10,040 research outputs found
The Complexity of Approximately Counting Stable Roommate Assignments
We investigate the complexity of approximately counting stable roommate
assignments in two models: (i) the -attribute model, in which the preference
lists are determined by dot products of "preference vectors" with "attribute
vectors" and (ii) the -Euclidean model, in which the preference lists are
determined by the closeness of the "positions" of the people to their
"preferred positions". Exactly counting the number of assignments is
#P-complete, since Irving and Leather demonstrated #P-completeness for the
special case of the stable marriage problem. We show that counting the number
of stable roommate assignments in the -attribute model () and the
3-Euclidean model() is interreducible, in an approximation-preserving
sense, with counting independent sets (of all sizes) (#IS) in a graph, or
counting the number of satisfying assignments of a Boolean formula (#SAT). This
means that there can be no FPRAS for any of these problems unless NP=RP. As a
consequence, we infer that there is no FPRAS for counting stable roommate
assignments (#SR) unless NP=RP. Utilizing previous results by the authors, we
give an approximation-preserving reduction from counting the number of
independent sets in a bipartite graph (#BIS) to counting the number of stable
roommate assignments both in the 3-attribute model and in the 2-Euclidean
model. #BIS is complete with respect to approximation-preserving reductions in
the logically-defined complexity class #RH\Pi_1. Hence, our result shows that
an FPRAS for counting stable roommate assignments in the 3-attribute model
would give an FPRAS for all of #RH\Pi_1. We also show that the 1-attribute
stable roommate problem always has either one or two stable roommate
assignments, so the number of assignments can be determined exactly in
polynomial time
Approximation Algorithms for Partially Colorable Graphs
Graph coloring problems are a central topic of study in the theory of algorithms. We study the problem of partially coloring partially colorable graphs. For alpha = alpha |V| such that the graph induced on S is k-colorable. Partial k-colorability is a more robust structural property of a graph than k-colorability. For graphs that arise in practice, partial k-colorability might be a better notion to use than k-colorability, since data arising in practice often contains various forms of noise.
We give a polynomial time algorithm that takes as input a (1 - epsilon)-partially 3-colorable graph G and a constant gamma in [epsilon, 1/10], and colors a (1 - epsilon/gamma) fraction of the vertices using O~(n^{0.25 + O(gamma^{1/2})}) colors. We also study natural semi-random families of instances of partially 3-colorable graphs and partially 2-colorable graphs, and give stronger bi-criteria approximation guarantees for these family of instances
Who witnesses The Witness? Finding witnesses in The Witness is hard and sometimes impossible
We analyze the computational complexity of the many types of
pencil-and-paper-style puzzles featured in the 2016 puzzle video game The
Witness. In all puzzles, the goal is to draw a simple path in a rectangular
grid graph from a start vertex to a destination vertex. The different puzzle
types place different constraints on the path: preventing some edges from being
visited (broken edges); forcing some edges or vertices to be visited
(hexagons); forcing some cells to have certain numbers of incident path edges
(triangles); or forcing the regions formed by the path to be partially
monochromatic (squares), have exactly two special cells (stars), or be singly
covered by given shapes (polyominoes) and/or negatively counting shapes
(antipolyominoes). We show that any one of these clue types (except the first)
is enough to make path finding NP-complete ("witnesses exist but are hard to
find"), even for rectangular boards. Furthermore, we show that a final clue
type (antibody), which necessarily "cancels" the effect of another clue in the
same region, makes path finding -complete ("witnesses do not exist"),
even with a single antibody (combined with many anti/polyominoes), and the
problem gets no harder with many antibodies. On the positive side, we give a
polynomial-time algorithm for monomino clues, by reducing to hexagon clues on
the boundary of the puzzle, even in the presence of broken edges, and solving
"subset Hamiltonian path" for terminals on the boundary of an embedded planar
graph in polynomial time.Comment: 72 pages, 59 figures. Revised proof of Lemma 3.5. A short version of
this paper appeared at the 9th International Conference on Fun with
Algorithms (FUN 2018
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