max relation for submodular functions on graphs. It proves a new combinatorial min-max equality that unifies and extends results including the matroid intersection theorem and the theorem of Lucchesi and Younger on the minimum number of edges, which meet every directed cut in a graph. The method of proof used in the chapter generalizes the method used to prove the polymatroid intersection theorem and the method used to prove the Lucchesi-Younger Theorem including an idea that Lovasz attributes to Neil Robertson.