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In the Maximum Independent Set problem we are asked to find a set of pairwise nonadjacent vertices in a given graph with the maximum possible cardinality. In general graphs, this classical problem is known to be NP-hard and hard to approximate within a factor of n1−ε for any ε > 0. Due to this, investigating the complexity of Maximum Independent Set in various graph classes in hope of finding better tractability results is an active research direction. In H-free graphs, that is, graphs not containing a fixed graph H as an induced subgraph, the problem is known to remain NP-hard and APX-hard whenever H contains a cycle, a vertex of degree at least four, or two vertices of degree at least three in one connected component. For the remaining cases, where every component of H is a path or a subdivided claw, the complexity of Maximum Independent Set remains widely open, with only a handful of polynomial-time solvability results for small graphs H such as P5, P6, the claw, or the fork. We prove that for every such “possibly tractable” graph H there exists an algorithm that, given an H-free graph G and an accuracy parameter ε > 0, finds an independent set in G of cardinality within a factor of (1 – ε) of the optimum in time exponential in a polynomial of log | V(G) | and ε−1. That is, we show that for every graph H for which Maximum Independent Set is not known to be APX-hard in H-free graphs, the problem admits a quasi-polynomial time approximation scheme in this graph class. Our algorithm works also in the more general weighted setting, where the input graph is supplied with a weight function on vertices and we are maximizing the total weight of an independent set.
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Independence number of hypergraphs under degree conditions
Abstract A well‐known result of Ajtai Komlós, Pintz, Spencer, and Szemerédi (J. Combin. Theory Ser. A32(1982), 321–335) states that every ‐graph on vertices, with girth at least five, and average degree contains an independent set of size for some . In this paper we show that an independent set of the same size can be found under weaker conditions allowing certain cycles of length 2, 3, and 4. Our work is motivated by a problem of Lo and Zhao, who asked for , how large of an independent set a ‐graph on vertices necessarily has when its maximum ‐degree . (The corresponding problem with respect to ‐degrees was solved by Kostochka, Mubayi, and Verstraëte (Random Struct. & Algorithms44(2014), 224–239).) In this paper we show that every ‐graph on vertices with contains an independent set of size , and under additional conditions, an independent set of size . The former assertion gives a new upper bound for the ‐degree Turán density of complete ‐graphs.
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- Award ID(s):
- 2300346
- PAR ID:
- 10419008
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Random Structures & Algorithms
- Volume:
- 63
- Issue:
- 3
- ISSN:
- 1042-9832
- Page Range / eLocation ID:
- p. 821-863
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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