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We present randomized algorithms for finding maximum matchings in general and bipartite graphs. Both algorithms have running time O(n ω), where ω is the exponent of the best known matrix multiplication algorithm. Since ω < 2.38, these algorithms break through the O(n 2.5) barrier for the matching problem. They both have a very simple implementation in time… (More)

In the maximum traveling salesman problem (Max TSP) we are given a complete undirected graph with nonnegative weights on the edges and we wish to compute a traveling salesman tour of maximum weight. We present a fast combinatorial 4 5-approximation algorithm for Max TSP. The previous best approximation for this problem was 7 9. The new algorithm is based on… (More)

The Travelling Salesman Problem is one the most fundamental and most studied problems in approximation algorithms. For more than 30 years, the best algorithm known for general metrics has been Christofides's algorithm with approximation factor of 3 2 , even though the so-called Held-Karp LP relaxation of the problem is conjectured to have the integrality… (More)

We present a randomized algorithm for finding maximum matchings in planar graphs in time O(n ω/2), where ω is the exponent of the best known matrix multiplication algorithm. Since ω < 2.38, this algorithm breaks through the O(n 1.5) barrier for the matching problem. This is the first result of this kind for general planar graphs. We also present an… (More)

- Marcin Mucha
- SODA
- 2013

In the shortest superstring problem, we are given a set of strings {s 1 ,. .. , s k } and want to find a string that contains all s i as substrings and has minimum length. This is a classical problem in approximation and the best known approximation factor is 2 1 2 , given by Sweedyk [19] in 1999. Since then no improvement has been made, howerever two other… (More)

We present the first 7/8-approximation algorithm for the maximum traveling salesman problem with triangle inequality. Our algorithm is deterministic. This improves over both the random-ized algorithm of Hassin and Rubinstein [3] with expected approximation ratio of 7/8 − O(n −1/2) and the deterministic (7/8 − O(n −1/3))-approximation algorithm of Chen and… (More)

- Marcin Mucha
- 2007

We describe a new approximation algorithm for the asymmetric maximum traveling salesman problem (ATSP) with triangle inequality. Our algorithm achieves approximation factor 35/44 which improves on the previous 31/40 factor of Bläser, Ram and Sviridenko [1].

The Travelling Salesman Problem is one the most fundamental and most studied problems in approximation algorithms. For more than 30 years, the best algorithm known for general metrics has been Christofides's algorithm with approximation factor of 3 2 , even though the so-called Held-Karp LP relaxation of the problem is conjectured to have the integrality… (More)