Integer Programming Approaches to Balanced Connected k-Partition

Theoretical Computer Science, 1998

This paper describes efficient algorithms for partitioning a k-edge-connected graph into k edge-disjoint connected subgraphs, each of which has a specified number of elements (vertices and edges). If each subgraph contains the specified element (called base), we call this problem the mixed k-partition problem with bases (called k-PART-WB), otherwise we call it the mixed k-partition problem without bases (called R-PART-WOB). In this paper, we show that K-PART-WB always has a solution for every k-edge-connected graph and we consider the problem without bases and we obtain the following results: (1) for any k 22, R-PART-WOB can be solved in O(l~I~~fI~I) t' lme for every 4-edge-connected graph G = (V,E), (2) 3-PART-WOB can be solved in 0(1 VI') for every 2-edge-connected graph G = (V,E) and (3) 4-PART-WOB can be solved in O(lE1*) for every 3-edge-connected graph G =(V,E).

Information Processing Letters, 1992

Journal of Graph Algorithms and Applications, 2000

This paper considers problems of the following type: given an edgeweighted k-colored input graph with maximum color class size c, find a minimum or maximum c-way cut such that each color class is totally partitioned. Equivalently, given a weighted complete k-partite graph, cover its vertices with a minimum number of disjoint cliques in such a way that the total weight of the cliques is maximized or minimized. Our study was motivated by some work called the index domain alignment problem [6], which shows its relevance to optimization of distributed computation. Solutions of these problems also have applications in logistics [3] and manufacturing systems [10]. In this paper, we design some approximation algorithms by extending the matching algorithms to these problems. Both theoretical and experimental results show that the algorithms we designed produce good approximations. Communicated by D. Eppstein.

Algorithms and Discrete Applied Mathematics, 2019

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.

2020

We develop an FPT algorithm and a bi-kernel for the Weighted Edge Clique Partition (WECP) problem, where a graph with $n$ vertices and integer edge weights is given together with an integer $k$, and the aim is to find $k$ cliques, such that every edge appears in exactly as many cliques as its weight. The problem has been previously only studied in the unweighted version called Edge Clique Partition (ECP), where the edges need to be partitioned into $k$ cliques. It was shown that ECP admits a kernel with~$k^2$ vertices [Mujuni and Rosamond, 2008], but this kernel does not extend to WECP. The previously fastest algorithm known for ECP has a runtime of $2^{\mathcal{O}(k^2)}n^{O(1)}$ [Issac, 2019]. For WECP we develop a bi-kernel with $4^k$ vertices, and an algorithm with runtime $2^{\mathcal{O}(k^{3/2}w^{1/2}\log(k/w))}n^{O(1)}$, where $w$ is the maximum edge weight. The latter in particular improves the runtime for ECP to~$2^{\mathcal{O}(k^{3/2}\log k)}n^{O(1)}$.

2012

In this paper we consider the classical combinatorial optimization graph partitioning problem with Sum-Max as objective function. Given a weighted graph G = (V,E) and a integer k, our objective is to find a k-partition (V1,..., Vk) of V that minimizes∑k−1 i=1 ∑k j=i+1maxu∈Vi,v∈Vj w(u, v), where w(u, v) denotes the weight of the edge {u, v} ∈ E. We prove, in addition to the NP and W [1] hardnesses (for the parameter k), that there is no ρ-approximation algorithm for any ρ ∈ O(n1−), given any fixed 0 < ≤ 1 (unless P = NP), improving the previous 1+ 1 k lower bound of [5]. Lastly, we present a natural greedy algorithm with an approximation ratio better than k/2

Journal of Algorithms, 2002

Consider the general partitioning (GP) problem defined as follows: Partition the vertices of a graph into k parts W 1 W k satisfying a polynomial time verifiable property. In particular, consider properties (introduced by T. Feder, P. Hell, S. Klein, and R. Motwani, in "Proceedings of the Annual ACM Symposium on Theory of Computing (STOC '99), 1999" and) specified by a pattern of requirements as to which W i forms a sparse or dense subgraph and which pairs W i , W j form a sparse or dense or an arbitrary (no restriction) bipartite subgraph. The sparsity or density is specified by upper or lower bounds on the edge density d ∈ 0 1 , which is the fraction of actual edges present to the maximum number of edges allowed. This problem is NP-hard even for some fixed patterns and includes as special cases well-known NP-hard problems like k-coloring (each d W i = 0; each d W i W j is arbitrary), bisection (k = 2; W 1 = W 2 ; d W 1 W 2 ≤ b), and also other problems like finding a clique/independent set of specified size. We show that GP is solvable in polynomial time almost surely over random instances with a planted partition of desired type, for several types of pattern requirement. The algorithm is based on the approach of growing BFS trees outlined by C. R. Subramanian (in "Proceedings of the 8th Annual European Symposium on Algorithms (ESA '00), 2000," pp. 415-426).

Discrete Optimization, 2016

We prove polynomial-time solvability of a large class of clustering problems where a weighted set of items has to be partitioned into clusters with respect to some balancing constraints. The data points are weighted with respect to different features and the clusters adhere to given lower and upper bounds on the total weight of their points with respect to each of these features. Further the weight-contribution of a vector to a cluster can depend on the cluster it is assigned to. Our interest in these types of clustering problems is motivated by an application in land consolidation where the ability to perform this kind of balancing is crucial. Our framework maximizes an objective function that is convex in the summed-up utility of the items in each cluster. Despite hardness of convex maximization and many related problems, for fixed dimension and number of clusters, we are able to show that our clustering model is solvable in time polynomial in the number of items if the weightbalancing restrictions are defined using vectors from a fixed, finite domain. We conclude our discussion with a new, efficient model and algorithm for land consolidation.

Graphs and Combinatorics, 2021

In the PARTITION INTO COMPLEMENTARY SUBGRAPHS (COMP-SUB) problem we are given a graph G ¼ ðV; EÞ, and an edge set property P, and asked whether G can be decomposed into two graphs, H and its complement H, for some graph H, in such a way that the edge cut-set (of the cut) ½VðHÞ; VðHÞ satisfies property P. Such a problem is motivated by the fact that several parameterized problems are trivially fixed-parameter tractable when the input graph G is decomposable into two complementary subgraphs. In addition, it generalizes the recognition of complementary prism graphs, and it is related to graph isomorphism when the desired cut-set is empty, COMP-SUB(;). In this paper we are particularly interested in the case COMP-SUB(;), where the decomposition also partitions the set of edges of G into E(H) and EðHÞ. When the input is a chordal graph, we show that COMP-SUB(;) is GI-complete, that is, polynomially equivalent to GRAPH ISOMORPHISM. But it becomes more tractable than GRAPH ISOMORPHISM for several subclasses of chordal graphs. We present structural characterizations for split, starlike, block, and unit interval graphs. We also obtain complexity results for permutation graphs, cographs, comparability graphs, co-comparability graphs, interval graphs, co-interval graphs and strongly chordal graphs. Furthermore, we present some remarks when P is a general edge set property and the case when the cut-set M induces a complete bipartite graph.

2012

In the k-arc connected subgraph problem, we are given a directed graph G and an integer k and the goal is the find a subgraph of minimum cost such that there are at least k-arc disjoint paths between any pair of vertices. We give a simple (1 + 1/k)-approximation to the unweighted variant of the problem, where all arcs of G have the same cost. This improves on the 1 + 2/k approximation of Gabow et al. .