Abstract
We consider an extended version of the classical Max-k-Cut problem in which we additionally require that the parts of the graph partition are connected. For this problem we study two alternative mixed-integer linear formulations and review existing as well as develop new branch-and-cut techniques like cuts, branching rules, propagation, primal heuristics, and symmetry breaking. The main focus of this paper is an extensive numerical study in which we analyze the impact of the different techniques for various test sets. It turns out that the techniques from the existing literature are not sufficient to solve an adequate fraction of the test sets. However, our novel techniques significantly outperform the existing ones both in terms of running times and the overall number of instances that can be solved.
Highlights
In this paper we study a special version of the graph partitioning problem in which all parts of the partition have to be connected
Our reason to study the C- Max-k-Cut stems from an application: it is a subproblem in computing a market splitting for electricity markets; see, e.g., [3,21,22,34]
Because dense constraints may slow down the solution process [40], it might be favorable to separate the local cuts instead of adding the global cut initially. These cuts can be separated by computing the graph gas instances (Gas) well as the corresponding value k, which can be done in O(k(|V | + |E|)) time, and adding the locally valid cut if (7) is violated
Summary
In this paper we study a special version of the graph partitioning problem in which all parts of the partition have to be connected. The objective is to maximize the number of edges between the parts We call it the connected Max-k-Cut problem or C- Maxk-Cut for short. Forest planning problems are studied in [8], where an important constraint is that, for a number of different planning problems, old tree populations must stay connected. Another related problem is that of finding a coloring of a graph such that each color induces a connected subgraph. We augment a general-purpose MILP solver by (i) tweaking known techniques from the existing literature for related problems and by (ii) developing new MILP-techniques that are tailored for the C- Max-k-Cut problem.
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