A Column Generation Algorithm for the Resource-Constrained Order Acceptance and Scheduling on Unrelated Parallel Machines

In this paper, we investigate the resource-constrained order acceptance and scheduling on unrelated parallel machines that arise in make-to-order systems. The objective of this problem is to simultaneously select a subset of orders to be processed and schedule the accepted orders on unrelated machines in such a way that the resources are not overutilized at any time. We first propose two formulations for the problem: mixed integer linear programming formulation and set partitioning. In view of the complexity of the problem, we then develop a column generation approach based on the set partitioning formulation. In the proposed column generation approach, a differential evolution algorithm is designed to solve subproblems efficiently. Extensive numerical experiments on different-sized instances are conducted, and the results demonstrate that the proposed column generation algorithm reports optimal or near-optimal solutions that are evidently better than the solutions obtained by solving the mixed integer linear programming formulation.

A Matheuristic approach for the maximum balanced subgraph of a signed graph

A graph G=(V,E) with its edges labeled in the set {+, -} is called a signed graph. It is balanced if its set of vertices V can be partitioned into two sets V 1 and V 2 , such that all positive edges connect nodes within V 1 or V 2 , and all negative edges connect nodes between V 1 and V 2 . The maximum balanced subgraph problem (MBSP) for a signed graph is the problem of finding a balanced subgraph with the maximum number of vertices. In this work, we present the first polynomial integer linear programming formulation for this problem and a matheuristic to obtain good quality solutions in a short time. The results obtained for different sets of instances show the effectiveness of the matheuristic, optimally solving several instances and finding better results than the exact method in a much shorter computational time.

Stratisfimal Layout: A modular optimization model for laying out layered node-link network visualizations

Node-link visualizations are a familiar and powerful tool for displaying the relationships in a network. The readability of these visualizations highly depends on the spatial layout used for the nodes. In this paper, we focus on computing layered layouts, in which nodes are aligned on a set of parallel axes to better expose hierarchical or sequential relationships. Heuristic-based layouts are widely used as they scale well to larger networks and usually create readable, albeit sub-optimal, visualizations. We instead use a layout optimization model that prioritizes optimality— as compared to scalability— because an optimal solution not only represents the best attainable result, but can also serve as a baseline to evaluate the effectiveness of layout heuristics. We take an important step towards powerful and flexible network visualization by proposing STRATISFIMAL LAYOUT, a modular integer-linear-programming formulation that can consider several important readability criteria simultaneously— crossing reduction, edge bendiness, and nested and multi-layer groups. The layout can be adapted to diverse use cases through its modularity. Individual features can be enabled and customized depending on the application. We provide open-source and documented implementations of the layout, both for web-based and desktop visualizations. As a proof-of-concept, we apply it to the problem of visualizing complicated SQL queries, which have features that we believe cannot be addressed by existing layout optimization models. We also include a benchmark network generator and the results of an empirical evaluation to assess the performance trade-offs of our design choices. A full version of this paper with all appendices, data, and source code is available at osf.io/3vqmswith live examples at https://visdunneright.github.io/stratisfimal/.

Algorithmic aspects of total Roman {3}-domination in graphs

For a simple, undirected, connected graph [Formula: see text], a function [Formula: see text] which satisfies the following conditions is called a total Roman {3}-dominating function (TR3DF) of [Formula: see text] with weight [Formula: see text]: (C1) For every vertex [Formula: see text] if [Formula: see text], then [Formula: see text] has [Formula: see text] ([Formula: see text]) neighbors such that whose sum is at least 3, and if [Formula: see text], then [Formula: see text] has [Formula: see text] ([Formula: see text]) neighbors such that whose sum is at least 2. (C2) The subgraph induced by the set of vertices labeled one, two or three has no isolated vertices. For a graph [Formula: see text], the smallest possible weight of a TR3DF of [Formula: see text] denoted [Formula: see text] is known as the total Roman[Formula: see text]-domination number of [Formula: see text]. The problem of determining [Formula: see text] of a graph [Formula: see text] is called minimum total Roman {3}-domination problem (MTR3DP). In this paper, we show that the problem of deciding if [Formula: see text] has a TR3DF of weight at most [Formula: see text] for chordal graphs is NP-complete. We also show that MTR3DP is polynomial time solvable for bounded treewidth graphs, chain graphs and threshold graphs. We design a [Formula: see text]-approximation algorithm for the MTR3DP and show that the same cannot have [Formula: see text] ratio approximation algorithm for any [Formula: see text] unless NP [Formula: see text]. Next, we show that MTR3DP is APX-complete for graphs with [Formula: see text]. We also show that the domination and total Roman {3}-domination problems are not equivalent in computational complexity aspects. Finally, we present an integer linear programming formulation for MTR3DP.