Combining NP-Hard Reduction Techniques and Strong Heuristics in an Exact Algorithm for the Maximum-Weight Connected Subgraph Problem

Abstract

Borne out of a surprising variety of practical applications, the maximum-weight connected subgraph problem has attracted considerable interest during the past years. This interest has not only led to notable research on theoretical properties, but has also brought about several (exact) solvers-with steadily increasing performance. Continuing along this path, the following article introduces several new algorithms such as reduction techniques and heuristics and describes their integration into an exact solver. Based on the presented new algorithms and a new formulation our solver is able to outperform previous methods by two orders of magnitude on average. Moreover, one large-scale benchmark instance from the 11th DIMACS Challenge can be solved for the first time to optimality and the primal-dual gap for two other ones can be significantly reduced. Although this article is set against the backdrop of improved practical solving, theoretical properties (such as N P-hardness) of the algorithmic components will receive considerable attention.

Solving Steiner trees: Recent advances, challenges, and perspectives

“…Several generic computational frameworks [95, 102, 153] emerged in the past few years, but none of them dominates the others. Some are stand‐alone open‐source implementations [153], the others rely on MIP solvers, like CPLEX [95] or SCIP [102] or can be applied only to specific problem variants, like the MWCS [198]. It is worth mentioning that the implementation of Polzin and Vahdati Daneshmand [187] from 2009 remains competitive or even state‐of‐the‐art for many instances from the SteinLib, but their code is unfortunately not publicly available.…”

“…Rehfeldt and Koch [198] studied the MWCS and provided a combination of (NP‐hard) reduction tests, primal heuristics, a problem transformation into the SAP and the integration of these techniques into a B&C framework. They further improved the computational times from Leitner et al [153] and solved one more difficult problem instance to optimality.…”

“…In contrast to the STP, relatively little work is devoted to studying reduction tests for node‐weighted (or directed) problem variants [62, 153, 198, 199]. For a detailed overview of reduction tests for node‐weighted versions of the STP until 2006, we refer to the survey by Costa et al [60].…”

“…Motivated by the work of Polzin [180] and Vahdati Daneshmand [228], a recent Networks contribution by Rehfeldt et al [199] introduced many new reduction tests for the PCSTP and MWCS on undirected graphs. In a follow‐up article, Rehfeldt and Koch [198] introduced additional NP‐hard reduction tests for the MWCS and exploited them in an exact solution framework.…”

Mixed-integer programming techniques for the connected max-k-cut problem

“…The multi-commodity formulation is larger than the single-commodity formulation and the authors observe that the larger formulation results in a stronger LP relaxation of the problem. Moreover, reduction techniques and heuristics for the maximum connected subgraph problem are investigated in [43].…”