Seminars and Colloquia by Series

First-Fit is Linear on (r+s)-free Posets

Series
Graph Theory Seminar
Time
Thursday, March 10, 2011 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Kevin MilansUniversity of South Carolina
First-Fit is an online algorithm that partitions the elements of a poset into chains. When presented with a new element x, First-Fit adds x to the first chain whose elements are all comparable to x. In 2004, Pemmaraju, Raman, and Varadarajan introduced the Column Construction Method to prove that when P is an interval order of width w, First-Fit partitions P into at most 10w chains. This bound was subsequently improved to 8w by Brightwell, Kierstead, and Trotter, and independently by Narayanaswamy and Babu. The poset r+s is the disjoint union of a chain of size r and a chain of size s. A poset is an interval order if and only if it does not contain 2+2 as an induced subposet. Bosek, Krawczyk, and Szczypka proved that if P is an (r+r)-free poset of width w, then First-Fit partitions P into at most 3rw^2 chains and asked whether the bound can be improved from O(w^2) to O(w). We answer this question in the affirmative. By generalizing the Column Construction Method, we show that if P is an (r+s)-free poset of width w, then First-Fit partitions P into at most 8(r-1)(s-1)w chains. This is joint work with Gwena\"el Joret.

Lecture series on the disjoint paths algorithm

Series
Graph Theory Seminar
Time
Monday, March 7, 2011 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 168
Speaker
Paul WollanSchool of Mathematics, Georgia Tech and University of Rome
The k-disjoint paths problem takes as input a graph G and k pairs of vertices (s_1, t_1),..., (s_k, t_k) and determines if there exist internally disjoint paths P_1,..., P_k such that the endpoints of P_i are s_i and t_i for all i=1,2,...,k. While the problem is NP-complete when k is allowed to be part of the input, Robertson and Seymour showed that there exists a polynomial time algorithm for fixed values of k. The existence of such an algorithm is the major algorithmic result of the Graph Minors series. The original proof of Robertson and Seymour relies on the whole theory of graph minors, and consequently is both quite technical and involved. Recent results have dramatically simplified the proof to the point where it is now feasible to present the proof in its entirety. This seminar series will do just that, with the level of detail aimed at a graduate student level.

Lecture series on the disjoint paths algorithm

Series
Graph Theory Seminar
Time
Monday, February 28, 2011 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 168
Speaker
Paul WollanSchool of Mathematics, Georgia Tech and University of Rome
The k-disjoint paths problem takes as input a graph G and k pairs of vertices (s_1, t_1),..., (s_k, t_k) and determines if there exist internally disjoint paths P_1,..., P_k such that the endpoints of P_i are s_i and t_i for all i=1,2,...,k. While the problem is NP-complete when k is allowed to be part of the input, Robertson and Seymour showed that there exists a polynomial time algorithm for fixed values of k. The existence of such an algorithm is the major algorithmic result of the Graph Minors series. The original proof of Robertson and Seymour relies on the whole theory of graph minors, and consequently is both quite technical and involved. Recent results have dramatically simplified the proof to the point where it is now feasible to present the proof in its entirety. This seminar series will do just that, with the level of detail aimed at a graduate student level.

Lecture series on the disjoint paths algorithm

Series
Graph Theory Seminar
Time
Monday, February 21, 2011 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 168
Speaker
Paul WollanMath, GT and University of Rome
The k-disjoint paths problem takes as input a graph G and k pairs of vertices (s_1, t_1),..., (s_k, t_k) and determines if there exist internally disjoint paths P_1,..., P_k such that the endpoints of P_i are s_i and t_i for all i=1,2,...,k. While the problem is NP-complete when k is allowed to be part of the input, Robertson and Seymour showed that there exists a polynomial time algorithm for fixed values of k. The existence of such an algorithm is the major algorithmic result of the Graph Minors series. The original proof of Robertson and Seymour relies on the whole theory of graph minors, and consequently is both quite technical and involved. Recent results have dramatically simplified the proof to the point where it is now feasible to present the proof in its entirety. This seminar series will do just that, with the level of detail aimed at a graduate student level.

Lecture series on the disjoint paths algorithm

Series
Graph Theory Seminar
Time
Monday, February 14, 2011 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 168
Speaker
Paul WollanSchool of Mathematics, Georgia Tech and University of Rome
The k-disjoint paths problem takes as input a graph G and k pairs of vertices (s_1, t_1),..., (s_k, t_k) and determines if there exist internally disjoint paths P_1,..., P_k such that the endpoints of P_i are s_i and t_i for all i=1,2,...,k. While the problem is NP-complete when k is allowed to be part of the input, Robertson and Seymour showed that there exists a polynomial time algorithm for fixed values of k. The existence of such an algorithm is the major algorithmic result of the Graph Minors series. The original proof of Robertson and Seymour relies on the whole theory of graph minors, and consequently is both quite technical and involved. Recent results have dramatically simplified the proof to the point where it is now feasible to present the proof in its entirety. This seminar series will do just that, with the level of detail aimed at a graduate student level.

The minimum number of edges in color-critical graphs

Series
Graph Theory Seminar
Time
Thursday, February 10, 2011 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Chun-Hung LiuMath, GT
A graph is k-critical if it is not (k-1)-colorable but every proper subgraph is. In 1963, Gallai conjectured that every k-critical graph G of order n has at least (k-1)n/2 + (k-3)(n-k)/(2k-2) edges. The currently best known results were given by Krivelevich for k=4 and 5, and by Kostochka and Stiebitz for k>5. When k=4, Krivelevich's bound is 11n/7, and the bound in Gallai's conjecture is 5n/3 -2/3. Recently, Farzad and Molloy proved Gallai's conjecture for k=4 under the extra condition that the subgraph induced by veritces of degree three is connected. We will review the proof given by Krivelevich, and the proof given by Farzad and Molloy in the seminar.

Lecture series on the disjoint paths algorithm

Series
Graph Theory Seminar
Time
Monday, February 7, 2011 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 168
Speaker
Paul WollanGT, Math and University of Rome
The k-disjoint paths problem takes as input a graph G and k pairs of vertices (s_1, t_1),..., (s_k, t_k) and determines if there exist internally disjoint paths P_1,..., P_k such that the endpoints of P_i are s_i and t_i for all i=1,2,...,k. While the problem is NP-complete when k is allowed to be part of the input, Robertson and Seymour showed that there exists a polynomial time algorithm for fixed values of k. The existence of such an algorithm is the major algorithmic result of the Graph Minors series. The original proof of Robertson and Seymour relies on the whole theory of graph minors, and consequently is both quite technical and involved. Recent results have dramatically simplified the proof to the point where it is now feasible to present the proof in its entirety. This seminar series will do just that, with the level of detail aimed at a graduate student level.

Decomposing an infinite matroid into its 3-connected minors

Series
Graph Theory Seminar
Time
Thursday, February 3, 2011 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Luke PostleMath, GT
This will be a continuation from last week. We extend the theory of infinite matroids recently developed by Bruhn et al to a well-known classical result in finite matroids while using the theory of connectivity for infinitematroids of Bruhn and Wollan. We prove that every infinite connected matroid M determines a graph-theoretic decomposition tree whose vertices correspond to minors of M that are3-connected, circuits, or cocircuits, and whose edges correspond to 2-separations of M. Tutte and many other authors proved such a decomposition for finite graphs; Cunningham andEdmonds proved this for finite matroids and showed that this decomposition is unique if circuits and cocircuits are also allowed. We do the same for infinite matroids. The knownproofs of these results, which use rank and induction arguments, do not extend to infinite matroids. Our proof avoids such arguments, thus giving a more first principles proof ofthe finite result. Furthermore, we overcome a number of complications arising from the infinite nature of the problem, ranging from the very existence of 2-sums to proving the treeis actually graph-theoretic.

Lecture series on the disjoint paths algorithm

Series
Graph Theory Seminar
Time
Monday, January 31, 2011 - 14:05 for 1 hour (actually 50 minutes)
Location
Skiles 168
Speaker
Paul WollanGT, Math and University of Rome
The k-disjoint paths problem takes as input a graph G and k pairs of vertices (s_1, t_1),..., (s_k, t_k) and determines if there exist internally disjoint paths P_1,..., P_k such that the endpoints of P_i are s_i and t_i for all i=1,2,...,k. While the problem is NP-complete when k is allowed to be part of the input, Robertson and Seymour showed that there exists a polynomial time algorithm for fixed values of k. The existence of such an algorithm is the major algorithmic result of the Graph Minors series. The original proof of Robertson and Seymour relies on the whole theory of graph minors, and consequently is both quite technical and involved. Recent results have dramatically simplified the proof to the point where it is now feasible to present the proof in its entirety. This seminar series will do just that, with the level of detail aimed at a graduate student level.

Traveling Salesman Problems

Series
Graph Theory Seminar
Time
Thursday, December 2, 2010 - 12:05 for 1 hour (actually 50 minutes)
Location
Skiles 114
Speaker
Bill CookISyE, GT
We discuss open research questions surrounding the traveling salesman problem. A focus will be on topics having potential impact on the computational solution of large-scale problem instances.

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