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

Abstract

The concept of jump system, introduced by Bouchet and Cunningham (1995), is a set of integer points with a certain exchange property. In this paper, we discuss several linear and convex optimization problems on jump systems and show that these problems can be solved in polynomial time under the assumption that a membership oracle for a jump system is available. We firstly present a polynomial-time implementation of the greedy algorithm for the minimization of a linear function. We then consider the minimization of a separable-convex function on a jump system, and propose the first polynomial-time algorithm for this problem. The algorithm is based on the domain reduction approach developed in Shioura (1998). We finally consider the concept of M-convex functions on constant-parity jump systems which has been recently proposed by Murota (2006). It is shown that the minimization of an M-convex function can be solved in polynomial time by the domain reduction approach.

A proof of Cunninghamʼs conjecture on restricted subgraphs and jump systems

“…M-concave functions on constant-parity jump systems appear in many combinatorial optimization problems such as the weighted matching problem [32], the minsquare factor problem [1], and the weighted even factor problem in odd-cycle-symmetric digraphs [7,8,26]. Some properties of M-concave functions are investigated in [24,25], and efficient algorithms for maximizing an M-concave function on a constant-parity jump system are given in [33,38].…”

“…We also remark that a general algorithm maximizing an M-concave function on a constant-parity jump system [32,33,38] cannot be applied directly to the weighted K t,t -free t-matching problem in this assumption. In such an algorithm, we compute the function value polynomially many times.…”

A simple algorithm for finding a maximum triangle-free 2-matching in subcubic graphs

“…Jump systems and M-concave functions are understood as a natural framework of efficiently solvable problems. Besides studies of these structures themselves [16,17,15,18], their relation to efficiently solvable combinatorial optimization problems has been revealed (see [19,6,[20][21][22]15,23]). In particular, on the basis of the theory of jump systems, an algorithm for the squarefree 2-matching problem in subcubic graphs is proposed in [6].…”

“…Note that an algorithm that runs in O(n 4 (log Φ(J)) 2 γ 0 ) time is given for the problem in [18]. However, in this paper we only deal with the case when Φ(J) is fixed.…”

Even factors, jump systems, and discrete convexity

“…A separable convex function on the degree sequences of an undirected graph is a typical example of an M-convex function on a constant-parity jump system. An M-convex function on a constant-parity jump system satisfies that global optimality (minimality) is guaranteed by local optimality in the neighborhood of 1 -distance two [26], and several efficient algorithms minimizing an M-convex function on a constant-parity jump system [27,31] follow from this optimality criterion. A recent work of Kobayashi, Murota, and Tanaka [20] showed that M-convex functions on constant-parity jump systems are closed under several basic operations including infimal convolution.…”