On Approximating Restricted Cycle Covers

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. 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 within a factor of 2 for undirected graphs and within a factor of 8/3 in the case of directed graphs. This holds for arbitrary sets L.Comment: To appear in SIAM Journal on Computing. Minor change

Minimum-weight Cycle Covers and Their Approximability

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. We investigate how well L-cycle covers of minimum weight can be approximated. For undirected graphs, we devise a polynomial-time approximation algorithm that achieves a constant approximation ratio for all sets L. On the other hand, we prove that the problem cannot be approximated within a factor of 2-eps for certain sets L. For directed graphs, we present a polynomial-time approximation algorithm that achieves an approximation ratio of O(n), where $n$ is the number of vertices. This is asymptotically optimal: We show that the problem cannot be approximated within a factor of o(n). To contrast the results for cycle covers of minimum weight, we show that the problem of computing L-cycle covers of maximum weight can, at least in principle, be approximated arbitrarily well.Comment: To appear in the Proceedings of the 33rd Workshop on Graph-Theoretic Concepts in Computer Science (WG 2007). Minor change

Applications of Discrepancy Theory in Multiobjective Approximation

We apply a multi-color extension of the Beck-Fiala theorem to show that the multiobjective maximum traveling salesman problem is randomized 1/2-approximable on directed graphs and randomized 2/3-approximable on undirected graphs. Using the same technique we show that the multiobjective maximum satisfiablilty problem is 1/2-approximable

Improved Approximation Algorithms for Cycle and Path Packings

Given an edge-weighted (metric/general) complete graph with $n$ vertices, the maximum weight (metric/general) $k$-cycle/path packing problem is to find a set of $\frac{n}{k}$ vertex-disjoint $k$-cycles/paths such that the total weight is maximized. In this paper, we consider approximation algorithms. For metric $k$-cycle packing, we improve the previous approximation ratio from $3/5$ to $7/10$ for $k=5$, and from $7/8\cdot(1-1/k)^2$ for $k>5$ to $(7/8-0.125/k)(1-1/k)$ for constant odd $k>5$ and to $7/8\cdot (1-1/k+\frac{1}{k(k-1)})$ for even $k>5$. For metric $k$-path packing, we improve the approximation ratio from $7/8\cdot (1-1/k)$ to $\frac{27k^2-48k+16}{32k^2-36k-24}$ for even $10\geq k\geq 6$. For the case of $k=4$, we improve the approximation ratio from $3/4$ to $5/6$ for metric 4-cycle packing, from $2/3$ to $3/4$ for general 4-cycle packing, and from $3/4$ to $14/17$ for metric 4-path packing.Comment: To appear in WALCOM 202