Sankar Veeramoni

On maximum differential graph coloring

We study the maximum differential graph coloring problem, in which the goal is to find a vertex labeling for a given undirected graph that maximizes the label difference along the edges. This problem has its origin in map coloring, where not all countries are necessarily contiguous. We define the differential chromatic number and establish the equivalence of the maximum differential coloring problem to that of k-Hamiltonian path. As computing the maximum differential coloring is NP-Complete, we describe an exact backtracking algorithm and a spectral-based heuristic. We also discuss lower bounds and upper bounds for the differential chromatic number for several classes of graphs.

Quantitative Measures for Cartogram Generation Techniques

Cartograms are used to visualize geographically distributed data by scaling the regions of a map (e.g., US states) such that their areas are proportional to some data associated with them (e.g., population). Thus the cartogram computation problem can be considered as a map deformation problem where the input is a planar polygonal map M and an assignment of some positive weight for each region. The goal is to create a deformed map M′, where the area of each region realizes the weight assigned to it (no cartographic error) while the overall map remains readable and recognizable (e.g., the topology, relative positions and shapes of the regions remain as close to those before the deformation as possible). Although several such measures of cartogram quality are well‐known, different cartogram generation methods optimize different features and there is no standard set of quantitative metrics. In this paper we define such a set of seven quantitative measures, designed to evaluate how faithfully a cartogram represents the desired weights and to estimate the readability of the final representation. We then study several cartogram‐generation algorithms and compare them in terms of these quantitative measures.

3D proportional contact representations of graphs

In 3D contact representations, the vertices of a graph are represented by 3D polyhedra and the edges are realized by non-zero-area common boundaries between corresponding polyhedra. While contact representations with cuboids have been studied in the literature, we consider a novel generalization of the problem in which vertices are represented by axis-aligned polyhedra that are union of two cuboids. In particular, we study the weighted (proportional) version of the problem, where the volumes of the polyhedra and the areas of the common boundaries realize prespecified vertex and edge weights. For some classes of graphs (e.g., outerplanar, planar bipartite, planar, complete), we provide algorithms to construct such representations for arbitrary given weights.We also show that not all graphs can be represented in 3D with axis-aligned polyhedra of constant complexity.

A note on maximum differential coloring of planar graphs

Abstract We study the maximum differential coloring problem , where the vertices of an n -vertex graph must be labeled with distinct numbers ranging from 1 to n , so that the minimum absolute difference between two labels of any two adjacent vertices is maximized. As the problem is NP -hard for general graphs [16] , we consider planar graphs and subclasses thereof. We prove that the maximum differential coloring problem remains NP -hard, even for planar graphs. We also present tight bounds for regular caterpillars and spider graphs. Using these new bounds, we prove that the Miller–Pritikin labeling scheme [19] for forests is optimal for regular caterpillars and for spider graphs.

Computing Consensus Curves

We study the problem of extracting accurate average ant trajectories from many (inaccurate) input trajectories contributed by citizen scientists. Although there are many generic software tools for motion tracking and specific ones for insect tracking, even untrained humans are better at this task. We consider several local (one ant at a time) and global (all ants together) methods. Our best performing algorithm uses a novel global method, based on finding edge-disjoint paths in a graph constructed from the input trajectories. The underlying optimization problem is a new and interesting network flow variant. Even though the problem is NP-complete, two heuristics work well in practice, outperforming all other approaches, including the best automated system.

The maximum k-differential coloring problem☆

Given an

The Maximum k-Differential Coloring Problem

n

Embedding, clustering and coloring for dynamic maps

-vertex graph

Approximating the Generalized Minimum Manhattan Network Problem

G

Polylogarithmic Approximation for Generalized Minimum Manhattan Networks

and two positive integers