Abstract
In the edge-disjoint cycle packing problem we are given a graph G and we have to find a largest set of edge-disjoint cycles in G. The problem of packing vertex-disjoint cycles in G is defined similarly. The best approximation algorithms for edge-disjoint cycle packing are due to Krivelevich et al. [16], where they give an \(O\sqrt{\rm log n}\)-approximation for undirected graphs and an \(O(\sqrt{n})\)-approximation for directed graphs. They also conjecture that the problem in directed case has an integrality gap of \(\Omega(\sqrt{\rm n})\). No non-trivial lower bound is known for the integrality gap of this problem. Here we show that both problems of packing edge-disjoint and packing vertex-disjoint cycles in a directed graph have an integrality gap of \(\Omega(\frac{log n}{log log n})\). This is the first super constant lower bound for the integrality gap of these problems. We also prove that both problems are quasi-NP-hard to approximate within a factor of Ω(log1 − − ε n), for any ε > 0. For the problem of packing vertex-disjoint cycles, we give the first approximation algorithms with ratios O(log n) (for undirected graphs) and \(O(\sqrt{n})\) (for directed graphs). Our algorithms work for the more general case where we have a capacity c v on every vertex v and we are seeking a largest set \(\mathcal{C}\) of cycles such that at most c v cycles of \(\mathcal{C}\) contain v.
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Salavatipour, M.R., Verstraete, J. (2005). Disjoint Cycles: Integrality Gap, Hardness, and Approximation. In: Jünger, M., Kaibel, V. (eds) Integer Programming and Combinatorial Optimization. IPCO 2005. Lecture Notes in Computer Science, vol 3509. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11496915_5
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DOI: https://doi.org/10.1007/11496915_5
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