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1997, Siam Journal on Discrete Mathematics
In this paper, we consider a large class of vertex partitioning problems and apply to them the theory of algorithm design for problems restricted to partial k-trees. We carefully describe the details of algorithms and analyze their complexity in an attempt to make the algorithms feasible as solutions for practical applications. We give a precise characterization of vertex partitioning problems,
Theoretical Computer Science - TCS, 2000
A c-vertex-ranking of a graph G for a positive integer c is a labeling of the vertices of G with integers such that, for any label i, deletion of all vertices with labels >i leaves connected components, each having at most c vertices with label i. A c-vertex-ranking is optimal if the number of labels used is as small as possible. We present sequential and parallel algorithms to find an optimal c-vertex-ranking of a partial k-tree, that is, a graph of treewidth bounded by a fixed integer k. The sequential algorithm takes polynomial-time for any positive integer c. The parallel algorithm takes O(logn) parallel time using a polynomial number of processors on the common CRCW PRAM, where n is the number of vertices in G.
Discrete Applied Mathematics, 1997
For a fixed positive integer k, the k-path partition problem is to partition the vertex set of a graph into the smallest number of paths such that each path has at most k vertices. The 2-path partition problem is equivalent to the edge-cover problem. This paper presents a linear-time algorithm for the k-path partition problem in trees. The algorithm is
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).
Lecture Notes in Computer Science, 1993
Many combinatorial problems can be efficiently solved for partial k-trees. The edge-coloring problem is one of a few combinatorial problems for which no linear-time algorithm has been obtained for partial k-trees. The best known algorithm solves the problem for partial k-trees G in time O(nA 2~r where n is the number of vertices and A is the maximum degree of G. This paper gives a linear algorithm which optimally edge-colors a given partial k-tree for fixed k.
Lecture Notes in Computer Science, 1997
A c-vertex-ranking of a graph G for a positive integer c is a labeling of the vertices of G with integers such that, for any label i, deletion of all vertices with labels > i leaves connected components, each having at most c vertices with label i. We present a polynomialtime algorithm to find a c-vertex-ranklng of a partial k-tree using the minimum number of ranks for any bounded integers c and k.
International journal of computer science, 2016
The problem of optimally removing a set of vertices from a graph to minimize the size of the largest resultant component is known to be NP-complete. Prior work has provided near optimal heuristics with a high time complexity that function on up to hundreds of nodes and less optimal but faster techniques that function on up to thousands of nodes. In this work, we analyze how to perform vertex partitioning on massive graphs of tens of millions of nodes. We use a previously known and very simple heuristic technique: iteratively removing the node of largest degree and all of its edges. This approach has an apparent quadratic complexity since, upon removal of a node and adjoining set of edges, the node degree calculations must be updated prior to choosing the next node. However, we describe a linear time complexity solution using an array whose indices map to node degree and whose values are hash tables indicating the presence or absence of a node at that degree value. This approach also...
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.
Journal of Global Optimization, 2010
The graph partitioning problem is to partition the vertex set of a graph into a number of nonempty subsets so that the total weight of edges connecting distinct subsets is minimized. Previous research requires the input of cardinalities of subsets or the number of subsets for equipartition. In this paper, the problem is formulated as a zero-one quadratic programming problem without the input of cardinalities. We also present three equivalent zero-one linear integer programming reformulations. Because of its importance in data biclustering, the bipartite graph partitioning is also studied. Several new methods to determine the number of subsets and the cardinalities are presented for practical applications. In addition, hierarchy partitioning and partitioning of bipartite graphs without reordering one vertex set, are studied.
2011
Graph partitioning is a classical graph theory problem that has proven to be NP-hard.
Discrete Optimization, 2019
Given an undirected graph G = (V, E), a vertex k-cut of G is a vertex subset of V the removing of which disconnects the graph in at least k connected components. Given a graph G and an integer k ≥ 2, the vertex k-cut problem consists in finding a vertex k-cut of G of minimum cardinality. We first prove that the problem is NP-hard for any fixed k ≥ 3. We then present a compact formulation, and an extended formulation from which we derive a column generation and a branching scheme. Extensive computational results prove the effectiveness of the proposed methods.
Discrete Applied Mathematics, 1994
In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives.
Lecture Notes in Computer Science, 2007
In this paper, we study the k-tree partition problem which is a partition of the set of edges of a graph into k edge-disjoint trees. This problem occurs at several places with applications e.g. in network reliability and graph theory. In graph drawing there is the still unbeaten (n − 2) × (n − 2) area planar straight line drawing of maximal planar graphs using Schnyder's realizers , which are a 3-tree partition of the inner edges. Maximal planar bipartite graphs have a 2-tree partition, as shown by Ringel [14]. Here we give a different proof of this result with a linear time algorithm. The algorithm makes use of a new ordering which is of interest of its own. Then we establish the NP-hardness of the k-tree partition problem for general graphs and k ≥ 2. This parallels NPhard partition problems for the vertices [3], but it contrasts the efficient computation of partitions into forests (also known as arboricity) by matroid techniques .
Information Processing Letters, 1992
We study the problem of finding the minimum number of edges that, when cut, form a partition of the vertices into k sets of equal size. This is called the k-BALANCED PARTITIONING problem. The problem is known to be inapproximable within any finite factor on general graphs, while little is known about restricted graph classes.
Discrete Applied Mathematics, 1989
We present and illustrate by a sequence of examples an algorithm paradigm for solving NPhard problems on graphs restricted to partial graphs of k-trees and given with an embedding in a k-tree. Such algorithms, linear in the size of the graph but exponential or superexponential in k, exist for most NP-hard problems that have liiear time algorithms for trees. The examples used are optimization problems involving independent sets, Zominating sets, graph coloring, Hamiltonian circuits, network reliabitity and minimum vertex deletion forbidden subgraphs. The results generalize previous results for series-parallel graphs, bandwidth-constrained graphs, and nonserial dynamic programming.
International Journal of Computer & Information Sciences, 1981
A matching and a dominating set in a graph G are related in that they determine diameter-bounded subtree partitions of G. For a maximum matching and a minimum dominating set, the associate partitions have the fewest numbers of trees. The problem of determining a minimum dominating set in an arbitrary graph G is known to be NP-complete. In this paper we present a linear algorithm for partitioning an arbitrary tree into a minimum number of subtrees, each having a diameter at most k, for a given k.
Theoretical Computer Science, 2000
The problem of determining the maximum number of node-disjoint subgraphs of a partial k-tree H on nH nodes that are isomorphic to a k-connected partial k-tree G on nG nodes is shown to be solvable in time O(n k+1 G nH + n k H).
Information Processing Letters, 2004
Journal of Algorithms, 1996
Many combinatorial problems can be efficiently solved for partial k-trees graphs . of treewidth bounded by k . The edge-coloring problem is one of the well-known combinatorial problems for which no efficient algorithms were previously known, except a polynomial-time algorithm of very high complexity. This paper gives a linear-time sequential algorithm and an optimal parallel algorithm which find an edge-coloring of a given partial k-tree with the minimum number of colors for fixed k.
Mathematical Programming, 1994
Let G = (N; E) be an edge-weighted undirected graph. The graph partitioning problem is the problem of partitioning the node set N into k disjoint subsets of speci ed sizes so as to minimize the total weight of the edges connecting nodes in distinct subsets of the partition. We present a numerical study on the use of eigenvalue-based techniques to nd upper and lower bounds for this problem. Results for bisecting graphs with up to several thousand nodes are given, and for small graphs some trisection results are presented. We show that the techniques are very robust and consistently produce upper and lower bounds having a relative gap of typically a few percentage points.
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