# On Approximating Restricted Cycle Covers

## Abstract

A cycle cover of a graph is a set of cycles such that every vertex is part of exactly one cycle. An *L*-cycle cover is a cycle cover in which the length of every cycle is in the set *L*. A special case of *L*-cycle covers are *k*-cycle covers for *k* ∈ ℕ, where the length of each cycle must be at least *k*. The weight of a cycle cover of an edge-weighted graph is the sum of the weights of its edges.

We come close to settling the complexity and approximability of computing *L*-cycle covers. On the one hand, we show that for almost all *L*, computing *L*-cycle covers of maximum weight in directed and undirected graphs is APX-hard and NP-hard. Most of our hardness results hold even if the edge weights are restricted to zero and one. On the other hand, we show that the problem of computing *L*-cycle covers of maximum weight can be approximated with factor 2.5 for undirected graphs and with factor 3 in the case of directed graphs. Finally, we show that 4-cycle covers of maximum weight in graphs with edge weights zero and one can be computed in polynomial time.

As a by-product, we show that the problem of computing minimum vertex covers in *λ*-regular graphs is APX-complete for every *λ*≥3.

## Keywords

Approximation Algorithm Directed Graph Undirected Graph Edge Weight Travel Salesman Problem## Preview

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