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Approximating maximum subgraphs without short cycles. (English) Zbl 1214.05057
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 M. Krivelevich [Discrete Math. 142, No.1–3, 281–286 (1995; Zbl 0920.05056)], 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 $$>k$$).

MSC:
 05C38 Paths and cycles 05C35 Extremal problems in graph theory 68W25 Approximation algorithms
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