8 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
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
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
A time- and space-optimal algorithm for the many-visits TSP
The many-visits traveling salesperson problem (MV-TSP) asks for an optimal
tour of cities that visits each city a prescribed number of
times. Travel costs may be asymmetric, and visiting a city twice in a row may
incur a non-zero cost. The MV-TSP problem finds applications in scheduling,
geometric approximation, and Hamiltonicity of certain graph families.
The fastest known algorithm for MV-TSP is due to Cosmadakis and Papadimitriou
(SICOMP, 1984). It runs in time and
requires space. An interesting feature of the
Cosmadakis-Papadimitriou algorithm is its \emph{logarithmic} dependence on the
total length of the tour, allowing the algorithm to handle
instances with very long tours. The \emph{superexponential} dependence on the
number of cities in both the time and space complexity, however, renders the
algorithm impractical for all but the narrowest range of this parameter.
In this paper we improve upon the Cosmadakis-Papadimitriou algorithm, giving
an MV-TSP algorithm that runs in time , i.e.\
\emph{single-exponential} in the number of cities, using \emph{polynomial}
space. Our algorithm is deterministic, and arguably both simpler and easier to
analyse than the original approach of Cosmadakis and Papadimitriou. It involves
an optimization over directed spanning trees and a recursive, centroid-based
decomposition of trees.Comment: Small fixes, journal versio
Studies in Efficient Discrete Algorithms
This thesis consists of five papers within the design and analysis of efficient algorithms.In the first paper, we consider the problem of computing all-pairs shortest paths in a directed graph with real weights assigned to vertices. We develop a combinatorial randomized algorithm that runs in subcubic time for a special class of graphs.In the second paper, we present a polynomial-time dynamic programming algorithm for optimal partitions of a complete edge-weighted graph, where the edges are weighted by the length of the unique shortest path connecting those vertices in the a priori given tree (shortest path metric induced by a tree). Our result resolves, in particular, the complexity status of the optimal partition problems in one-dimensional geometric (Euclidean) setting.In the third paper, we study the NP-hard problem of partitioning an orthogonal polyhedron P into a minimum number of 3D rectangles. We present an approximation algorithm with the approximation ratio 4 for the special case of the problem in which P is a so-called 3D histogram. We then apply it to compute the exact arithmetic matrix product of two matrices with non-negative integer entries. The computation is time-efficient if the 3D histograms induced by the input matrices can be partitioned into relatively few 3D rectangles.In the fourth paper, we present the first quasi-polynomial approximation schemes for the base of the number of triangulations of a planar point set and the base of the number of crossing-free spanning trees on a planar point set, respectively.In the fifth paper, we study the complexity of detecting monomials with special properties in the sum-product expansion of a polynomial represented by an arithmetic circuit of size polynomial in the number of input variables and using only multiplication and addition. We present a fixed-parameter tractable algorithms for the detection of monomial having at least k distinct variables, parametrized with respect to k. Furthermore, we derive several hardness results on the detection of monomials with such properties within exact, parametrized and approximation complexity