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  • 1
    Publication Date: 2021-10-19
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
    Library Location Call Number Volume/Issue/Year Availability
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  • 2
    Publication Date: 2021-08-03
    Description: The covering of a graph with (possibly disjoint) connected subgraphs is a fundamental problem in graph theory. In this paper, we study a version to cover a graph's vertices by connected subgraphs subject to lower and upper weight bounds, and propose a column generation approach to dynamically generate feasible and promising subgraphs. Our focus is on the solution of the pricing problem which turns out to be a variant of the NP-hard Maximum Weight Connected Subgraph Problem. We compare different formulations to handle connectivity, and find that a single-commodity flow formulation performs best. This is notable since the respective literature seems to have dismissed this formulation. We improve it to a new coarse-to-fine flow formulation that is theoretically and computationally superior, especially for large instances with many vertices of degree 2 like highway networks, where it provides a speed-up factor of 10 over the non-flow-based formulations. We also propose a preprocessing method that exploits a median property of weight constrained subgraphs, a primal heuristic, and a local search heuristic. In an extensive computational study we evaluate the presented connectivity formulations on different classes of instances, and demonstrate the effectiveness of the proposed enhancements. Their speed-ups essentially multiply to an overall factor of 20. Overall, our approach allows the reliabe solution of instances with several hundreds of nodes in a few minutes. These findings are further corroborated in a comparison to existing districting models on a set of test instances from the literature.
    Language: English
    Type: reportzib , doc-type:preprint
    Format: application/pdf
    Library Location Call Number Volume/Issue/Year Availability
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  • 3
    Publication Date: 2022-08-31
    Description: Balanced separators are node sets that split the graph into size bounded components. They find applications in different theoretical and practical problems. In this paper we discuss how to find a minimum set of balanced separators in node weighted graphs. Our contribution is a new and exact algorithm that solves Minimum Balanced Separators by a sequence of Hitting Set problems. The only other exact method appears to be a mixed-integer program (MIP) for the edge weighted case. We adapt this model to node weighted graphs and compare it to our approach on a set of instances, resembling transit networks. It shows that our algorithm is far superior on almost all test instances.
    Language: English
    Type: conferenceobject , doc-type:conferenceObject
    Format: application/pdf
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  • 4
    Publication Date: 2022-11-11
    Description: The covering of a graph with (possibly disjoint) connected subgraphs is a funda-mental problem in graph theory. In this paper, we study a version to cover a graph’svertices by connected subgraphs subject to lower and upper weight bounds, and pro-pose a column generation approach to dynamically generate feasible and promisingsubgraphs. Our focus is on the solution of the pricing problem which turns out to bea variant of the NP-hard Maximum Weight Connected Subgraph Problem. We com-pare different formulations to handle connectivity, and find that a single-commodityflow formulation performs best. This is notable since the respective literature seemsto have widely dismissed this formulation. We improve it to a new coarse-to-fine flowformulation that is theoretically and computationally superior, especially for largeinstances with many vertices of degree 2 like highway networks, where it provides aspeed-up factor of 5 over the non-flow-based formulations. We also propose a pre-processing method that exploits a median property of weight-constrained subgraphs,a primal heuristic, and a local search heuristic. In an extensive computational studywe evaluate the presented connectivity formulations on different classes of instances,and demonstrate the effectiveness of the proposed enhancements. Their speed-upsessentially multiply to an overall factor of well over 10. Overall, our approach allowsthe reliable solution of instances with several hundreds of vertices in a few min-utes. These findings are further corroborated in a comparison to existing districtingmodels on a set of test instances from the literature
    Language: English
    Type: article , doc-type:article
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  • 5
    Publication Date: 2022-11-24
    Description: Finding connected subgraphs of maximum weight subject to additional constraints on the subgraphs is a common (sub)problem in many applications. In this paper, we study the Maximum Weight Connected Subgraph Problem with a given root node and a lower and upper capacity constraint on the chosen subgraph. In addition, the nodes of the input graph are colored blue and red, and the chosen subgraph is required to be balanced regarding its cumulated blue and red weight. This problem arises as an essential subproblem in district planning applications. We show that the problem is NP-hard and give an integer programming formulation. By exploiting the capacity and balancing condition, we develop a powerful reduction technique that is able to significantly shrink the problem size. In addition, we propose a method to strengthen the LP relaxation of our formulation by identifying conflict pairs, i.e., nodes that cannot be both part of a chosen subgraph. Our computational study confirms the positive impact of the new preprocessing technique and of the proposed conflict cuts.
    Language: English
    Type: conferenceobject , doc-type:conferenceObject
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