New lower bounds for Hopcroft's problem

New Lower Bounds for Hopcroft's Problem

We establish new lower bounds on the complexity of the following basic geometric problem, attributed to John Hopcroft: Given a set of n points and m hyperplanes in IR, is any point contained in any hyperplane? We de ne a general class of partitioning algorithms, and show that in the worst case, for all m and n, any such algorithm requires time (n logm + nm + m logn) in two dimensions, or (n log...

New Lower Bounds for the Split Delivery Vehicle Routing Problem

This paper presents an algorithm to obtain lower bounds for the Split Delivery Vehicle Routing Problem. An extended formulation over a large set of variables is provided and valid inequalities are identified. The algorithm combines column and cut generation and was able to improve the lower bound for almost all instances tested.

Some Lower Bounds for Estrada Index

New Lower Bounds for -Nets

Following groundbreaking work by Haussler and Welzl (1987), the use of small -nets has become a standard technique for solving algorithmic and extremal problems in geometry and learning theory. Two significant recent developments are: (i) an upper bound on the size of the smallest -nets for set systems, as a function of their so-called shallow-cell complexity (Chan, Grant, Könemann, and Sharpe)...

Lower Bounds for the Graph Homomorphism Problem

The graph homomorphism problem (HOM) asks whether the vertices of a given n-vertex graph G can be mapped to the vertices of a given h-vertex graph H such that each edge of G is mapped to an edge of H. The problem generalizes the graph coloring problem and at the same time can be viewed as a special case of the 2-CSP problem. In this paper, we prove several lower bounds for HOM under the Exponen...