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### Nonoverlapping local alignments (weighted independent sets of axis-parallel rectangles)

Vineet Bafna, Babu Narayanan, R. Ravi
1996 Discrete Applied Mathematics
We extend the heuristic to the problem of finding a maximum-weight independent set in (d + 1)-claw-free graphs, and show a tight performance ratio of d -1 -t l/d.  ...  We are given a set of weighted axis-parallel rectangles such that for any pair of rectangles and either axis, the projection of one rectangle does not enclose that of the other.  ...  for carefully reading the manuscript and pointing out errors.  ...

### Approximations of independent sets in graphs [chapter]

1998 Lecture Notes in Computer Science
Any 2-optimal solution is (~ + 1)~2-approximate. Proof. Let us argue in terms of independent sets in a + 1-claw free graphs. Let I be a 2-optimal solution, and let O be any optimal independent set.  ...  Demange and Paschos  have parametrized the strategy of  to give a ratio A/c for every c, that holds for every value of A in time O(nC). + l-claw-free graphs Claw-free graphs appear considerably  ...

### Nearly Optimal Scheduling of Wireless Ad Hoc Networks in Polynomial Time [article]

Alper Kose, Noyan Evirgen, Hakan Gokcesu, Kaan Gokcesu, Muriel Medard
2017 arXiv   pre-print
It is possible to formulate a scheduling problem of network coded flows as finding maximum weighted independent set (MWIS) in the conflict graph of the network.  ...  In a claw-free conflict graph, MWIS can be found in polynomial time leading to a throughput-optimal scheduling. We show that the conflict graph of certain wireless ad hoc networks are claw-free.  ...  One of these transmissions can be selected for a possible claw, say v 3 . Since, it is not possible to find another independent transmission v 4 , we have a claw-free conflict graph for the network.  ...

### From matchings to independent sets

2017 Discrete Applied Mathematics
It is well-known that finding a maximum matching in G is equivalent to finding a maximum independent set in the line graph of G. For general graphs, the maximum independent set problem is NP-hard.  ...  We also review various techniques that may lead to efficient algorithms for the maximum independent set problem in restricted graph families, with a focus given to augmenting graphs and graph transformations  ...  The maximum independent set problem Finding a maximum independent set in a graph is one of the central problems of combinatorial optimization and theoretical computer science with numerous applications  ...

### On local search and LP and SDP relaxations for k-Set Packing [article]

Tim Oosterwijk
2015 arXiv   pre-print
It is equivalent to hypergraph matching and it is strongly related to the maximum independent set problem.  ...  Set packing is a fundamental problem that generalises some well-known combinatorial optimization problems and knows a lot of applications.  ...  They reduce the problem to the independent set problem in a k + 1-claw free graph G, and their local search finds an improving claw.  ...

### Page 5939 of Mathematical Reviews Vol. , Issue 89K [page]

1989 Mathematical Reviews
The author proves the theorem: Let k be an integer greater than 1 and let G be a k-connected claw-free graph of order n such that the degree sum of any k + 1 independent vertices is at least n —k; then  ...  This theorem verifies for 2-connected claw-free graphs a conjecture of J. A. Bondy [in The theory and applications of graphs (Kalamazoo, MI, 1980), 117-125, Wiley, New York, 1981; MR 83e:05070].  ...

### Improved Approximations for k-Exchange Systems [chapter]

Moran Feldman, Joseph Naor, Roy Schwartz, Justin Ward
2011 Lecture Notes in Computer Science
, b-matching, matroid k-parity in strongly base orderable matroids, and additional combinatorial optimization problems such as: independent set in (k + 1)-claw free graphs, asymmetric TSP, job interval  ...  Note that these last 3 problems, like independent set in (k + 1)-claw free graphs, do not belong to the k-intersection class.  ...  Acknowledgements: The last author thanks Julián Mestre and Allan Borodin for providing comments on a preliminary version of the paper.  ...

### Nonoverlapping local alignments (weighted independent sets of axis parallel rectangles) [chapter]

Vineet Bafna, Babu Narayanan, R. Ravi
1995 Lecture Notes in Computer Science
We extend the heuristic to the problem of nding a maximum-weight independent s e t i n ( d+1)claw free graphs, and show a tight performance ratio of d;1+ 1 d .  ...  W e are given a set of weighted axis-parallel rectangles such that for any pair of rectangles and either axis, the projection of one rectangle does not enclose that of the other.  ...  for carefully reading the manuscript and pointing out errors.  ...

### Page 1423 of Mathematical Reviews Vol. , Issue 99b [page]

1991 Mathematical Reviews
(1-TX-IS; Austin, TX); Goldschmidt, Olivier (1-TX-ME; Austin, TX) Local optimality and its application on independent sets for k-claw free graphs. (English summary) J. Comb.  ...  In this paper, we demon- strate that for k-claw free graphs the r-locally optimal independent sets are found in polynomial time and the worst case is bounded by I*|< WIS3|S e-po where J and /* are a locally  ...

### Performance of Distributed Algorithms for QoS in Wireless Ad Hoc Networks [article]

Ashwin Ganesan
2019 arXiv   pre-print
At one extreme, each node knows the demands of only its neighbors; at the other extreme, there exists an optimal, centralized scheduler that has global information.  ...  Given the conflict graph and link bandwidth requirements, the objective is to determine, using only localized information, whether the demands of all the links can be satisfied.  ...  The graph K 1,3 is called a claw. A graph is claw-free if it does not contain a claw as an induced subgraph. Equivalently, G C is claw-free if and only if σ(G C ) ≤ 2.  ...

### Independence complexes of claw-free graphs

Alexander Engström
2008 European journal of combinatorics (Print)
We study the class of independence complexes of claw-free graphs. The main theorem gives good bounds on the connectivity of these complexes, given bounds for a few subcomplexes of the same class.  ...  Two applications are presented. Firstly, we show that the independence complex of a claw-free graph with n vertices and maximal degree d is (cn/d + ε)-connected, where c = 2/3.  ...  Acknowledgements The author would like to thank the referees for their valuable comments.  ...

### Densest k-Subgraph Approximation on Intersection Graphs [chapter]

Danny Z. Chen, Rudolf Fleischer, Jian Li
2011 Lecture Notes in Computer Science
This concept allows us to derive constant factor approximation algorithms for DS-k on many intersection graph classes, such as chordal graphs, circular-arc graphs, claw-free graphs, line graphs of -hypergraphs  ...  We study approximation solutions for the densest k-subgraph problem (DS-k) on several classes of intersection graphs.  ...  B114), and the Shanghai Committee of Science and Technology (no. 09DZ2272800).  ...

### Resource allocation for V2X communications: A local search based 3D matching approach

Qing Wei, Wanlu Sun, Bo Bai, Li Wang, Erik G. Strom, Mei Song
2017 2017 IEEE International Conference on Communications (ICC)
Due to its NP-hardness, we then exploit hypergraph theory and propose a local search based approximation algorithm to solve it.  ...  requirements on C-UEs and on safety V-UEs.  ...  Specially, a graph is k-claw free if it possesses no k-claws.  ...

### An algorithmic weakening of the Erdős-Hajnal conjecture [article]

Édouard Bonnet, Stéphan Thomassé, Xuan Thang Tran, Rémi Watrigant
2020 arXiv   pre-print
in C_4-free graphs), and, while there is a simple √(n)-approximation in triangle-free graphs, it cannot be improved to n^1/4-ε for any ε > 0 unless NP ⊆ BPP.  ...  We then focus on improving the constant δ for some graph classes: we prove that the classical Local Search algorithm provides an OPT^1-1/t-approximation in K_t,t-free graphs (hence a √(OPT)-approximation  ...  We would like to thank Colin Geniet for pointing out to us the remark on graph products mentioned in the conclusion.  ...

### Approximating the Maximum Internal Spanning Tree problem

Gábor Salamon
2009 Theoretical Computer Science
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.  ...  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.  ...  The author's research was partially supported by the Hungarian National Research Fund and by the National Office for Research and Technology (Grant Number OTKA 67651).  ...
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