## Archive for the ‘2014’ Category

### O-Joung Kwon, Upper bounds on the size of obstructions for graphs of linear rank-width at most k

Monday, November 17th, 2014
Upper bounds on the size of obstructions for graphs of linear rank-width at most k
2014/12/02 Tuesday 4PM-5PM
Room 1409
Graph layout problems are a class of optimization problems whose goal is to find a linear ordering of an input graph in such a way that a certain objective function is optimized. The matrix rank function has been studied as an objective function. The linear rank-width of a graph G is the minimum integer k such that G admits a linear ordering $$v_1, v_2, \ldots , v_n$$ satisfying that the maximum over all values $$\operatorname{rank}A_G[\{v_1, v_2, \ldots, v_t\}, \{v_{t+1}, \ldots, v_n\}]$$ is k, where $$A_G$$ is the adjacency matrix of G and the rank is computed over the binary field.

In this talk, we present a result that for every graph G that is vertex-minor minimal with the property having linear rank-width larger than p, the number of vertices in G is at most doubly exponential in $$\mathcal{O}(p)$$. The number of vertex-minor obstructions for linear rank-width at most p is of interest because the only known fixed parameter tractable algorithm to test whether linear rank-width is at most p is using the finiteness of the number of forbidden vertex-minor obstructions. Our result gives an upper bound of the complexity on this algorithm. Our basic tools are the algebraic operations on labelled graphs introduced by Kanté and Rao, and we extend the notion of vertex-minors in our purpose. This is joint work with Mamadou Moustapha Kanté.

### Hana Kim, Enumeration of symmetric hex trees and the related polynomials

Tuesday, November 11th, 2014
Enumeration of symmetric hex trees and the related polynomials
Hana Kim
NIMS
2014/11/25 Tuesday 4:30PM-5:30PM
Room 1409
A hex tree is an ordered tree of which each vertex has updegree 0, 1, or 2, and an edge from a vertex of updegree 1 is either left, median, or right. In this talk, we introduce an enumeration problem of symmetric hex trees and describe a bijection between symmetric hex trees and a certain class of supertrees. Some algebraic properties of the polynomials arising in this procedure also will be discussed.

### Bryan Wilkinson, Generalized Davenport-Schinzel Sequences: Regaining Linearity

Monday, November 10th, 2014
Generalized Davenport-Schinzel Sequences: Regaining Linearity
Bryan Wilkinson
Aarhus University, Danmark
2014/11/18 Tuesday 4PM-5PM
Room 1409
We prove the linearity (of the lengths) of some generalized Davenport-Schinzel sequences. Standard Davenport-Schinzel sequences of order 2 (avoiding abab) are linear, while those of order 3 (avoiding ababa) and higher can be superlinear. Our goal is to determine what pattern(s), in addition to ababa, must be forbidden to regain linearity. This work is motivated by an intriguing open problem: does the lower envelope of any set of degree 3 polynomials have linear complexity?

### (CS Colloquium) Jeong Han Kim, How to Find Counterfeit Coins? An Algorithmic Version

Sunday, November 9th, 2014

FYI (CS Colloquium)

How to Find Counterfeit Coins?
An Algorithmic Version
2014/11/17 Monday 4PM – 5:30PM
E3-1 CS Bldg. Room 1501
In this talk, we consider a well-known combinatorial search problem. Suppose that there are n identical looking coins and some of them are counterfeit. The weights of all authentic coins are the same and known a
priori. The weights of counterfeit coins vary but different from the weight of an authentic coin. Without loss of generality, we may assume the weight of authentic coins is 0. The problem is to find all counterfeit coins by weighing (queries) sets of coins on a spring scale. Finding the optimal number of queries is difficult even when there are only 2 counterfeit coins. We introduce a polynomial time randomized algorithm to find all counterfeit coins when the number of them is known to be at most $$m \geq 2$$ and the weight $$w(c)$$ of each counterfeit coin c satisfies $$\alpha \leq |w(c)| \leq \beta$$ for fixed constants $$\alpha, \beta>0$$. The query complexity of the algorithm is $$O(\frac{m \log n }{\log m})$$, which is optimal up to a constant factor. The algorithm uses, in part, random walks. The algorithm may be generalized to find all edges of a hidden weighted graph using a similar type of queries. This graph finding algorithm has various applications including DNA sequencing.

### JiYoon Jung, The topology of restricted partition posets

Saturday, October 18th, 2014
The topology of restricted partition posets
2014/11/04 Tuesday 4PM-5PM
Room 1409
For each composition $$\vec{c}$$ we show that the order complex of the poset of pointed set partitions $$\Pi_{\vec{c}}^{\bullet}$$ is a wedge of spheres of the same dimension with the multiplicity given by the number of permutations with descent composition $$\vec{c}$$. Furthermore, the action of the symmetric group on the top homology is isomorphic to the Specht module $$S^B$$ where $$B$$ is a border strip associated to the composition. We also study the filter of pointed set partitions generated by a knapsack integer partition and show the analogous results on homotopy type and action on the top homology.