Piotr Sankowski

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.

Mathematical Foundations of Computer Science 2011

This volume constitutes the refereed proceedings of the 36th International Symposium on Mathematical Foundations of Computer Science, MFCS 2011, held in Warsaw, Poland, in August 2011. The 48 revised full papers presented together with 6 invited talks were carefully reviewed and selected from 129 submissions. Topics covered include algorithmic game theory, algorithmic learning theory, algorithms and data structures, automata, grammars and formal languages, bioinformatics, complexity, computational geometry, computer-assisted reasoning, concurrency theory, cryptography and security, databases and knowledge-based systems, formal specifications and program development, foundations of computing, logic in computer science, mobile computing, models of computation, networks, parallel and distributed computing, quantum computing, semantics and verification of programs, and theoretical issues in artificial intelligence.

Dynamic transitive closure via dynamic matrix inverse: extended abstract

We consider dynamic evaluation of algebraic functions such as computing determinant, matrix adjoint, matrix inverse and solving linear system of equations. We show that in the dynamic setup the above problems can be solved faster than evaluating everything from scratch. In the case when rows and columns of the matrix can change we show an algorithm that achieves O(n/sup 2/) arithmetic operations per update and O(1) arithmetic operations per query. Next, we describe two algorithms, with different tradeoffs, for updating the inverse and determinant when single entries of the matrix are changed. The fastest update for the first tradeoff is O(n/sup 1.575/) arithmetic operations per update and O(n/sup 0.575/) arithmetic operations per query. The second tradeoff gives O(n/sup 1.495/) arithmetic operations per update and O(n/sup 1.495/) arithmetic operations per query. We also consider the case when some number of columns or rows can change. We use dynamic determinant computations to solve the following problems in the dynamic setup: computing the number of spanning trees in a graph and testing if an edge in a graph is contained in some perfect matching. These are the first dynamic algorithms for these problems. Next, with the use of dynamic matrix inverse, we solve fully dynamic transitive closure in general directed graphs. The bounds on arithmetic operations for dynamic matrix inverse translate directly to time bounds for dynamic transitive closure. Thus we obtain the first known algorithm with O(n/sup 2/) worst-case update time and constant query time and two algorithms for transitive closure in general digraphs with subquadratic update and query times. Our algorithms for transitive closure are randomized with one-sided error. We also consider for the first time the case when the edges incident with a part of vertices of the graph can be changed.

Improved algorithms for min cut and max flow in undirected planar graphs

We study the min st-cut and max st-flow problems in planar graphs, both in static and in dynamic settings. First, we present an algorithm that given an undirected planar graph and two vertices s and t computes a min st-cut in O(n log log n) time. Second, we show how to achieve the same bound for the problem of computing a max st-flow in an undirected planar graph. These are the first algorithms breaking the O(n log n) barrier for those two problems, which has been standing for more than 25 years. Third, we present a fully dynamic algorithm maintaining the value of the min st-cuts and the max st-flows in an undirected plane graph (i.e., a planar graph with a fixed embedding): our algorithm is able to insert and delete edges and answer queries for min st-cut/max st-flow values between any pair of vertices s and t in O(n(2/3) log(8/3) n) time per operation. This result is based on a new dynamic shortest path algorithm for planar graphs which may be of independent interest. We remark that this is the first known non-trivial dynamic algorithm for min st-cut and max st-flow.

Single valued combinatorial auctions with budgets

We consider budget constrained combinatorial auctions where each bidder has a private value for each of the items in some subset of the items and an overall budget constraint. Such auctions capture adword auctions, where advertisers offer a bid for those adwords that (hopefully) target their intended audience, and advertisers also have budgets. It is known that even if all items are identical and all budgets are public it is not possible to be truthful and efficient. Our main result is a novel auction that runs in polynomial time, is incentive compatible, and ensures Pareto-optimality. The auction is incentive compatible with respect to the private valuations whereas the budgets and the sets of interest are assumed to be public knowledge. This extends the result of Dobzinski, Lavi and Nisan (FOCS 2008) for auctions of multiple identical items with bugets to single-valued combinatorial auctions and address one of the basic challenges on auctioning web ads (see Nisan et al, 2009, Google auctions for tv ads).

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.

Min st -Cut Oracle for Planar Graphs with Near-Linear Preprocessing Time

For an undirected

Interlayer exchange coupling in (Ga,Mn)As-based superlattices

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Algorithmic Applications of Baur-Strassen’s Theorem: Shortest Cycles, Diameter, and Matchings

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Maximum matchings in planar graphs via gaussian elimination

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