1,572,480 research outputs found
On -Simple -Path
An -simple -path is a {path} in the graph of length that passes
through each vertex at most times. The -SIMPLE -PATH problem, given a
graph as input, asks whether there exists an -simple -path in . We
first show that this problem is NP-Complete. We then show that there is a graph
that contains an -simple -path and no simple path of length greater
than . So this, in a sense, motivates this problem especially
when one's goal is to find a short path that visits many vertices in the graph
while bounding the number of visits at each vertex.
We then give a randomized algorithm that runs in time that solves the -SIMPLE -PATH on a graph with
vertices with one-sided error. We also show that a randomized algorithm
with running time with gives a
randomized algorithm with running time \poly(n)\cdot 2^{cn} for the
Hamiltonian path problem in a directed graph - an outstanding open problem. So
in a sense our algorithm is optimal up to an factor
Fast Algorithms for Parameterized Problems with Relaxed Disjointness Constraints
In parameterized complexity, it is a natural idea to consider different
generalizations of classic problems. Usually, such generalization are obtained
by introducing a "relaxation" variable, where the original problem corresponds
to setting this variable to a constant value. For instance, the problem of
packing sets of size at most into a given universe generalizes the Maximum
Matching problem, which is recovered by taking . Most often, the
complexity of the problem increases with the relaxation variable, but very
recently Abasi et al. have given a surprising example of a problem ---
-Simple -Path --- that can be solved by a randomized algorithm with
running time . That is, the complexity of the
problem decreases with . In this paper we pursue further the direction
sketched by Abasi et al. Our main contribution is a derandomization tool that
provides a deterministic counterpart of the main technical result of Abasi et
al.: the algorithm for -Monomial
Detection, which is the problem of finding a monomial of total degree and
individual degrees at most in a polynomial given as an arithmetic circuit.
Our technique works for a large class of circuits, and in particular it can be
used to derandomize the result of Abasi et al. for -Simple -Path. On our
way to this result we introduce the notion of representative sets for
multisets, which may be of independent interest. Finally, we give two more
examples of problems that were already studied in the literature, where the
same relaxation phenomenon happens. The first one is a natural relaxation of
the Set Packing problem, where we allow the packed sets to overlap at each
element at most times. The second one is Degree Bounded Spanning Tree,
where we seek for a spanning tree of the graph with a small maximum degree
On r-Simple k-Path and Related Problems Parameterized by k/r
Abasi et al. (2014) and Gabizon et al. (2015) studied the following problems.
In the -Simple -Path problem, given a digraph on vertices and
integers , decide whether has an -simple -path, which is a walk
where every vertex occurs at most times and the total number of vertex
occurrences is . In the -Monomial Detection problem, given an
arithmetic circuit that encodes some polynomial on variables and
integers , decide whether has a monomial of degree where the
degree of each variable is at most~. In the -Set -Packing problem,
given a universe , positive integers , and a collection of
sets of size whose elements belong to , decide whether there exists a
subcollection of of size where each element occurs in
at most sets of . Abasi et al. and Gabizon et al. proved that
the three problems are single-exponentially fixed-parameter tractable (FPT)
when parameterized by , where for -Set -Packing
and asked whether the factor in the exponent can be avoided.
We consider their question from a wider perspective: are the above problems
FPT when parameterized by only? We resolve the wider question by (a)
obtaining a -time algorithm for
-Simple -Path on digraphs and a -time
algorithm for -Simple -Path on undirected graphs (i.e., for undirected
graphs we answer the original question in affirmative), (b) showing that
-Set -Packing is FPT, and (c) proving that -Monomial Detection
is para-NP-hard. For -Set -Packing, we obtain a polynomial kernel for
any fixed , which resolves a question posed by Gabizon et al. regarding the
existence of polynomial kernels for problems with relaxed disjointness
constraints
On the path-avoidance vertex-coloring game
For any graph and any integer , the \emph{online vertex-Ramsey
density of and }, denoted , is a parameter defined via a
deterministic two-player Ramsey-type game (Painter vs.\ Builder). This
parameter was introduced in a recent paper \cite{mrs11}, where it was shown
that the online vertex-Ramsey density determines the threshold of a similar
probabilistic one-player game (Painter vs.\ the binomial random graph
). For a large class of graphs , including cliques, cycles,
complete bipartite graphs, hypercubes, wheels, and stars of arbitrary size, a
simple greedy strategy is optimal for Painter and closed formulas for
are known. In this work we show that for the case where
is a (long) path, the picture is very different. It is not hard to see that
for an appropriately defined integer
, and that the greedy strategy gives a lower bound of
. We construct and analyze Painter strategies that
improve on this greedy lower bound by a factor polynomial in , and we
show that no superpolynomial improvement is possible
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