Approximation Algorithm For Graphs Research Articles

Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs

We study the family of intersection graphs of low density objects in low dimensional Euclidean space. This family is quite general, and includes planar graphs. We prove that such graphs have small separators. Next, we present efficient $(1+\varepsilon)$-approximation algorithms for these graphs, for Independent Set, Set Cover, and Dominating Set problems, among others. We also prove corresponding hardness of approximation for some of these optimization problems, providing a characterization of their intractability in terms of density.

Min-Max Graph Partitioning and Small Set Expansion

We study graph partitioning problems from a min-max perspective, in which an input graph on $n$ vertices should be partitioned into $k$ parts, and the objective is to minimize the maximum number of edges leaving a single part. The two main versions we consider are where the $k$ parts need to be of equal size, and where they must separate a set of $k$ given terminals. We consider a common generalization of these two problems, and design for it an $O(\sqrt{\log n\log k})$ approximation algorithm. This improves over an $O(\log^2 n)$ approximation for the second version due to Svitkina and Tardos [Min-max multiway cut, in APPROX-RANDOM, 2004, Springer, Berlin, 2004], and roughly $O(k\log n)$ approximation for the first version that follows from other previous work. We also give an $O(1)$ approximation algorithm for graphs that exclude any fixed minor. Our algorithm uses a new procedure for solving the small-set expansion problem. In this problem, we are given a graph $G$ and the goal is to find a nonempty set...

Approximating the Maximum Internal Spanning Tree problem

Given an undirected connected graph G we consider the problem of finding a spanning tree of G which has a maximum number of internal (non-leaf) vertices among all spanning trees of G . This problem, called Maximum Internal Spanning Tree problem, is clearly NP-hard since it is a generalization of the Hamiltonian Path problem. From the optimization point of view the Maximum Internal Spanning Tree problem is equivalent to the Minimum Leaf Spanning Tree problem. However, the two problems have different approximability properties. Lu and Ravi proved that the latter has no constant factor approximation–unless P = NP–, while Salamon and Wiener gave a linear-time 2-approximation algorithm for the Maximum Internal Spanning Tree problem. In this paper, we improve this approximation ratio by giving an O ( | V ( G ) | 4 ) -time 7 / 4 -approximation algorithm for graphs without pendant vertices. Our approach is based on the successive execution of local improvement steps. We use a linear programming formulation and a primal–dual technique to prove the approximation ratio. We also investigate the vertex-weighted case, that is to find a spanning tree of a vertex-weighted graph G in which the weight sum of internal vertices is maximal among all spanning trees of G . For this problem we present a ( 2 Δ ( G ) − 3 ) -approximation algorithm, where Δ ( G ) is the maximum vertex-degree of G . A slight modification of this algorithm ensures a 2-approximation whenever the input graph is claw-free. Both algorithms run in O ( | V ( G ) | 4 ) time for graphs with no pendant vertices.

Approximating the Spanning Star Forest Problem and Its Application to Genomic Sequence Alignment

This paper studies the algorithmic issues of the spanning star forest problem. We prove the following results: (1) There is a polynomial-time approximation scheme for planar graphs; (2) there is a polynomial-time $\frac{3}{5}$-approximation algorithm for graphs; (3) it is NP-hard to approximate the problem within ratio $\frac{259}{260} + \epsilon$ for graphs; (4) there is a linear-time algorithm to compute the maximum star forest of a weighted tree; (5) there is a polynomial-time $\frac{1}{2}$-approximation algorithm for weighted graphs. We also show how to apply this spanning star forest model to aligning multiple genomic sequences over a tandem duplication region.

Approximating the Minimum Number of Maximum Power Users in Ad hoc Networks

Topology control is the problem of assigning transmission power values to the nodes of an ad hoc network so that the induced graph satisfies some specified property. The most fundamental such property is that the network/graph be connected. For connectivity, prior work on topology control gave a polynomial time algorithm for minimizing the maximum power assigned to any node (such that the induced graph is connected). In this paper we study the problem of minimizing the number of maximum power nodes. After establishing that this minimization problem is NP-complete, we focus on approximation algorithms for graphs with symmetric power thresholds. We first show that the problem is reducible in an approximation preserving manner to the problem of assigning power values so that the sum of the powers is minimized. Using known results for that problem, this provides a family of approximation algorithms for the problem of minimizing the number of maximum power nodes with approximation ratios of 5/3 + e for every e > 0. Unfortunately, these algorithms, based on solving large linear programming problems, are not practical. The main results of this paper are practical algorithms with approximation ratios of 7/4 and 5/3 (exactly). In addition, we present experimental results, both on randomly generated networks, and on two networks derived from proximity data associated with the TRANSIMS project of Los Alamos National Labs. Finally, based on the reduction to the problem of minimizing the total power, we describe some additional results for minimizing the number of maximum power users, both for graph properties other than connectivity and for graphs with asymmetric power thresholds.

Approximation Algorithms for the Achromatic Number

The achromatic number for a graph G=〈V,E〉 is the largest integer m such that there is a partition of V into disjoint independent sets {V1,…,Vm} such that for each pair of distinct sets Vi, Vj, Vi∪Vj is not an independent set in G. Yannakakis and Gavril (1980, SIAM J. Appl. Math.38, 364–372) proved that determining this value for general graphs is NP-complete. For n-vertex graphs we present the first o(n) approximation algorithm for this problem. We also present an O(n5/12) approximation algorithm for graphs with girth at least 5 and a constant approximation algorithm for trees.