The greedy algorithm for shortest superstrings
Information Processing Letters
Complexity of the directed spanning cactus problem
Discrete Applied Mathematics
A Recursive Greedy Algorithm for Walks in Directed Graphs
FOCS '05 Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science
An improved approximation algorithm for the asymmetric TSP with strengthened triangle inequality
Journal of Discrete Algorithms
LP-based solution methods for the asymmetric TSP
Information Processing Letters
On the Integrality Ratio for the Asymmetric Traveling Salesman Problem
Mathematics of Operations Research
The on-line asymmetric traveling salesman problem
Journal of Discrete Algorithms
A Simple LP Relaxation for the Asymmetric Traveling Salesman Problem
APPROX '08 / RANDOM '08 Proceedings of the 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization: Algorithms and Techniques
Complexity of the directed spanning cactus problem
Discrete Applied Mathematics
The greedy algorithm for shortest superstrings
Information Processing Letters
Improved approximation algorithms for metric max TSP
ESA'05 Proceedings of the 13th annual European conference on Algorithms
Approximate fair cost allocation in metric traveling salesman games
WAOA'05 Proceedings of the Third international conference on Approximation and Online Algorithms
The on-line asymmetric traveling salesman problem
WADS'05 Proceedings of the 9th international conference on Algorithms and Data Structures
Improved approximation algorithms for metric maximum ATSP and maximum 3-cycle cover problems
WADS'05 Proceedings of the 9th international conference on Algorithms and Data Structures
An improved approximation algorithm for TSP with distances one and two
FCT'05 Proceedings of the 15th international conference on Fundamentals of Computation Theory
The minimum spanning tree problem with non-terminal set
Information Processing Letters
The asymmetric bottleneck traveling salesman problem: Algorithms, complexity and empirical analysis
Computers and Operations Research
Hi-index | 0.00 |
A directed multigraph is said to be d-regular if the indegree and outdegree of every vertex is exactly d. By Hall's theorem one can represent such a multigraph as a combinationof at most n2 cycle covers each taken with an appropriate multiplicity. We prove that if the d-regular multigraph does not contain more than \left\lfloor {{d \mathord{\left/ {\vphantom {d 2}} \right. \kern-\nulldelimiterspace} 2}} \right\rfloor copies of any 2-cycle then we can find a similar decomposition into 0(n2) pairs of cycle covers where each 2-cycle occurs in at most one component of each pair. Our proof is construtive and gives a polynomial algorithm to .nd such a decomposition. Since our applications only need one such a pair of cycle covers whose weight is at least the average weight of all pairs, we also give a simpler algorithm to extract a single such pair.This combinatorial theorem then comes handy in rounding a fractional solution of an LP relaxation of the maximum and minimum TSP problems. For maximum TSP, we obtain a tour whose weight is at least 2/3 of the weight of the longest tour, improving a previous 5/8 approximation. For minimum TSP we obtain a tour whose weight is at most 0.842log2n times the optimal, improving a previous 0.999log2n approximation. Utilizing a reduction from maximum TSP to the shortest superstring problem we obtain a 2.5-approximation algorithm for the latter problem which is again much simpler than the previous one. Other applications of the rounding procedure are approximation algorithms for maximum 3-cycle cover (factor 2/3, previously 3/5) and maximum asymmetric TSP with triangle inequality (factor 10/13, previously 3/4).