Marcin Mucha

Maximum matchings via Gaussian elimination

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 implementation in time O(n/sup 3/) and the only non-trivial element of the O(n/sup w/) bipartite matching algorithm is the fast matrix multiplication algorithm. Our results resolve a long-standing open question of whether Lovaszs randomized technique of testing graphs for perfect matching in time O(n/sup w/) can be extended to an algorithm that actually constructs a perfect matching.

Maximum Matchings in Planar Graphs via Gaussian Elimination

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.

Maximum matchings in planar graphs via gaussian elimination

We present a randomized algorithm for finding maximum matchings in planar graphs in timeO(nω/2), whereω is the exponent of the best known matrix multiplication algorithm. Sinceω<2.38, this algorithm breaks through theO(n1.5) barrier for the matching problem. This is the first result of this kind for general planar graphs. We also present an algorithm for generating perfect matchings in planar graphs uniformly at random usingO(nω/2) arithmetic operations. Our algorithms are based on the Gaussian elimination approach to maximum matchings introduced in [16].

Fast Dynamic Transitive Closure with Lookahead

AbstractIn 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(nω(1,1,ε)−ε) time given a lookahead of nε operations, where ω(1,1,ε) is the exponent of multiplication of n×n matrix by n×nε matrix. For ε≤0.294 we obtain an algorithm with queries and updates in O(n2−ε) time, whereas for ε=1 the time is O(nω−1). This is essentially optimal as it implies an O(nω) algorithm for boolean matrix multiplication. We also consider the offline transitive closure in planar graphs. For this problem, we show an algorithm that requires

On Problems Equivalent to (min, +)-Convolution.

O(n^{\frac{\omega}{2}})

Deterministic 7/8-Approximation for the Metric Maximum TSP

time to process

Dynamic Beats Fixed: On Phase-Based Algorithms for File Migration.

n^{\frac{1}{2}}

No-Wait Flowshop Scheduling Is as Hard as Asymmetric Traveling Salesman Problem

operations. We also show a modification of these algorithms that gives faster amortized queries. Finally, we give faster algorithms for restricted type of updates, so called element updates. All of the presented algorithms are randomized with one-sided error.All our algorithms are based on dynamic algorithms with lookahead for matrix inverse, which are of independent interest.

Catch them if you can: how to serve impatient users

In the recent years, significant progress has been made in explaining apparent hardness of improving over naive solutions for many fundamental polynomially solvable problems. This came in the form of conditional lower bounds -- reductions from a problem assumed to be hard. These include 3SUM, All-Pairs Shortest Paths, SAT and Orthogonal Vectors, and others. In the (min,+)-convolution problem, the goal is to compute a sequence c, where c[k] = min_i a[i]+b[k-i], given sequences a and b. This can easily be done in O(n^2) time, but no O(n^{2-eps}) algorithm is known for eps > 0. In this paper we undertake a systematic study of the (min,+)-convolution problem as a hardness assumption. As the first step, we establish equivalence of this problem to a group of other problems, including variants of the classic knapsack problem and problems related to subadditive sequences. The (min,+)-convolution has been used as a building block in algorithms for many problems, notably problems in stringology. It has also already appeared as an ad hoc hardness assumption. We investigate some of these connections and provide new reductions and other results.

A 9 k kernel for nonseparating independent set in planar graphs

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 randomized algorithm of Hassin and Rubinstein [2] with expected approximation ratio of 7/8 i¾? O(ni¾? 1/2) and the deterministic (7/8 i¾? O(ni¾? 1/3))-approximation algorithm of Chen and Nagoya [1]. In the new algorithm, we extend the approach of processing local configurations using so-called loose-ends, which we introduced in [4].