A Greedy Algorithm for Minimizing a Separable Convex Function Over a Finite Jump System

Abstract

We present a greedy algorithm for minimizing a separable convex function over a finite jump system (E,F), where E is a noBempty finite set amd lr is a nonempty finite set of integral points in ZE satisfying a certain exchange axiom. The concept ofjump system was introduced by A. Bouchet and W. H. Cunningham. A jump system is a genera}ization of an integral bisubmodulam polyhedron, alt integral polymatroid, a <poly-)pseudomatroid and a delttumatroid, and has combinatorially nice properties. The algorithm starts with an arbitrary feasible solution and a current feasible solution incrementally moves toward an optimal one in a greedy way. We also show that the greedy algorithm terminates after changing an initial feasible solution at most

Δ-matroid and jump system

“…In fact, it is easy to modify these algorithms to apply directly to problem (5.1). Such a modification of algorithm GGA(III) is considered in [3]. Their results are precisely Corollary 2 and Theorem 3.6 applied to problem (5.2), and hence to problem (5.1).…”

“…We start with the results of Ando et al [3], on maximization of a separable concave function over a jump system. Thus, for J ⊆ Z n , consider the following problem:…”

“…Using this equivalence and the ∆-matroid theory, we give simple proofs and extensions of many of the results on jump systems in [3,6,12].…”

A Steepest Descent Algorithm for M-Convex Functions on Jump Systems

“…Fundamental properties of M-convex functions on constantparity jump systems are investigated in [12], such as equivalence between different exchange axioms, a local optimality criterion guaranteeing global optimality, and some ideas for minimization algorithms. Minimization of a separable convex function over a jump system has been studied in [1], where a local criterion for optimality as well as a greedy algorithm is given.…”

Polynomial-Time Algorithms for Linear and Convex Optimization on Jump Systems

“…A canonical example of this problem arises from the minimization of a separable-convex function on the degree sequences of an undirected graph; a related problem called the minsquare factor problem is discussed in [4,5]. The problem (ScFMin) is studied in [3], where a local criterion for optimality as well as a greedy algorithm is given. Although it is shown that the greedy algorithm runs in pseudo-polynomial time, it is not known so far whether the problem (ScFMin) can be solved in polynomial time.…”