Inapproximability of Feedback Vertex Set for Bounded Length Cycles

The Feedback Vertex Set problem (FVS), where the goal is to find a small subset of ver-tices that intersects every cycle in an input directed graph, is among the fundamental prob-lems whose approximability is not well-understood. One can efficiently find an Õ(log n)factor approximation, and while a constant-factor approximation is ruled out under theUnique Games Conjecture (UGC), the best NP-hardness result is only a factor of≈ 1.36 (viaa simple reduction from Vertex Cover).This work studies a natural variant of the Feedback Vertex Set problem (FVS), where thegoal is to find a small subset of vertices that intersects every cycle of bounded length. For thisvariant, we prove strong NP-hardness of approximation results: For any integer constantk > 3 and > 0, it is hard to find a (k − 1 − )-approximate solution to the problem ofintersecting every cycle of length at most k. The hardness result almost matches the trivialfactor k approximation algorithm for the problem. In fact, the hardness holds also for theproblem of hitting every cycle of length at most a parameter k′ > k where k′ can be takento be Ω( lognlog logn ). Taking k ′ = ω(log n log log n) would be enough to prove a hardness forFVS (for arbitrary length cycles). Our work thus identifies the problem of hitting cycles oflength ≈ log n as the key towards resolving the approximability of FVS.Our result is based on reductions from k-uniform Hypergraph Vertex Cover with ran-dom matching and labeling techniques. As byproducts of our techniques, we also prove afactor (k−1− ) hardness of approximation result for k-Clique Transversal, where one musthit every k-clique in the graph with fewest possible vertices, and a factor Ω(k) hardness re-sult for finding a minimum-sized set of edges to hit all k-cycles. We also obtain almost tightΩ̃(k) factor hardness results for the dual problem of packing vertex-disjoint k-cycles andk-cliques in a graph, albeit relying on the UGC for k-Cycle Packing (but we do get a weakerfactor Ω̃(√k) NP-hardness result). ∗Supported in part by NSF grant CCF-1115525. [email protected]†Supported by a Samsung Fellowship, US-Israel BSF grant 2008293, and NSF [email protected] ISSN 1433-8092Electronic Colloquium on Computational Complexity, Report No. 6 (2014)