Abstract We study the use of flow‐based algorithmic and proof techniques applied to preemptive scheduling of jobs on parallel identical machines. For the classical problem in which the jobs have individual release dates and must be finished by a common deadline, we present and prove unified necessary and sufficient conditions for the existence of a feasible schedule by examining the structure of minimum cuts in a special network. We then study an enhanced model that allows the presence of additional resources, provided that some jobs at any time of their processing require one unit of a particular resource. We extend our argument developed for the classical case to this enhanced model. The generalized necessary and sufficient conditions for the existence of a feasible schedule are presented and proved using the max‐flow/min‐cut reasoning.

We consider a range of scheduling problems with controllable processing times, in which the jobs must be completed by a common deadline by compressing appropriately their processing times. The objective is to minimize the maximum compression cost. We present a number of algorithms based on common general principles adapted with a purpose of reducing the resulting running times.

In this paper, we propose a new methodology for the speed-scaling problem based on its link to scheduling with controllable processing times and submodular optimization. It results in faster algorithms for traditional speed-scaling models, characterized by a common speed/energy function. Additionally, it efficiently handles the most general models with job-dependent speed/energy functions with single and multiple machines. To the best of our knowledge, this has not been addressed prior to this study. In particular, the general version of the single-machine case is solvable by the new technique in O(n2) time. The online appendix is available at https://doi.org/10.1287/ijoc.2017.0758 .

In this paper, we study a scheduling problem on a single machine, provided that the jobs have individual release dates and deadlines, and the processing times are controllable. The objective is to find a feasible schedule that minimizes the total cost of reducing the processing times. We reformulate the problem in terms of maximizing a linear function over a submodular polyhedron intersected with a box. For the latter problem of submodular optimization, we develop a recursive decomposition algorithm and apply it to solving the single machine scheduling problem to achieve the best possible running time.

In this paper, we present a general methodology for designing polynomial-time algorithms for bicriteria scheduling problems on parallel machines with controllable processing times. For each considered problem, the two criteria are the makespan and the total compression cost, and the solution is delivered in the form of the break points of the efficient frontier. We reformulate the scheduling problems in terms of optimization over submodular polyhedra and give efficient procedures for computing the corresponding rank functions. As a result, for two of the considered problems we obtain the first polynomial-time algorithms, while for the third problem we considerably improve the known running time.

The concept of neighbor system, introduced by Hartvigsen in 2010, is a set of integral vectors satisfying a certain combinatorial property. In this paper, we reveal the relationship of neighbor systems with jump systems and with bisubmodular polyhedra. We first prove that for every neighbor system, there exists a jump system which has the same neighborhood structure as the original neighbor system. This shows that the concept of neighbor system is essentially equivalent to that of jump system. We next show that the convex closure of a neighbor system is an integral bisubmodular polyhedron. In addition, we give a characterization of neighbor systems using bisubmodular polyhedra. Finally, we consider the problem of minimizing a separable convex function on a neighbor system. It is shown that the problem can be solved in weakly polynomial time for a class of neighbor systems.

The concept of a jump system, introduced by Bouchet and Cunningham [SIAM J. Discrete Math., 8 (1995), pp. 17–32], 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 first 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 [Discrete Appl. Math., 84 (1998), pp. 215–220]. We finally consider the concept of M-convex functions on constant-parity jump systems which has been recently proposed by Murota [SIAM J. Discrete Math., 20 (2006), pp. 213–226]. It is shown that the minimization of an M-convex function can be solved in polynomial time by the domain reduction approach.

The tree center problems are designed to find a subtree minimizing the maximum distance from any vertex. This paper shows that these problems in a tree network are related to the bottleneck knapsack problems and presents linear-time algorithms for the tree center problems by using the relation. © 1997 John Wiley & Sons, Inc. Networks, 29: 107–110, 1997

Let G be an undirected graph with V vertices and E edges. Many algorithms have been developed for enumerating all spanning trees in G. Most of the early algorithms use a technique called "backtracking." Recently, several algorithms using a different technique have been proposed by Kapoor and Ramesh (1992), Matsui (1993), and Shioura and Tamura (1993). They find a new spanning tree by exchanging one edge of a current one. This technique has the merit of enabling us to compress the whole output of all spanning trees by outputting only relative changes of edges. Kapoor and Ramesh first proposed an O(N + V + E)-time algorithm by adopting such a "compact" output, where N is the number of spanning trees. Another algorithm with the same time complexity was constructed by Shioura and Tamura. These are optimal in the sense of time complexity but not in terms of space complexity because they take O(VE) space. We refine Shioura and Tamura's algorithm and decrease the space complexity from O(VE) to O(V + E) while preserving the time complexity. Therefore, our algorithm is optimal in the sense of both time and space complexities.