Implementing path coloring algorithms on planar graphs

Master's Project (M.S.) University of Alaska Fairbanks, 2017A path coloring of a graph partitions its vertex set into color classes such that each class induces a disjoint union of paths. In this project we implement several algorithms to compute path colorings of graphs embedded in the plane. We present two algorithms to path color plane graphs with 3 colors based on a proof by Poh in 1990. First we describe a naive algorithm that directly follows Poh's procedure, then we give a modified algorithm that runs in linear time. Independent results of Hartman and Skrekovski describe a procedure that takes a plane graph G and a list of 3 colors for each vertex, and computes a path coloring of G such that each vertex receives a color from its list. We present a linear time implementation based on Hartman and Skrekovski's proofs. A C++ implementation is provided for all three algorithms, utilizing the Boost Graph Library. Instructions are given on how to use the implementation to construct colorings for plane graphs represented by Boost data structures

Decremental Single-Source Reachability in Planar Digraphs

In this paper we show a new algorithm for the decremental single-source reachability problem in directed planar graphs. It processes any sequence of edge deletions in $O(n\log^2{n}\log\log{n})$ total time and explicitly maintains the set of vertices reachable from a fixed source vertex. Hence, if all edges are eventually deleted, the amortized time of processing each edge deletion is only $O(\log^2 n \log \log n)$, which improves upon a previously known $O(\sqrt{n})$ solution. We also show an algorithm for decremental maintenance of strongly connected components in directed planar graphs with the same total update time. These results constitute the first almost optimal (up to polylogarithmic factors) algorithms for both problems. To the best of our knowledge, these are the first dynamic algorithms with polylogarithmic update times on general directed planar graphs for non-trivial reachability-type problems, for which only polynomial bounds are known in general graphs

Structure of conflict graphs in constraint alignment problems and algorithms

We consider the constrained graph alignment problem which has applications in biological network analysis. Given two input graphs $G_1=(V_1,E_1), G_2=(V_2,E_2)$, a pair of vertex mappings induces an {\it edge conservation} if the vertex pairs are adjacent in their respective graphs. %In general terms The goal is to provide a one-to-one mapping between the vertices of the input graphs in order to maximize edge conservation. However the allowed mappings are restricted since each vertex from $V_1$ (resp. $V_2$) is allowed to be mapped to at most $m_1$ (resp. $m_2$) specified vertices in $V_2$ (resp. $V_1$). Most of results in this paper deal with the case $m_2=1$ which attracted most attention in the related literature. We formulate the problem as a maximum independent set problem in a related {\em conflict graph} and investigate structural properties of this graph in terms of forbidden subgraphs. We are interested, in particular, in excluding certain wheals, fans, cliques or claws (all terms are defined in the paper), which corresponds in excluding certain cycles, paths, cliques or independent sets in the neighborhood of each vertex. Then, we investigate algorithmic consequences of some of these properties, which illustrates the potential of this approach and raises new horizons for further works. In particular this approach allows us to reinterpret a known polynomial case in terms of conflict graph and to improve known approximation and fixed-parameter tractability results through efficiently solving the maximum independent set problem in conflict graphs. Some of our new approximation results involve approximation ratios that are function of the optimal value, in particular its square root; this kind of results cannot be achieved for maximum independent set in general graphs.Comment: 22 pages, 6 figure

Canonical ordering for graphs on the cylinder, with applications to periodic straight-line drawings on the flat cylinder and torus

We extend the notion of canonical ordering (initially developed for planar triangulations and 3-connected planar maps) to cylindric (essentially simple) triangulations and more generally to cylindric (essentially internally) $3$-connected maps. This allows us to extend the incremental straight-line drawing algorithm of de Fraysseix, Pach and Pollack (in the triangulated case) and of Kant (in the $3$-connected case) to this setting. Precisely, for any cylindric essentially internally $3$-connected map $G$ with $n$ vertices, we can obtain in linear time a periodic (in $x$) straight-line drawing of $G$ that is crossing-free and internally (weakly) convex, on a regular grid $\mathbb{Z}/w\mathbb{Z}\times[0..h]$, with $w\leq 2n$ and $h\leq n(2d+1)$, where $d$ is the face-distance between the two boundaries. This also yields an efficient periodic drawing algorithm for graphs on the torus. Precisely, for any essentially $3$-connected map $G$ on the torus (i.e., $3$-connected in the periodic representation) with $n$ vertices, we can compute in linear time a periodic straight-line drawing of $G$ that is crossing-free and (weakly) convex, on a periodic regular grid $\mathbb{Z}/w\mathbb{Z}\times\mathbb{Z}/h\mathbb{Z}$, with $w\leq 2n$ and $h\leq 1+2n(c+1)$, where $c$ is the face-width of $G$. Since $c\leq\sqrt{2n}$, the grid area is $O(n^{5/2})$.Comment: 37 page

On Semantic Word Cloud Representation

We study the problem of computing semantic-preserving word clouds in which semantically related words are close to each other. While several heuristic approaches have been described in the literature, we formalize the underlying geometric algorithm problem: Word Rectangle Adjacency Contact (WRAC). In this model each word is associated with rectangle with fixed dimensions, and the goal is to represent semantically related words by ensuring that the two corresponding rectangles touch. We design and analyze efficient polynomial-time algorithms for some variants of the WRAC problem, show that several general variants are NP-hard, and describe a number of approximation algorithms. Finally, we experimentally demonstrate that our theoretically-sound algorithms outperform the early heuristics