February 28th, 2014
F-Matchings in a tree
2014/04/28 Monday 4PM-5PM
Given trees F and T, a F-matching is a collection of disjoint copies F in T. Generalizing results from Wagner (2009), Alon et al (2011) proved that the number of F-matchings for fixed F in a random tree T whose size tends to infinity is a.a.s. a multiple of any given integer m>0. We will discuss inverse and extremal problems on the number of F-matchings.
February 27th, 2014
Graph polynomials from simple graph sequences
2014/04/14 Monday 4PM-5PM
The chromatic polynomial evaluated at a positive integer n is equal to the number of homomorphisms to the complete graph on n vertices. Many important graph polynomials are likewise determined by counting homomorphisms to a sequence of (multi) graphs, such as the Tutte polynomial and the independence polynomial. We give a powerful construction that produces graph polynomials in this way from sequences of simple graphs. It uses just two fundamental operations (disjoint union and interpretations of relational structures) and starts from a simple set of building blocks (coloured transitive tournaments).This will be illustrated by many examples, some of whose combinatorial properties have been well explored (such as that of the chromatic polynomial), but most of which are new. This is joint work with Jarik Nesetril and Patrice Ossona de Mendez.
February 26th, 2014
Pfaffian Sum formula of the double Schubert polynomials for the symplectic Grassmannians
2014/04/08 Tuesday 4PM-5PM
It is well-known that the ring of symmetric polynomials has a special basis, consisting of Schur polynomials and being indexed by Young diagrams. There underlies the representation theory of general linear groups and the combinatorics of Young diagrams. The geometry behind it is the geometry of Schubert varieties of Grassmannians of subspaces in a complex vector space. If we change the geometric setup to the case of Grassmannians of isotropic subspaces of a symplectic vector space, we have analogous functions called Schur Q-functions and their slight generalization. I will explain an explicit closed formula to express those functions as sums of Pfaffians and also the geometry and combinatorics behind it. If time allows, I will mention the technique of how we obtained the formula, that involves the so-called divided difference operators. This is a joint work with Takeshi Ikeda.
February 25th, 2014
A conditional lower bound for computing the Gromov hyperbolicity of a discrete metric space
2014/03/31 Monday 4PM-5PM
We show that the Gromov hyperbolicity of a discrete metric space at a fixed base-point cannot be computed in O(n2.05) time, unless there exists a faster algorithm for (max,min) matrix multiplication algorithm than is currently known.
February 25th, 2014
Determining the Winner of a Dodgson Election is Hard
2014/03/25 Tuesday 4PM -5PM
Computational social choice is a growing discipline at the interface of social choice theory and computer science. It is concerned with the application of computational techniques to the study of social choice mechanisms, and with the integration of social choice paradigms into computing. Several problems have been investigated in the framework of parameterized complexity. This talk will describe the election scheme of Charles Dodgson. In 1876 the mathematician Charles Dodgson (better known as Lewis Carroll) formulated a rule that defines the winner of an election even if there is no Condorcet winner. A candidate who beats every other candidate in pairwise comparison is said to be a Condorcet winner. Unfortunately, an election may have a cyclic structure (candidate a beats b, candidate b beats c and c beats a), and therefore no candidate who beats all others in pairwise comparison. The Dodgson Score measures how close a candidate is to being a Condorcet winner. Candidates with a minimum Dodgson Score are the election winners. As are many election methods, Dodgson Score is NP-hard. This talk discusses the complexity of Dodgson Score in the parameterized framework. We give a reduction from Multi-Colored Clique to show that Dodgson Score, parameterized by the number of votes, is W-hard. When parameterized by the number of swaps, Dodgson Score is FPT, but we show by polynomial parameter transformation that it has no polynomial kernel.