Suil O (오수일), Interlacing families and the Hermitian spectral norm of digraphs

April 20th, 2016
Interlacing families and the Hermitian spectral norm of digraphs
Suil O (오수일)
Department of Mathematics, Simon Fraser University, Burnaby, B.C., Canada
2016/6/1 Wed 4PM-5PM
Recently, Marcus, Spielman, and Srivastava proved the existence of infinite families of bipartite Ramanujan graphs of every degree at least 3 by using the method of interlacing families of polynomials. In this talk, we apply their method to prove that for any connected graph G, there exists an orientation of G such that the spectral radius of the corresponding Hermitian adjacency matrix is at most that of the universal cover of G.

(Colloquium) András Sebő, Classical tools and recent advances in combinatorial optimization

April 4th, 2016

FYI: Colloquium of Dept. of Mathematical Sciences.

Classical tools and recent advances in combinatorial optimization
András Sebő
CNRS, Laboratoire G-SCOP, Université Grenoble-Alpes, France
2016/04/28 Thu 4:15PM-5:15PM (Room 1501, Bldg. E6-1)
Combinatorial optimization has recently discovered new usages of probability theory, information theory, algebra, semidefinit programming, etc. This allows addressing the problems arising in new application areas such as the management of very large networks, which require new tools. A new layer of results make use of several classical methods at the same time, in new ways, combined with newly developed arguments.
After a brief panorama of this evolution, I would like to show the new place of the best-known, classical combinatorial optimization tools in this jungle: matroids, matchings, elementary probabilities, polyhedra, linear programming.
More concretely, I try to demonstrate on the example of the Travelling Salesman Problem, how strong meta-methods may predict possibilities, and then be replaced by better suited elementary methods. The pillars of combinatorial optimization such as matroid intersection, matchings, T-joins, graph connectivity, used in parallel with elements of freshmen’s probabilities, and linear programming, appropriately merged with newly developed ideas tailored for the problems, may not only replace difficult generic methods, but essentially improve the results. I would like to show how this has happened with various versions of the TSP problem in the past years (see results of Gharan, Saberi, Singh, Mömke and Svensson, and several recent results of Anke van Zuylen, Jens Vygen and the speaker), essentially improving the approximation ratios of algorithms.

András Sebő, The Salesman’s Improved Paths

April 4th, 2016
The Salesman’s Improved Paths
András Sebő
CNRS, Laboratoire G-SCOP, Université Grenoble-Alpes, France
2016/05/04 Wed 4PM-5PM
A new algorithm will be presented for the path tsp, with an improved analysis and ratio. After the starting idea of deleting some edges of Christofides’ trees, we do parity correction and eventual reconnection, taking the salesman to travel through a linear program determining the conditional probabilities for some of his choices; through matroid partition of a set of different matroids for a better choice of his initial spanning trees; and through some other adventures and misadventures.
The proofs proceed by global and intuitively justified steps, where the trees do not hide the forests.
One more pleasant piece of news is that we get closer to the conjectured approximation ratio of 3/2, and a hopefully last misadventure before finishing up this problem is that we still have to add 1/34 to this ratio, and also for the integrality gap. (The previous result was 8/5 with slight improvements.)
This is joint work with Anke van Zuylen.

Hyung-Chan An (안형찬), LP-based Algorithms for Capacitated Facility Location

March 12th, 2016
LP-based Algorithms for Capacitated Facility Location
Hyung-Chan An (안형찬)
Department of Computer Science, Yonsei University, Seoul
2016/4/6 Wed 4PM-5PM (Room 3433 of Bldg. E6-1)
Linear programming has played a key role in the study of algorithms for combinatorial optimization problems. In the field of approximation algorithms, this is well illustrated by the uncapacitated facility location problem. A variety of algorithmic methodologies, such as LP-rounding and primal-dual method, have been applied to and evolved from algorithms for this problem. Unfortunately, this collection of powerful algorithmic techniques had not yet been applicable to the more general capacitated facility location problem. In fact, all of the known algorithms with good performance guarantees were based on a single technique, local search, and no linear programming relaxation was known to efficiently approximate the problem.
In this paper, we present a linear programming relaxation with constant integrality gap for capacitated facility location. We demonstrate that the fundamental theories of multi-commodity flows and matchings provide key insights that lead to the strong relaxation. Our algorithmic proof of integrality gap is obtained by finally accessing the rich toolbox of LP-based methodologies: we present a constant factor approximation algorithm based on LP-rounding.
Joint work with Mohit Singh and Ola Svensson.

Ilkyoo Choi (최일규), Improper coloring graphs on surfaces

March 7th, 2016
Improper coloring graphs on surfaces
Ilkyoo Choi (최일규)
Department of Mathematical Sciences, KAIST
2016/3/9 Wed 4PM-5PM
A graph is (d1, …, dr)-colorable if its vertex set can be partitioned into r sets V1, …, Vr where the maximum degree of the graph induced by Vi is at most di for each i in {1, …, r}.
Given r and d1, …, dr, determining if a (sparse) graph is (d1, …, dr)-colorable has attracted much interest.
For example, the Four Color Theorem states that all planar graphs are 4-colorable, and therefore (0, 0, 0, 0)-colorable.
The question is also well studied for partitioning planar graphs into three parts.
For two parts, it is known that for given d1 and d2, there exists a planar graph that is not (d1, d2)-colorable.
Therefore, it is natural to study the question for planar graphs with girth conditions.
Namely, given g and d1, determine the minimum d2=d2(g, d1) such that planar graphs with girth g are (d1, d2)-colorable. We continue the study and ask the same question for graphs that are embeddable on a fixed surface.
Given integers k, γ, g we completely characterize the smallest k-tuple (d1, …, dk) in lexicographical order such that each graph of girth at least g that is embeddable on a surface of Euler genus γ is (d1, …, dk)-colorable.
All of our results are tight, up to a constant multiplicative factor for the degrees di depending on g.
In particular, we show that a graph embeddable on a surface of Euler genus γ is (0, 0, 0, K1(γ))-colorable and (2, 2, K2(γ))-colorable, where K1(γ) and K2(γ) are linear functions in γ.This talk is based on results and discussions with H. Choi, F. Dross, L. Esperet, J. Jeong, M. Montassier, P. Ochem, A. Raspaud, and G. Suh.