Probabilistic checking of proofs: a new characterization of NP
Journal of the ACM (JACM)
Proof verification and the hardness of approximation problems
Journal of the ACM (JACM)
SIAM Journal on Computing
Randomized metarounding (extended abstract)
STOC '00 Proceedings of the thirty-second annual ACM symposium on Theory of computing
On the inapproximability of disjoint paths and minimum Steiner forest with bandwidth constraints
Journal of Computer and System Sciences
Approximating Minimum Subset Feedback Sets in Undirected Graphs with Applications
SIAM Journal on Discrete Mathematics
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
Constant Ratio Approximations of the Weighted Feedback Vertex Set Problem for Undirected Graphs
ISAAC '95 Proceedings of the 6th International Symposium on Algorithms and Computation
Journal of Computer and System Sciences
Packing Digraphs with Directed Closed Trails
Combinatorics, Probability and Computing
Packing cycles in undirected graphs
Journal of Algorithms - Special issue: Twelfth annual ACM-SIAM symposium on discrete algorithms
An Approximate Max-Steiner-Tree-Packing Min-Steiner-Cut Theorem
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Approximation algorithms for cycle packing problems
SODA '05 Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms
Approximation algorithms for the loop cutset problem
UAI'94 Proceedings of the Tenth international conference on Uncertainty in artificial intelligence
Combination can be hard: approximability of the unique coverage problem
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Efficient approximation algorithms for shortest cycles in undirected graphs
Information Processing Letters
Approximability of packing disjoint cycles
ISAAC'07 Proceedings of the 18th international conference on Algorithms and computation
Efficient approximation algorithms for shortest cycles in undirected graphs
LATIN'08 Proceedings of the 8th Latin American conference on Theoretical informatics
The transposition median problem is NP-complete
Theoretical Computer Science
Exact and approximation algorithms for DNA tag set design
CPM'05 Proceedings of the 16th annual conference on Combinatorial Pattern Matching
Packing cycles exactly in polynomial time
Journal of Combinatorial Optimization
Parameterized approximability of the disjoint cycle problem
ICALP'07 Proceedings of the 34th international conference on Automata, Languages and Programming
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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−− εn), 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 cv on every vertex v and we are seeking a largest set $\mathcal{C}$ of cycles such that at most cv cycles of $\mathcal{C}$ contain v.