KAIST Discrete Math Seminar

A phylogenetic (i.e evolutionary) tree can be interpreted as a compatible split system, that is a collection of bipartitions of a finite set X such that, for all four elements of X, there are no two bipartitions in the collection that induce different splits of those four elements into two pairs. Such a split of a 4-set into two 2-sets is called a quartet, and a split system is said to display a quartet, if there is at least one split in the system that induces this quartet. In phylogenetics, it is often useful to allow more general than compatible split systems, in order to display contradicting signals in the data or to find evidence for reticulate evolution. One natural such generalization are weakly compatible split systems, where for every 4-set at most two of the three possible quartets are allowed to be displayed. The split decomposition algorithm (implemented in the Splitstree software) is a successful tool to construct weakly compatible split systems from distance data. However, weakly compatible split systems are not as well understood as compatible ones. For example, maximal compatible split systems, i.e. compatible split systems which become incompatible whenever a new split is added, correspond to binary trees and display one quartet for every 4-set. In contrast, maximal weakly compatible split systems often display less than the two quartets per 4-set that are allowed by definition. Indeed there are examples where no quartet is displayed for almost all 4-sets. This leaves the question what is the minimum cardinality of maximal weakly compatible split systems for given cardinality of X.
In my talk I will introduce weakly compatible split systems and explain their relevance for phylogenetics, and I will present upper and lower bounds for the smallest number of quartets in maximal weakly compatible split systems.

We introduce a new class of rate one half codes, called complementary information set codes. A binary linear code of length 2n and dimension n is called a complementary information set code (CIS code for short) if it has two disjoint information sets. This class of codes contains self-dual codes as a subclass. It is connected to graph correlation immune functions of use in the security of hardware implementations of  cryptographic primitives. In this talk, we give optimal or best known CIS codes of length <132. We  derive general constructions based on cyclic codes, double circulant codes, strongly regular graphs, and doubly regular tournaments. We derive a Varshamov-Gilbert bound for long CIS codes, and show that they can all be classified in small lengths up to 12 by the building up construction. This is a joint work with Claude Carlet, Philippe Gaborit, and Patrick Sole.

Given an even subset T of the vertices of an undirected graph, a T-join is a subgraph in which the subset of vertices with odd degree is exactly T. Given edge weights, the weighted T-join problem is to find a T-join of minimum weight. With nonnegative edge weights, the problem can be reduced to finding a minimum weight perfect matching on the metric completion of the vertices in T.
Given an undirected graph with nonnegative edge weights but no specific T, the Max-Min Weighted T-join problem is to find an even cardinality vertex subset T such that the minimum weight T-join for this set is maximum. The unweighted case of the problem when all weights are either unit or infinity has been well characterized by a decomposition of the underlying graph into factor critical and matching-covered bipartite subgraphs (Frank1993). We consider the weighted version which is NP-hard even on a cycle. After showing a simple exact solution on trees, we present a 2/3-approximation algorithm for the general case. Our algorithm is based on a natural cut packing upper bound obtained using an LP relaxation and uncrossing, and relating it to the T-join problem using duality.
This is a joint work with R. Ravi.

Submodular functions are discrete analogues of convex functions. Examples include cut capacity functions, matroid rank functions, and entropy functions. Submodular functions can be minimized in polynomial time, which provides a fairly general framework of efficiently solvable combinatorial optimization problems. In contrast, the maximization problems are NP-hard and several approximation algorithms have been developed so far.
In this talk, I will review the above results in submodular optimization and present recent approximation algorithms for combinatorial optimization problems described in terms of submodular functions.

Many problems that are intractable on general graphs become linear time solvable on graphs of bounded treewidth. The constant factor however of such algorithms is exponential or worse in the treewidth of the tree decomposition that is used. In this talk, a number of known and some new results are surveyed. In particular, it is shown how speed improvements can be obtained using convolutions, and how a recent technique called “cut-and-count” can be used to obtain fast probabilistic algorithms for problems like Hamiltonian Circuit.