Oriented coloring on recursively defined digraphs

Coloring is one of the most famous problems in graph theory. The coloring problem on undirected graphs has been well studied, whereas there are very few results for coloring problems on directed graphs. An oriented k-coloring of an oriented graph G=(V,A) is a partition of the vertex set V into k independent sets such that all the arcs linking two of these subsets have the same direction. The oriented chromatic number of an oriented graph G is the smallest k such that G allows an oriented k-coloring. Deciding whether an acyclic digraph allows an oriented 4-coloring is NP-hard. It follows, that finding the chromatic number of an oriented graph is an NP-hard problem. This motivates to consider the problem on oriented co-graphs. After giving several characterizations for this graph class, we show a linear time algorithm which computes an optimal oriented coloring for an oriented co-graph. We further prove how the oriented chromatic number can be computed for the disjoint union and order composition from the oriented chromatic number of the involved oriented co-graphs. It turns out that within oriented co-graphs the oriented chromatic number is equal to the length of a longest oriented path plus one. We also show that the graph isomorphism problem on oriented co-graphs can be solved in linear time.Comment: 14 page

Two novel evolutionary formulations of the graph coloring problem

We introduce two novel evolutionary formulations of the problem of coloring the nodes of a graph. The first formulation is based on the relationship that exists between a graph's chromatic number and its acyclic orientations. It views such orientations as individuals and evolves them with the aid of evolutionary operators that are very heavily based on the structure of the graph and its acyclic orientations. The second formulation, unlike the first one, does not tackle one graph at a time, but rather aims at evolving a `program' to color all graphs belonging to a class whose members all have the same number of nodes and other common attributes. The heuristics that result from these formulations have been tested on some of the Second DIMACS Implementation Challenge benchmark graphs, and have been found to be competitive when compared to the several other heuristics that have also been tested on those graphs.Comment: To appear in Journal of Combinatorial Optimizatio

Reduction Techniques for Graph Isomorphism in the Context of Width Parameters

We study the parameterized complexity of the graph isomorphism problem when parameterized by width parameters related to tree decompositions. We apply the following technique to obtain fixed-parameter tractability for such parameters. We first compute an isomorphism invariant set of potential bags for a decomposition and then apply a restricted version of the Weisfeiler-Lehman algorithm to solve isomorphism. With this we show fixed-parameter tractability for several parameters and provide a unified explanation for various isomorphism results concerned with parameters related to tree decompositions. As a possibly first step towards intractability results for parameterized graph isomorphism we develop an fpt Turing-reduction from strong tree width to the a priori unrelated parameter maximum degree.Comment: 23 pages, 4 figure

On space efficiency of algorithms working on structural decompositions of graphs

Dynamic programming on path and tree decompositions of graphs is a technique that is ubiquitous in the field of parameterized and exponential-time algorithms. However, one of its drawbacks is that the space usage is exponential in the decomposition's width. Following the work of Allender et al. [Theory of Computing, '14], we investigate whether this space complexity explosion is unavoidable. Using the idea of reparameterization of Cai and Juedes [J. Comput. Syst. Sci., '03], we prove that the question is closely related to a conjecture that the Longest Common Subsequence problem parameterized by the number of input strings does not admit an algorithm that simultaneously uses XP time and FPT space. Moreover, we complete the complexity landscape sketched for pathwidth and treewidth by Allender et al. by considering the parameter tree-depth. We prove that computations on tree-depth decompositions correspond to a model of non-deterministic machines that work in polynomial time and logarithmic space, with access to an auxiliary stack of maximum height equal to the decomposition's depth. Together with the results of Allender et al., this describes a hierarchy of complexity classes for polynomial-time non-deterministic machines with different restrictions on the access to working space, which mirrors the classic relations between treewidth, pathwidth, and tree-depth.Comment: An extended abstract appeared in the proceedings of STACS'16. The new version is augmented with a space-efficient algorithm for Dominating Set using the Chinese remainder theore

Upper and Lower Bounds on Long Dual-Paths in Line Arrangements

Given a line arrangement $\cal A$ with $n$ lines, we show that there exists a path of length $n^2/3 - O(n)$ in the dual graph of $\cal A$ formed by its faces. This bound is tight up to lower order terms. For the bicolored version, we describe an example of a line arrangement with $3k$ blue and $2k$ red lines with no alternating path longer than $14k$. Further, we show that any line arrangement with $n$ lines has a coloring such that it has an alternating path of length $\Omega (n^2/ \log n)$. Our results also hold for pseudoline arrangements.Comment: 19 page