Maximum packing for -connected partial -trees in polynomial time

International Journal of Foundations of Computer Science, 1999

A total coloring of a graph G is a coloring of all elements of G, i.e. vertices and edges, in such a way that no two adjacent or incident elements receive the same color. Many combinatorial problems can be efficiently solved for partial k-trees, that is, graphs of treewidth bounded by a constant k. However, no polynomial-time algorithm has been known for the problem of finding a total coloring of a given partial k-tree with the minimum number of colors. This paper gives such a first polynomial-time algorithm.

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.

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 Mathematics, 1984

1999

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.

Graphs and Combinatorics, 2010

Let H be a set of undirected graphs. The induced H-packing problem in an input graph G is to find a subgraph Q of G of maximum size such that each connected component of Q is an induced subgraph of G and is isomorphic to some member of H. In this paper we focus on the case when H consists of factor-critical graphs and a certain family of 'propellers'. Clarifying the methods developed in the related theory of non-induced graph packings, we show a Gallai-Edmonds type structure theorem and a Berge-Tutte type minimax formula. We also give an Edmonds type alternating forest algorithm for the case when H consists of a sequential set of stars and factor-critical graphs. This simplifies the related result of Egawa, Kano and Kelmans.

Discrete Applied Mathematics, 2010

For all integers k ≥ 3, we give an O(n 4 )-time algorithm for the problem whose instance is a graph G of girth at least k together with k vertices and whose question is ''Does G contains an induced subgraph containing the k vertices and isomorphic to a tree?''.

Algorithmica, 2012

The cycle packing number νe(G) of a graph G is the maximum number of pairwise edge-disjoint cycles in G. Computing νe(G) is an NP-hard problem. We present approximation algorithms for computing νe(G) in both undirected and directed graphs. In the undirected case we analyze a variant of the modified greedy algorithm suggested by Caprara et al. [J. of Algorithms 48 (2003), [239][240][241][242][243][244][245][246][247][248][249][250][251][252][253][254][255][256] and show that it has approximation ratio Θ( log n) where n = |V (G)|. This improves upon the previous O(log n) upper bound for the approximation ratio of this algorithm. In the directed case we present a √ n-approximation algorithm. Finally we give an O(n 2/3 )approximation algorithm for the problem of finding a maximum number of edge-disjoint cycles that intersect a specified subset S of vertices. We also study generalizations of these problems. Our approximation ratios are the currently best known ones and, in addition, provide upper bounds on the integrality gap of standard LP-relaxations of these problems. In addition, we give lower bounds for the integrality gap and approximability of νe(G) in directed graphs. Specifically, we prove a lower bound of Ω( log n log log n ) for the integrality gap of edge-disjoint cycle packing. We also show that it is quasi-NP-hard to approximate νe(G) within a factor of O(log 1−ǫ n) for any constant ǫ > 0. This improves upon the previously known APX-hardness result for this problem. A natural generalization of EDC that we consider is the problem of S-cycle packing (denote by S-EDC). In S-EDC we are given a directed graph G and a subset S of its vertices and the goal is to find among the cycles that intersect S (henceforth, Scycles) a maximum number ν e (G, S) of edge-disjoint ones. We note that on directed simple graphs, S-EDC is a special case of the extensively studied edge-disjoint paths problem. See for an O(n 4/5 )-approximation algorithm · 3 and [Varadarajan and Venkataraman 2004] for an O(n 2/3 log 2/3 n)-approximation algorithm for the edge-disjoint paths problem in directed graphs.

2015

In this thesis, we define the spectrum problem for packings (coverings) of G to be the problem of finding all graphs H such that a maximum G-packing (minimum G- covering) of the complete graph with the leave (excess) graph H exists. The set of achievable leave (excess) graphs in G-packings (G-coverings) of the complete graph is called the spectrum of leave (excess) graphs for G. Then, we consider this problem for trees with up to five edges. We will prove that for any tree T with up to five edges, if the leave graph in a maximum T-packing of the complete graph Kn has i edges, then the spectrum of leave graphs for T is the set of all simple graphs with i edges. In fact, for these T and i and H any simple graph with i edges, we will construct a maximum T-packing of Kn with the leave graph H. We will also show that for any tree T with k ≤ 5 edges, if the excess graph in a minimum T-covering of the complete graph Kn has i edges, then the spectrum of excess graphs for T is the set of all...

1986

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.

Graphs and Combinatorics, 2008

If m is a positive integer then we call a tree on at least 2 vertices an m-tree if no vertex is adjacent to more than m leaves. Kaneko proved that a connected, undirected graph G = (V, E) has a spanning m-tree if and only if for every X ⊆ V the number of isolated vertices of G − X is at most m|X | + max{0, |X | − 1}-unless we have the exceptional case of G K 3 and m = 1. As an attempt to integrate this result into the theory of graph packings, in this paper we consider the problem of packing a graph with m-trees. We use an approach different from that of Kaneko, and we deduce Gallai-Edmonds and Berge-Tutte type theorems and a matroidal result for the m-tree packing problem.

Journal of Combinatorial Optimization, 2015

Given a graph G = (V, E) and a non-negative integer c u for each u ∈ V , Partial Degree Bounded Edge Packing (PDBEP) problem is to find a subgraph G = (V, E) with maximum |E | such that for each edge (u, v) ∈ E , either deg G (u) ≤ c u or deg G (v) ≤ c v. The problem has been shown to be NPhard even for uniform degree constraint (i.e., all c u being equal). In this work we study the general degree constraint case (arbitrary degree constraint for each vertex) and present two combinatorial approximation algorithms with approximation factors 4 and 2. Then we give a log 2 n approximation algorithm for edge-weighted version of the problem and an efficient exact algorithm for edge-weighted trees with time complexity O(n log n). We also consider a generalization of this problem to k-uniform hypergraphs and present a constant factor approximation algorithm based on linear programming using Lagrangian relaxation.

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 .