Gyárfás and Sumner independently conjectured that for every tree T and integer k there is an integer f(k, T ) such that every graph G with χ(G) > f(k, T ) contains either Kk or an induced copy of T . We prove a ‘topological’ version of the conjecture: for every tree T and integer k there is g(k, T ) such that every graph G with χ(G) > g(k, T ) contains either Kk or an induced copy of a subdivis...

We show how to efficiently model binary constraint problems (BCP) as integer programs. After considering tree-structured BCPs first, we show that a Sherali-Adams-like procedure results in a polynomial-size linear programming description of the convex hull of all integer feasible solutions when the BCP that is given has bounded tree-width.

For two given graphs G1 and G2, the Ramseynumber R(G1,G2) is the smallest integer n such that for anygraph G of order n, either $G$ contains G1 or the complementof G contains G2. Let Tn denote a tree of order n andWm a wheel of order m+1. To the best of our knowledge, only R(Tn,Wm) with small wheels are known.In this paper, we show that R(Tn,Wm)=3n-2 for odd m with n>756m^{10}.

A branch and cut algorithm is developed for solving 0-1 MINLP problems. The algorithm integrates Branch and Bound, Outer Approximation and Gomory Cutting Planes. Only the initial Mixed Integer Linear Programming (MILP) master problem is considered. At integer solutions Nonlinear Programming (NLP) problems are solved, using a primal-dual interior point algorithm. The objective and constraints ar...

Abstract Regression analysis is a statistical procedure that fits mathematical function to set of data in order capture the relationship between dependent and independent variables. In tree regression, structures are constructed by repeated splits input space into two subsets, creating if-then-else rules. Such models popular literature due their ability be computed quickly simple interpretation...

Suppose we label the vertices of a tree by positive integers. The weight of an edge is defined by a monotonically increasing function of the absolute value of the difference of the labels of its endpoints. We define the total cost of the labeling to be the sum of weights of all the edges. The problem we consider is that of determining for a given tree G and given a labeling of the leaves of G t...

In this survey paper, we discuss the development of the Generalized Minimum Spanning Tree Problem, denoted by GMSTP, and we focus on the integer programming formulations of the problem. The GMSTP is a variant of the classical minimum spanning tree problem (MST), in which the nodes of an undirected graph are partitioned into node sets (clusters) and we are looking for a minimum cost tree spannin...

Methods for constructing binary exhaustive prefix codes with certain synchronizing properties are given. In particular it is shown how to formulate the requirement that a code be anagrammatic and the requirement for minimal synchronizing delay as linear integer programming problems. The tree structure of the code is given and the program optimizes over all possible labelings of the tree. Severa...