Efficient Algorithms for Finding Maximal Matching in Graphs

This paper surveys the techniques used for designing the most efficient algorithms for finding a maximum cardinality or weighted matching in (general or bipartite) graphs. It also lists some open problems concerning possible improvements in existing algorithms and the existence of fast parallel algorithms for these problems

The Distance and Median Problems in the Single-Cut-Or-Join Model with Single-Gene Duplications

Background. In the field of genome rearrangement algorithms, models accounting for gene duplication lead often to hard problems. For example, while computing the pairwise distance is tractable in most duplication-free models, the problem is NP-complete for most extensions of these models accounting for duplicated genes. Moreover, problems involving more than two genomes, such as the genome median and the Small Parsimony problem, are intractable for most duplication-free models, with some exceptions, for example the Single-Cut-or-Join (SCJ) model. Results. We introduce a variant of the SCJ distance that accounts for duplicated genes, in the context of directed evolution from an ancestral genome to a descendant genome where orthology relations between ancestral genes and their descendant are known. Our model includes two duplication mechanisms: single-gene tandem duplication and the creation of single-gene circular chromosomes. We prove that in this model, computing the directed distance and a parsimonious evolutionary scenario in terms of SCJ and single-gene duplication events can be done in linear time. We also show that the directed median problem is tractable for this distance, while the rooted median problem, where we assume that one of the given genomes is ancestral to the median, is NP-complete. We also describe an Integer Linear Program for solving this problem. We evaluate the directed distance and rooted median algorithms on simulated data. Conclusion. Our results provide a simple genome rearrangement model, extending the SCJ model to account for single-gene duplications, for which we prove a mix of tractability and hardness results. For the NP-complete rooted median problem, we design a simple Integer Linear Program. Our publicly available implementation of these algorithms for the directed distance and median problems allow to solve efficiently these problems on large instances

Efficient Algorithms for Large Scale Network Problems

In recent years, the growing scale of data has renewed our understanding of what is an efficient algorithm and poses many essential challenges for the algorithm designers. This thesis aims to improve our understanding of many algorithmic problems in this context. These include problems in communication complexity, matching theory, and approximate query processing for database systems. We first study the fundamental and well-known question of {SetIntersection} in communication complexity. We give a result that incorporates the error probability as an independent parameter into the classical trade-off between round complexity and communication complexity. We show that any $r$-round protocol that errs with error probability $2^{-E}$ requires $Omega(Ek^{1/r})$ bits of communication. We also give several almost matching upper bounds. In matching theory, we first study several generalizations of the ordinary matching problem, namely the $f$-matching and $f$-edge cover problem. We also consider the problem of computing a minimum weight perfect matching in a metric space with moderate expansion. We give almost linear time approximation algorithms for all these problems. Finally, we study the sample-based join problem in approximate query processing. We present a result that improves our understanding of the effectiveness and limitations in using sampling to approximate join queries and provides a guideline for practitioners in building AQP systems from a theory perspective.PHDComputer Science & EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/155263/1/hdawei_1.pd