5,866 research outputs found
Polynomial-time algorithm for Maximum Weight Independent Set on -free graphs
In the classic Maximum Weight Independent Set problem we are given a graph
with a nonnegative weight function on vertices, and the goal is to find an
independent set in of maximum possible weight. While the problem is NP-hard
in general, we give a polynomial-time algorithm working on any -free
graph, that is, a graph that has no path on vertices as an induced
subgraph. This improves the polynomial-time algorithm on -free graphs of
Lokshtanov et al. (SODA 2014), and the quasipolynomial-time algorithm on
-free graphs of Lokshtanov et al (SODA 2016). The main technical
contribution leading to our main result is enumeration of a polynomial-size
family of vertex subsets with the following property: for every
maximal independent set in the graph, contains all maximal
cliques of some minimal chordal completion of that does not add any edge
incident to a vertex of
Meta-Kernelization with Structural Parameters
Meta-kernelization theorems are general results that provide polynomial
kernels for large classes of parameterized problems. The known
meta-kernelization theorems, in particular the results of Bodlaender et al.
(FOCS'09) and of Fomin et al. (FOCS'10), apply to optimization problems
parameterized by solution size. We present the first meta-kernelization
theorems that use a structural parameters of the input and not the solution
size. Let C be a graph class. We define the C-cover number of a graph to be a
the smallest number of modules the vertex set can be partitioned into, such
that each module induces a subgraph that belongs to the class C. We show that
each graph problem that can be expressed in Monadic Second Order (MSO) logic
has a polynomial kernel with a linear number of vertices when parameterized by
the C-cover number for any fixed class C of bounded rank-width (or
equivalently, of bounded clique-width, or bounded Boolean width). Many graph
problems such as Independent Dominating Set, c-Coloring, and c-Domatic Number
are covered by this meta-kernelization result. Our second result applies to MSO
expressible optimization problems, such as Minimum Vertex Cover, Minimum
Dominating Set, and Maximum Clique. We show that these problems admit a
polynomial annotated kernel with a linear number of vertices
Relativistic MHD with Adaptive Mesh Refinement
This paper presents a new computer code to solve the general relativistic
magnetohydrodynamics (GRMHD) equations using distributed parallel adaptive mesh
refinement (AMR). The fluid equations are solved using a finite difference
Convex ENO method (CENO) in 3+1 dimensions, and the AMR is Berger-Oliger.
Hyperbolic divergence cleaning is used to control the
constraint. We present results from three flat space tests, and examine the
accretion of a fluid onto a Schwarzschild black hole, reproducing the Michel
solution. The AMR simulations substantially improve performance while
reproducing the resolution equivalent unigrid simulation results. Finally, we
discuss strong scaling results for parallel unigrid and AMR runs.Comment: 24 pages, 14 figures, 3 table
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