NDLI: Approximating Maximum Subgraphs without Short Cycles

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Nutov, Zeev Kortsarz, Guy Langberg, Michael
Copyright Year	2010
Abstract	We study approximation algorithms, integrality gaps, and hardness of approximation of two problems related to cycles of small length k in a given (undirected) graph. The instance for these problems consists of a graph $G=(V,E)$ and an integer k. The k-Cycle Transversal problem is to find a minimum edge subset of E that intersects every k-cycle. The k-Cycle-Free Subgraph problem is to find a maximum edge subset of E without k-cycles. Our main result is for the k-Cycle-Free Subgraph problem with even values of k. For any $k=2r$, we give an $\Omega(n^{-\frac{1}{r}+\frac{1}{r(2r-1)}-\varepsilon})$-approximation scheme with running time $(1/\varepsilon)^{O(1/\varepsilon)}\mathsf{poly}(n)$, where $n=\|V\|$ is the number of vertices in the graph. This improves upon the ratio $\Omega(n^{-1/r})$ that can be deduced from extremal graph theory. In particular, for $k=4$ the improvement is from $\Omega(n^{-1/2})$ to $\Omega(n^{-1/3-\varepsilon})$. Our additional result is for odd k. The 3-Cycle Transversal problem (covering all triangles) was studied by Krivelevich [Discrete Math., 142 (1995), pp. 281286], who presented an LP-based 2-approximation algorithm. We show that k-Cycle Transversal admits a $(k-1)$-approximation algorithm, which extends to any odd k the result that Krivelevich proved for $k=3$. Based on this, for odd k we give an algorithm for k-Cycle-Free Subgraph with ratio $\frac{k-1}{2k-3}=\frac{1}{2}+\frac{1}{4k-6}$; this improves upon the trivial ratio of $1/2$. For $k=3$, the integrality gap of the underlying LP was posed as an open problem in the work of Krivelevich. We resolve this problem by showing a sequence of graphs with integrality gap approaching 2. In addition, we show that if k-Cycle Transversal admits a $(2-\varepsilon)$-approximation algorithm, then so does the Vertex-Cover problem; thus improving the ratio 2 is unlikely. Similar results are shown for the problem of covering cycles of length $\leq k$ or finding a maximum subgraph without cycles of length $\leq k$ (i.e., with girth $>k$).
Starting Page	255
Ending Page	269
Page Count	15
File Format	PDF
ISSN	08954801
DOI	10.1137/09074944X
e-ISSN	10957146
Journal	SIAM Journal on Discrete Mathematics (SJDMEC)
Issue Number	1
Volume Number	24
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2010-03-10
Access Restriction	Subscribed
Subject Keyword	approximation cycles of length k LP girth packing
Content Type	Text
Resource Type	Article
Subject	Mathematics

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