Jan Arne Telle

Boolean-width of graphs

Abstract We introduce the graph parameter boolean-width, related to the number of different unions of neighborhoods–Boolean sums of neighborhoods–across a cut of a graph. For many graph problems, this number is the runtime bottleneck when using a divide-and-conquer approach. For an n -vertex graph given with a decomposition tree of boolean-width k , we solve Maximum Weight Independent Set in time O ( n 2 k 2 2 k ) and Minimum Weight Dominating Set in time O ( n 2 + n k 2 3 k ) . With an additional n 2 factor in the runtime, we can also count all independent sets and dominating sets of each cardinality. Boolean-width is bounded on the same classes of graphs as clique-width. boolean-width is similar to rank-width, which is related to the number of G F ( 2 ) -sums of neighborhoods instead of the Boolean sums used for boolean-width. We show for any graph that its boolean-width is at most its clique-width and at most quadratic in its rank-width. We exhibit a class of graphs, the Hsu-grids, having the property that a Hsu-grid on Θ ( n 2 ) vertices has boolean-width Θ ( log n ) and rank-width, clique-width, tree-width, and branch-width Θ ( n ) .

Practical Algorithms on Partial k-Trees with an Application to Domination-like Problems

Many NP-hard problems on graphs have polynomial, in fact usually linear, dynamic programming algorithms when restricted to partial k-trees (graphs of treewidth bounded by k), for fixed values of k. We investigate the practicality of such algorithms, both in terms of their complexity and their derivation, and account for the dependency on the treewidth k. We define a general procedure to derive the details of table updates in the dynamic programming solution algorithms. This procedure is based on a binary parse tree of the input graph. We give a formal description of vertex subset optimization problems in a class that includes several variants of domination, independence, efficiency and packing. We give algorithms for any problem in this class, which take a graph G, integer k and a width k tree-decomposition of G as input, and solve the problem on G in O(n24k) steps.

OREGAMI: Tools for mapping parallel computations to parallel architectures

The OREGAMI project involves the design, implementation, and testing of algorithms for mapping parallel computations to message-passing parallel architectures. OREGAMI addresses the mapping problem by exploiting regularity and by allowing the user to guide and evaluate mapping decisions made by OREGAMIs efficient combinatorial mapping algorithms. OREGAMIs approach to mapping is based on a new graph theoretic model of parallel computation called the Temporal Communication Graph. The OREGAMI software tools include three components: (1) LaRCS is a graph description language which allows the user to describe regularity in the communication topology as well as the temporal communication behavior (the pattern of message-passing over time). (2) MAPPER is our library of mapping algorithms which utilize information provided by LaRCS to perform contraction, embedding, and routing. (3) METRICS is an interactive graphics tool for display and analysis of mappings. This paper gives an overview of the OREGAMI project, the software tools, and OREGAMIs mapping algorithms.

Interval Completion Is Fixed Parameter Tractable

We present an algorithm with runtime

Covering Regular Graphs

O(k^{2k}n^3m)

The Treewidth of Java Programs

for the following NP-complete problem [M. Garey and D. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman and Co., San Francisco, 1979, problem GT35]: Given an arbitrary graph

Faster Algorithms for the Nonemptiness of Streett Automata and for Communication Protocol Pruning

G

Interval completion with few edges

on

Multicoloring trees

n

Generalized H-Coloring of Graphs

vertices and