Multigrid solver for axisymmetrical 2D fluid equations

We have developed an efficient algorithm for steady axisymmetrical 2D fluid equations. The algorithm employs multigrid method as well as standard implicit discretization schemes for systems of partial differential equations. Linearity of the multigrid method with respect to the number of grid points allowed us to use $256\times 256$ grid, where we could achieve solutions in several minutes. Time limitations due to nonlinearity of the system are partially avoided by using multi level grids(the initial solution on $256\times 256$ grid was extrapolated steady solution from $128\times 128$ grid which allowed using "long" integration time steps). The fluid solver may be used as the basis for hybrid codes for DC discharges.Comment: preliminary version; presented at 28 ICPIG, July 15-20, 2007, Prague, Czech Republi

Fast algorithms for min independent dominating set

We first devise a branching algorithm that computes a minimum independent dominating set on any graph with running time O*(2^0.424n) and polynomial space. This improves the O*(2^0.441n) result by (S. Gaspers and M. Liedloff, A branch-and-reduce algorithm for finding a minimum independent dominating set in graphs, Proc. WG'06). We then show that, for every r>3, it is possible to compute an r-((r-1)/r)log_2(r)-approximate solution for min independent dominating set within time O*(2^(nlog_2(r)/r))

Tur\'an Graphs, Stability Number, and Fibonacci Index

The Fibonacci index of a graph is the number of its stable sets. This parameter is widely studied and has applications in chemical graph theory. In this paper, we establish tight upper bounds for the Fibonacci index in terms of the stability number and the order of general graphs and connected graphs. Tur\'an graphs frequently appear in extremal graph theory. We show that Tur\'an graphs and a connected variant of them are also extremal for these particular problems.Comment: 11 pages, 3 figure

Feedback Vertex Sets in Tournaments

We study combinatorial and algorithmic questions around minimal feedback vertex sets in tournament graphs. On the combinatorial side, we derive strong upper and lower bounds on the maximum number of minimal feedback vertex sets in an n-vertex tournament. We prove that every tournament on n vertices has at most 1.6740^n minimal feedback vertex sets, and that there is an infinite family of tournaments, all having at least 1.5448^n minimal feedback vertex sets. This improves and extends the bounds of Moon (1971). On the algorithmic side, we design the first polynomial space algorithm that enumerates the minimal feedback vertex sets of a tournament with polynomial delay. The combination of our results yields the fastest known algorithm for finding a minimum size feedback vertex set in a tournament

Branch and Recharge: Exact algorithms for generalized domination

Let σ and ϱ be two sets of nonnegative integers. A vertex subset S ⊆ V of an undirected graph G =(V,E) is called a (σ, ϱ)-dominating set of G if |N(v) ∩ S | ∈σ for all v ∈ S and |N(v) ∩ S | ∈ϱ for all v ∈ V \ S. This notion introduced by J. A. Telle generalizes many domination-type graph invariants. We show a general algorithm enumerating all (σ, ϱ)-dominating sets of an input graph G in time O ∗ (c n)forsomec<2 using only polynomial space, if σ is successorfree, i.e., it does not contain two consecutive integers, and either both σ and ϱ are finite, or one of them is finite and σ ∩ ϱ = ∅. Thus in this case one can find maximum and minimum (σ, ϱ)-dominating sets in time o(2 n), though for many particular choices of σ and ϱ already the existence of a (σ, ϱ)-dominating set is NP-complete. Our algorithm straightforwardly implies a non trivial upper bound c n with c<2forthenumberof (σ, ϱ)-dominating sets in an n-vertex graph under the above conditions on σ and ϱ. Finally, we also present algorithms using a Sort & Search paradigm to find maximum and minimum ({p}, {q})-dominating sets and to count the ({p}, {q})-dominating sets of a graph in time O ∗ (2 n/2).

Exact algorithms for exact satisfiability and number of perfect matchings

We present exact algorithms with exponential running times for variants of n-element set cover problems, based on divide-and-conquer and on inclusion exclusion characterizations. We show that the Exact Satisfiability problem of size l with m clauses can be solved in time 2(m)l(O(1)) and polynomial space. The same bounds hold for counting the number of solutions. As a special case, we can count the number of perfect matchings in an n-vertex graph in time 2(n)n(O(1)) and polynomial space. We also show how to count the number of perfect matchings in time O(1.732(n)) and exponential space. We give a number of examples where the running time can be further improved if the hypergraph corresponding to the set cover instance has low pathwidth. This yields exponential-time algorithms for counting k-dimensional matchings, Exact Uniform Set Cover, Clique Partition, and Minimum Dominating Set in graphs of degree at most three. We extend the analysis to a number of related problems such as TSP and Chromatic Number

Partitioning based algorithms for some colouring problems

We discuss four variants of the graph colouring problem, and present algorithms for solving them. The problems are k-COLOURABILITY, MAX IND k-COL, MAX VAL k-COL, and, finally, MAX k-COL, which is the unweighted case of the MAX k-CUT problem. The algorithms are based on the idea of partitioning the domain of the problems into disjoint subsets, and then considering all possible instances were the variables are restricted to values from these partitions. If a pair of variables have been restricted to different partitions, then the constraint between them is always satisfied since the only allowed constraint is disequality

Computing branchwidth via Efficient Triangulations and blocks

Minimal triangulations and potential maximal cliques are main ingredients for a number of polynomial time algorithms computing the treewidth for different graph classes. Based on the recent results of Mazoit, we define the structures that can be regarded as minimal triangulations and potential maximal cliques for branchwidth: efficient triangulations and blocks. We show how blocks can be used for computing the branchwidth in O*((2 + \sqrt{3})^n) time