Marcin Mucha

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We present randomized algorithms for finding maximum matchings in general and bipartite graphs. Both algorithms have running time O(n/sup w/), where w is the exponent of the best known matrix multiplication algorithm. Since w < 2.38, these algorithms break through the O(n/sup 2.5/) barrier for the matching problem. They both have a very simple(More)
The Travelling Salesman Problem is one of 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 2 , even though the socalled 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 timeO(n ω/2), whereω is the exponent of the best known matrix multiplication algorithm. Sinceω<2.38, this algorithm breaks through theO(n 1.5) barrier for the matching problem. This is the first result of this kind for general planar graphs. We also present an algorithm for(More)
In the shortest superstring problem, we are given a set of strings {s1, . . . , sk} and want to find a string that contains all si 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 prob-<lb>lem with triangle inequality. Our algorithm is deterministic. This improves over both the random-<lb>ized algorithm of Hassin and Rubinstein [3] with expected approximation ratio of 7/8−O(n)<lb>and the deterministic (7/8 −O(n))-approximation algorithm of Chen and(More)
This is a tutorial on approximation algorithms for the Shortest Superstring Problem (SSP). My intention when writing it was to provide the foundations for actually doing research on this topic. In Section 2 I cover the basic definitions and observations and introduce the ”Greedy Conjecture”. In Section 3 I describe the standard framework for approximating(More)
In this paper we consider the problem of dynamic transitive closure with lookahead. We present a randomized one-sided error algorithm with updates and queries in O(n ω(1,1,ε)−ε ) time given a lookahead of n ε operations, where ω(1,1,ε) is the exponent of multiplication of n×n matrix by n×n ε matrix. For ε≤0.294 we obtain an algorithm with queries and(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 HeldKarp LP relaxation of the problem is conjectured to have the integrality gap(More)