The Balanced Connected Subgraph Problem

Bhore, Sujoy; Chakraborty, Sourav; Jana, Satyabrata; Mitchell, Joseph S. B.; Pandit, Supantha; Roy, Sasanka

doi:10.1007/978-3-030-11509-8_17

Cited by 8 publications

(14 citation statements)

References 28 publications

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“…This section is devoted to showing that BCS can be solved in O(n 2 ) time for trees, which improves upon the previous running time O(n 3 ) of [3]. We also give an algorithm for BCS on bounded treewidth graphs whose running time is O(n 2 ) as well.…”

Section: Trees and Bounded Treewidth Graphsmentioning

confidence: 83%

“…The essential idea behind our algorithm is the same as one in [3]. Let T be a bicolored rooted tree.…”

Section: Trees and Bounded Treewidth Graphsmentioning

confidence: 99%

“…In this section, we discuss the hardness of the weighted counterpart of BCS, namely WBCS. Bhore et al [3] proved that BCS is respectively solvable in polynomial time on trees, split graphs, and properly colored bipartite graphs. However, we prove in this section that WBCS is hard even on those graph classes.…”

Section: Hardness For Wbcsmentioning

confidence: 99%

“…The objective of the problem is to find a balanced connected subgraph with the maximum number of vertices. This problem is called the balanced connected subgraph problem (BCS), recently introduced by Bhore et al [3]. Although finding a maximum size connected subgraph is trivially solvable in linear time, they proved that BCS is NP-hard even on bipartite graphs, on chordal graphs, and on planar graphs.…”

Section: Introductionmentioning

confidence: 99%

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Algorithms and Hardness Results for the Maximum Balanced Connected Subgraph Problem

Kobayashi

Kojima

Matsubara

et al. 2019

Combinatorial Optimization and Applications

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The Balanced Connected Subgraph problem (BCS) was recently introduced by Bhore et al. (CALDAM 2019). In this problem, we are given a graph G whose vertices are colored by red or blue. The goal is to find a maximum connected subgraph of G having the same number of blue vertices and red vertices. They showed that this problem is NP-hard even on planar graphs, bipartite graphs, and chordal graphs. They also gave some positive results: BCS can be solved in O(n 3 ) time for trees and O(n + m) time for split graphs and properly colored bipartite graphs, where n is the number of vertices and m is the number of edges.In this paper, we show that BCS can be solved in O(n 2 ) time for trees and O(n 3 ) time for interval graphs. The former result can be extended to bounded treewidth graphs. We also consider a weighted version of BCS (WBCS). We prove that this variant is weakly NP-hard even on star graphs and strongly NP-hard even on split graphs and properly colored bipartite graphs, whereas the unweighted counterpart is tractable on those graph classes. Finally, we consider an exact exponential-time algorithm for general graphs. We show that BCS can be solved in 2 n/2 n O(1) time. This algorithm is based on a variant of Dreyfus-Wagner algorithm for the Steiner tree problem.

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Section: Trees and Bounded Treewidth Graphsmentioning

confidence: 83%

“…The essential idea behind our algorithm is the same as one in [3]. Let T be a bicolored rooted tree.…”

Section: Trees and Bounded Treewidth Graphsmentioning

confidence: 99%

Section: Hardness For Wbcsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Algorithms and Hardness Results for the Maximum Balanced Connected Subgraph Problem

Kobayashi

Kojima

Matsubara

et al. 2019

Combinatorial Optimization and Applications

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“…In this section, we study the BCS problem on unit-disk graphs. It has been shown that this problem is NP-hard on planar graphs [2]. Besides, we know that every planar graph can be represented as a disk graph (due to Koebe's kissing disk embedding theorem [9]).…”

Section: Unit Disk Graphsmentioning

confidence: 99%

Balanced Connected Subgraph Problem in Geometric Intersection Graphs

Bhore

Jana

Pandit

et al. 2019

Combinatorial Optimization and Applications

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We study the Balanced Connected Subgraph (shortly, BCS) problem on geometric intersection graphs such as interval, circulararc, permutation, unit-disk, outer-string graphs, etc. Given a vertexcolored graph G = (V, E), where each vertex in V is colored with either "red " or "blue", the BCS problem seeks a maximum cardinality induced connected subgraph H of G such that H is color-balanced , i.e., H contains an equal number of red and blue vertices. We study the computational complexity landscape of the BCS problem while considering geometric intersection graphs. On one hand, we prove that the BCS problem is NP-hard on the unit disk, outer-string, complete grid, and unit square graphs. On the other hand, we design polynomial-time algorithms for the BCS problem on interval, circular-arc and permutation graphs. In particular, we give algorithm for the Steiner Tree problem on both the interval graphs and circular arc graphs, that is used as a subroutine for solving BCS problem on same graph classes. Finally, we present a FPT algorithm for the BCS problem on general graphs.

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Vertex covering with capacitated trees

2022

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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 widely 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 5 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 well over 10. Overall, our approach allows the reliable solution of instances with several hundreds of vertices 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.

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The Balanced Connected Subgraph Problem

Cited by 8 publications

References 28 publications

Algorithms and Hardness Results for the Maximum Balanced Connected Subgraph Problem

Algorithms and Hardness Results for the Maximum Balanced Connected Subgraph Problem

Balanced Connected Subgraph Problem in Geometric Intersection Graphs

Vertex covering with capacitated trees

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