KAIST Discrete Math Seminar

Belief propagation (BP) is a popular message-passing algorithm for computing a maximum-a-posteriori assignment in a graphical model. It has been shown that BP can solve a few classes of Linear Programming (LP) formulations to combinatorial optimization problems including maximum weight matching and shortest path. However, it has been not clear what extent these results can be generalized to. In this talk, I first present a generic criteria that BP converges to the optimal solution of given LP, and show that it is satisfied in LP formulations associated to many classical combinatorial optimization problems including maximum weight perfect matching, shortest path, network flow, traveling salesman, cycle packing and vertex cover. Using the criteria, we also construct the exact distributed algorithm, called Blossom-BP, solving the maximum weight matching problem over arbitrary graphs. In essence, Blossom-BP offers a distributed version of the celebrated Blossom algorithm (Edmonds 1965) jumping at once over many sub-steps of the Blossom-V (most recent implementation of the Blossom algorithm due to Kolmogorov, 2011). Finally, I report the empirical performance of BP for solving large-scale combinatorial optimization problems. This talk is based on a series of joint works with Sungsoo Ahn (KAIST), Michael Chertkov (LANL), Inho Cho (KAIST), Dongsu Han (KAIST) and Sejun Park (KAIST).

The celebrated theorem of Kuratowski characterizes those graphs that require at least one crossing when drawn in the plane, by exhibiting the complete list of topologically-minimal such graphs. As it is very well known, this list contains precisely two such 1-crossing-critical graphs: K₅ and K_3,3. The analogous problem of producing the complete list of 2-crossing-critical graphs is significantly harder. In fact, in 1987, Kochol exhibited an infinite family of 3-connected 2-crossing-critical graphs. In the talk, I will discuss the current status of the problem, including our recent work, which includes: (i) a description of all 3-connected 2-crossing-critical graphs that contain a subdivision of the Möbius Ladder V₁₀; (ii) a proof that there are only finitely many 3-connected 2-crossing-critical graphs not containing a subdivision of V₁₀; (iii) a description of all 2-crossing-critical graphs that are not 3-connected; and (iv) a recipe on how to construct all 3-connected 2-crossing-critical graphs that do not contain a subdivision of V₈.
This is a joint work with Drago Bokal, Bruce Richter, and Gelasio Salazar.

The clique-width parameter provides a rough measure of the complexity of structure in (classes of) graphs. A well-known result of Courcelle, Makowsky and Rotics shows that many problems on graphs which are NP-hard in general can be solved in polynomial time in any class of graphs of bounded clique-width. Unlike the better-known treewidth graph parameter, clique-width respects the induced subgraph ordering, and in particular it can handle dense graphs. However, also unlike treewidth there is no known characterisation of the minimal classes of graphs which have unbounded clique-width.
In this talk, I will survey a number of results and techniques for studying the interface between bounded and unbounded clique-width. Of particular interest are insights from the combinatorial study of permutations (“permutation patterns”), which has brought to light several more minimal graph classes with unbounded clique-width, and also suggests that a restricted version of the parameter, called linear clique-width, often appears to characterise the interface. Time-permitting, I will also discuss recent developments and open problems in the relationship between clique-width and well-quasi-ordering.

In Bondy and Murty’s book the authors wrote that Tutte conjectured the wheels have the fewest spanning trees out of all 3-connected graphs on fixed number of vertices. The statement can easily be shown to be false and the corrected version, where we fix the number of edges and consider only the planar graphs, were also found to be false. We prove that if we consider the cycles instead of spanning trees then the wheels are indeed extremal. We also establish a lower bound for the number of spanning trees and suggest the prisms as possible extremal graphs.

A k-colouring of a graph G is a partition of V(G) into k independent sets. The chromatic number χ(G) is defined as the smallest k so that G has a k-colouring. Alternatively, we can view colourings as homomorphisms to complete graphs, and define χ(G) to be the smallest k so that there is a homomorphism from G to K_k. The variants of χ(G) (for example, fractional chromatic number) are defined by replacing the complete graph K_k with a representative of a different class of graphs (for example, Kneser graphs).

In this talk, we will consider the vector chromatic number of a graph. A vector colouring of a G is a map from V(G) to vectors in R^m (for some m). The goal is to map adjacent vertices to vectors that are far apart. We will look at representations of a graph on its least eigenspace as examples of vector colourings. For distance-regular graphs, these colourings are optimal. Furthermore, we give a method for determining if these colourings are unique. This leads to proofs that certain classes of distance-regular graphs are all cores.