Approximating Maximum Subgraphs without Short Cycles

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. 281-286], 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 $>k$).