A Fast and Scalable Graph Coloring Algorithm for Multi-core and Many-core Architectures

Irregular computations on unstructured data are an important class of problems for parallel programming. Graph coloring is often an important preprocessing step, e.g. as a way to perform dependency analysis for safe parallel execution. The total run time of a coloring algorithm adds to the overall parallel overhead of the application whereas the number of colors used determines the amount of exposed parallelism. A fast and scalable coloring algorithm using as few colors as possible is vital for the overall parallel performance and scalability of many irregular applications that depend upon runtime dependency analysis. Catalyurek et al. have proposed a graph coloring algorithm which relies on speculative, local assignment of colors. In this paper we present an improved version which runs even more optimistically with less thread synchronization and reduced number of conflicts compared to Catalyurek et al.'s algorithm. We show that the new technique scales better on multi-core and many-core systems and performs up to 1.5x faster than its predecessor on graphs with high-degree vertices, while keeping the number of colors at the same near-optimal levels.Comment: To appear in the proceedings of Euro Par 201

Conflict-Free Coloring of Planar Graphs

A conflict-free k-coloring of a graph assigns one of k different colors to some of the vertices such that, for every vertex v, there is a color that is assigned to exactly one vertex among v and v's neighbors. Such colorings have applications in wireless networking, robotics, and geometry, and are well-studied in graph theory. Here we study the natural problem of the conflict-free chromatic number chi_CF(G) (the smallest k for which conflict-free k-colorings exist). We provide results both for closed neighborhoods N[v], for which a vertex v is a member of its neighborhood, and for open neighborhoods N(v), for which vertex v is not a member of its neighborhood. For closed neighborhoods, we prove the conflict-free variant of the famous Hadwiger Conjecture: If an arbitrary graph G does not contain K_{k+1} as a minor, then chi_CF(G) <= k. For planar graphs, we obtain a tight worst-case bound: three colors are sometimes necessary and always sufficient. We also give a complete characterization of the computational complexity of conflict-free coloring. Deciding whether chi_CF(G)<= 1 is NP-complete for planar graphs G, but polynomial for outerplanar graphs. Furthermore, deciding whether chi_CF(G)<= 2 is NP-complete for planar graphs G, but always true for outerplanar graphs. For the bicriteria problem of minimizing the number of colored vertices subject to a given bound k on the number of colors, we give a full algorithmic characterization in terms of complexity and approximation for outerplanar and planar graphs. For open neighborhoods, we show that every planar bipartite graph has a conflict-free coloring with at most four colors; on the other hand, we prove that for k in {1,2,3}, it is NP-complete to decide whether a planar bipartite graph has a conflict-free k-coloring. Moreover, we establish that any general} planar graph has a conflict-free coloring with at most eight colors.Comment: 30 pages, 17 figures; full version (to appear in SIAM Journal on Discrete Mathematics) of extended abstract that appears in Proceeedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2017), pp. 1951-196

Edge and total choosability of near-outerplanar graphs

It is proved that, if G is a K4-minor-free graph with maximum degree ∆ ≥ 4, then G is totally (∆ + 1)-choosable; that is, if every element (vertex or edge) of G is assigned a list of ∆ + 1 colours, then every element can be coloured with a colour from its own list in such a way that every two adjacent or incident elements are coloured with different colours. Together with other known results, this shows that the List-Total-Colouring Conjecture, that ch’’(G) = χ’(G) for every graph G, is true for all K4-minor-free graphs. The List-Edge-Colouring Conjecture is also known to be true for these graphs. As a fairly straightforward consequence, it is proved that both conjectures hold also for all K2,3-minor free graphs and all ( K2 + (K1 U K2))-minor-free graphs

On edge-group choosability of graphs

In this paper, we study the concept of edge-group choosability of graphs. We say that G is edge k-group choosable if its line graph is k-group choosable. An edge-group choosability version of Vizing conjecture is given. The evidence of our claim are graphs with maximum degree less than 4, planar graphs with maximum degree at least 11, planar graphs without small cycles, outerplanar graphs and near-outerplanar graphs