Bisimplicial edges in bipartite graphs

Bisimplicial edges in bipartite graphs are closely related to pivots in Gaussian elimination that avoid turning zeroes into non-zeroes. We present a new deterministic algorithm to nd such edges in bipartite graphs. The expected time complexity of our new algorithm is $O(n^2 \log n)$ on random bipartite graphs in which each edge is present with a fixed probability p, a polynomial improvement over the fastest algorithm found in the existing literature

Recognizing sparse perfect elimination bipartite graphs

When applying Gaussian elimination to a sparse matrix, it is desirable to avoid turning zeros into non-zeros to preserve the sparsity. The class of perfect elimination bipartite graphs is closely related to square matrices that Gaussian elimination can be applied to without turning any zero into a non-zero. Existing literature on the recognition of this class and finding suitable pivots mainly focusses on time complexity. For $n \times n$ matrices with m non-zero elements, the currently best known algorithm has a time complexity of $O(n^3/\log n)$. However, when viewed from a practical perspective, the space complexity also deserves attention: it may not be worthwhile to look for a suitable set of pivots for a sparse matrix if this requires $\Omega(n^2)$ space. We present two new algorithms for the recognition of sparse instances: one with a $O(n m)$ time complexity in $\Theta(n^2)$ space and one with a $O(m^2)$ time complexity in $\Theta(m)$ space. Furthermore, if we allow only pivots on the diagonal, our second algorithm can easily be adapted to run in time $O(n m)$

Some recent results in the analysis of greedy algorithms for assignment problems

We survey some recent developments in the analysis of greedy algorithms for assignment and transportation problems. We focus on the linear programming model for matroids and linear assignment problems with Monge property, on general linear programs, probabilistic analysis for linear assignment and makespan minimization, and on-line algorithms for linear and non-linear assignment problems

On the Parameterized Complexity of Biclique Cover and Partition

Given a bipartite graph G, we consider the decision problem called BicliqueCover for a fixed positive integer parameter k where we are asked whether the edges of G can be covered with at most k complete bipartite subgraphs (a.k.a. bicliques). In the BicliquePartition problem, we have the additional constraint that each edge should appear in exactly one of the k bicliques. These problems are both known to be NP-complete but fixed parameter tractable. However, the known FPT algorithms have a running time that is doubly exponential in k, and the best known kernel for both problems is exponential in k. We build on this kernel and improve the running time for BicliquePartition to O*(2^{2k^2+k*log(k)+k}) by exploiting a linear algebraic view on this problem. On the other hand, we show that no such improvement is possible for BicliqueCover unless the Exponential Time Hypothesis (ETH) is false by proving a doubly exponential lower bound on the running time. We achieve this by giving a reduction from 3SAT on n variables to an instance of BicliqueCover with k=O(log(n)). As a further consequence of this reduction, we show that there is no subexponential kernel for BicliqueCover unless P=NP. Finally, we point out the significance of the exponential kernel mentioned above for the design of polynomial-time approximation algorithms for the optimization versions of both problems. That is, we show that it is possible to obtain approximation factors of n/log(n) for both problems, whereas the previous best approximation factor was n/sqrt(log(n))

Algorithmic aspects of bipartite graphs

We generalize previous work done by Donald J. Rose and Robert E. Tarjan [2], who developed efficient algorithms for use on directed graphs. This paper considers an edge elimination process on bipartite graphs, presenting several theorems which lead to an algorithm for computing the minimal fill-in of a given ordered graph