Approximation Algorithms for Some Graph Partitioning Problems

Journal of Graph Algorithms and Applications, 1997

We present practical algorithms for constructing partitions of graphs into a fixed number of vertex-disjoint subgraphs that satisfy particular degree constraints. We use this in particular to find k-cuts of graphs of maximum degree ∆ that cut at least a k−1 k (1 + 1 2∆+k−1) fraction of the edges, improving previous bounds known. The partitions also apply to constraint networks, for which we give a tight analysis of natural local search heuristics for the maximum constraint satisfaction problem. These partitions also imply efficient approximations for several problems on weighted bounded-degree graphs. In particular, we improve the best performance ratio for the weighted independent set problem to 3 ∆+2 , and obtain an efficient algorithm for coloring 3-colorable graphs with at most 3∆+2 4 colors.

Journal of Graph Algorithms and …, 1997

We present practical algorithms for constructing partitions of graphs into a fixed number of vertex-disjoint subgraphs that satisfy particular degree constraints. We use this in particular to find k-cuts of graphs of maximum degree ∆ that cut at least a k−1 k (1 + 1 2∆+k−1 ) fraction of the edges, improving previous bounds known. The partitions also apply to constraint networks, for which we give a tight analysis of natural local search heuristics for the maximum constraint satisfaction problem.

Information Processing Letters, 1992

Graph algorithms and applications I, 2002

We present practical algorithms for constructing partitions of graphs into a fixed number of vertex-disjoint subgraphs that satisfy particular degree constraints. We use this in particular to find k-cuts of graphs of maximum degree ∆ that cut at least a k−1 k (1 + 1 2∆+k−1 ) fraction of the edges, improving previous bounds known. The partitions also apply to constraint networks, for which we give a tight analysis of natural local search heuristics for the maximum constraint satisfaction problem.

Discrete Applied Mathematics, 2013

This paper considers the problem of clustering the vertices of a complete edge-weighted graph. The objective is to maximize the sum of the edge weights within the clusters (also called cliques). This so-called Clique Partitioning Problem (CPP) is NP-complete, and has several real-life applications such as groupings in flexible manufacturing systems, in biology, in flight gate assignment, etc. Numerous heuristics and exact approaches as well as benchmark tests have been presented in the literature. Most exact methods use branch and bound with branching over edges. We present tighter upper bounds for each search tree node than those known from the literature, improve the constraint propagation techniques for fixing edges in each node, and present a new branching scheme. The theoretical improvements are reflected by computational tests with real-life data. Although a standard solver delivers best results on randomly generated data, the runtime of the proposed algorithm is very low when being applied to instances on object clustering.

Mathematical Programming, 1998

In this paper we consider the problem of k-partitioning the nodes of a graph with capacity restrictions on the sum of the node weights in each subset of the partition, and the objective of minimizing the sum of the costs of the edges between the subsets of the partition. Based on a study of valid inequalities, we present a variety of separation heuristics for cycle, cycle with ears, knapsack tree and path-block-cycle inequalities among others. The separation heuristics, plus primal heuristics, have been implemented in a branch-andcut routine using a formulation including variables for the edges with nonzero costs and node partition variables. Results are presented for three classes of problems: equipartitioning problems arising in nite element methods and partitioning problems associated with electronic circuit layout and compiler design.

2021

Partitioning a connected graph into k vertex-disjoint connected subgraphs of similar (or given) orders is a classical problem that has been intensively investigated since late seventies. Given a connected graph G = (V,E) and a weight function w : V → Q≥, a connected k-partition of G is a partition of V such that each class induces a connected subgraph. The balanced connected k-partition problem consists in finding a connected k-partition in which every class has roughly the same weight. To model this concept of balance, one may seek connected k-partitions that either maximize the weight of a lightest class (max-min BCPk) or minimize the weight of a heaviest class (min-max BCPk). Such problems are equivalent when k = 2, but they are different when k ≥ 3. In this work, we propose a simple pseudo-polynomial k 2 -approximation algorithm for min-max BCPk which runs in time O(W |V ||E|), where W = ∑ v∈V w(v). Based on this algorithm and using a scaling technique, we design a (polynomial) ...

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

Networks, 2009

Let G = (V, E, Q) be a undirected graph, where V is the set of vertices, E is the set of edges, and Q = {Q 1 ,. .. , Q q } is a partition of V into q subsets. We refer to Q 1 ,. .. , Q q as the components of the partition. The Partition Coloring Problem (PCP) consists of finding a subset V ′ of V with exactly one vertex from each component Q 1 ,. .. , Q q and such that the chromatic number of the graph induced in G by V ′ is minimum. This problem is a generalization of the graph coloring problem. This work presents a branchand-cut algorithm proposed for PCP. An integer programming formulation and valid inequalities are proposed. A tabu search heuristic is used for providing primal bounds. Computational experiments are reported for random graphs and for PCP instances originating from the problem of routing and wavelength assignment in all-optical WDM networks.

1992