The problem of computing induced subgraphs that satisfy some specified restrictions arises in various applications of graph algorithms and has been well studied. In this paper, we consider the following Balanced Connected Subgraph (shortly, BCS) problem. The input is a graph G = (V, E), with each vertex in the set V having an assigned color, "red" or "blue". We seek a maximum-cardinality subset V ′ ⊆ V of vertices that is color-balanced (having exactly |V ′ |/2 red nodes and |V ′ |/2 blue nodes), such that the subgraph induced by the vertex set V ′ in G is connected. We show that the BCS problem is NP-hard, even for bipartite graphs G (with red/blue color assignment not necessarily being a proper 2-coloring). Further, we consider this problem for various classes of the input graph G, including, e.g., planar graphs, chordal graphs, trees, split graphs, bipartite graphs with a proper red/blue 2coloring, and graphs with diameter 2. For each of these classes either we prove NP-hardness or design a polynomial time algorithm.
Classical separability problem involving multi-color point sets is an important area of study in computational geometry. In this paper, we study different separability problems for bichromatic point set P = Pr ∪ P b on a plane, where Pr and P b represent the set of n red points and m blue points respectively, and the objective is to compute a monochromatic object of the desired type and of maximum size. We propose in-place algorithms for computing (i) an arbitrarily oriented monochromatic rectangle of maximum size in R 2 , and (ii) an axis-parallel monochromatic cuboid of maximum size in R 3 . The time complexities of the algorithms for problems (i) and (ii) are O(m(m+n)(m √ n+m log m+n log n)) and O(m 3 √ n+m 2 n log n), respectively. As a prerequisite, we propose an in-place construction of the classic data structure the k-d tree, which was originally invented by J. L. Bentley in 1975. Our in-place variant of the k-d tree for a set of n points in R k supports both orthogonal range reporting and counting query using O(1) extra workspace, and these query time complexities are same as the classical complexities, i.e., O(n 1−1/k + µ) and O(n 1−1/k ), respectively, where µ is the output size of the reporting query. The construction time of this data structure is O(n log n). Both the construction and query algorithms are nonrecursive in nature that do not need O(log n) size recursion stack compared to the previously known construction algorithm for in-place k-d tree and query in it. We believe that this result is of independent interest. We also propose an algorithm for the problem of computing an arbitrarily oriented rectangle of maximum weight among a point set P = Pr ∪ P b , where each point in P b (resp. Pr) is associated with a negative (resp. positive) real-valued weight that runs in O(m 2 (n + m) log(n + m)) time using O(n) extra space.
Maximum independent set from a given set D of unit disks intersecting a horizontal line can be solved in O(n 2 ) time and O(n 2 ) space. As a corollary, we design a factor 2 approximation algorithm for the maximum independent set problem on unit disk graph which takes both time and space of O(n 2 ). The best known factor 2 approximation algorithm for this problem runs in O(n 2 log n) time and takes O(n 2 ) space [1,2].
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.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.
hi@scite.ai
10624 S. Eastern Ave., Ste. A-614
Henderson, NV 89052, USA
Copyright © 2024 scite LLC. All rights reserved.
Made with 💙 for researchers
Part of the Research Solutions Family.