14,199 research outputs found
Faster Detours in Undirected Graphs
The -Detour problem is a basic path-finding problem: given a graph on
vertices, with specified nodes and , and a positive integer , the
goal is to determine if has an -path of length exactly , where is the length of a shortest path from to
. The -Detour problem is NP-hard when is part of the input, so
researchers have sought efficient parameterized algorithms for this task,
running in time, for as slow-growing as possible.
We present faster algorithms for -Detour in undirected graphs, running in
randomized and deterministic
time. The previous fastest algorithms for this problem took randomized and deterministic time
[Bez\'akov\'a-Curticapean-Dell-Fomin, ICALP 2017]. Our algorithms use the fact
that detecting a path of a given length in an undirected graph is easier if we
are promised that the path belongs to what we call a "bipartitioned" subgraph,
where the nodes are split into two parts and the path must satisfy constraints
on those parts. Previously, this idea was used to obtain the fastest known
algorithm for finding paths of length in undirected graphs
[Bj\"orklund-Husfeldt-Kaski-Koivisto, JCSS 2017].
Our work has direct implications for the -Longest Detour problem: in this
problem, we are given the same input as in -Detour, but are now tasked with
determining if has an -path of length at least
Our results for k-Detour imply that we can solve -Longest Detour in randomized and deterministic time.
The previous fastest algorithms for this problem took
randomized and deterministic time [Fomin et al.,
STACS 2022]
Parameterized Distributed Algorithms
In this work, we initiate a thorough study of graph optimization problems parameterized by the output size in the distributed setting. In such a problem, an algorithm decides whether a solution of size bounded by k exists and if so, it finds one. We study fundamental problems, including Minimum Vertex Cover (MVC), Maximum Independent Set (MaxIS), Maximum Matching (MaxM), and many others, in both the LOCAL and CONGEST distributed computation models. We present lower bounds for the round complexity of solving parameterized problems in both models, together with optimal and near-optimal upper bounds.
Our results extend beyond the scope of parameterized problems. We show that any LOCAL (1+epsilon)-approximation algorithm for the above problems must take Omega(epsilon^{-1}) rounds. Joined with the (epsilon^{-1}log n)^{O(1)} rounds algorithm of [Ghaffari et al., 2017] and the Omega (sqrt{(log n)/(log log n)}) lower bound of [Fabian Kuhn et al., 2016], the lower bounds match the upper bound up to polynomial factors in both parameters. We also show that our parameterized approach reduces the runtime of exact and approximate CONGEST algorithms for MVC and MaxM if the optimal solution is small, without knowing its size beforehand. Finally, we propose the first o(n^2) rounds CONGEST algorithms that approximate MVC within a factor strictly smaller than 2
On the Equivalence among Problems of Bounded Width
In this paper, we introduce a methodology, called decomposition-based
reductions, for showing the equivalence among various problems of
bounded-width.
First, we show that the following are equivalent for any :
* SAT can be solved in time,
* 3-SAT can be solved in time,
* Max 2-SAT can be solved in time,
* Independent Set can be solved in time, and
* Independent Set can be solved in time, where
tw and cw are the tree-width and clique-width of the instance, respectively.
Then, we introduce a new parameterized complexity class EPNL, which includes
Set Cover and Directed Hamiltonicity, and show that SAT, 3-SAT, Max 2-SAT, and
Independent Set parameterized by path-width are EPNL-complete. This implies
that if one of these EPNL-complete problems can be solved in time,
then any problem in EPNL can be solved in time.Comment: accepted to ESA 201
Faster Deterministic Algorithms for Packing, Matching and -Dominating Set Problems
In this paper, we devise three deterministic algorithms for solving the
-set -packing, -dimensional -matching, and -dominating set
problems in time , and ,
respectively. Although recently there has been remarkable progress on
randomized solutions to those problems, our bounds make good improvements on
the best known bounds for deterministic solutions to those problems.Comment: ISAAC13 Submission. arXiv admin note: substantial text overlap with
arXiv:1303.047
Fast Witness Extraction Using a Decision Oracle
The gist of many (NP-)hard combinatorial problems is to decide whether a
universe of elements contains a witness consisting of elements that
match some prescribed pattern. For some of these problems there are known
advanced algebra-based FPT algorithms which solve the decision problem but do
not return the witness. We investigate techniques for turning such a
YES/NO-decision oracle into an algorithm for extracting a single witness, with
an objective to obtain practical scalability for large values of . By
relying on techniques from combinatorial group testing, we demonstrate that a
witness may be extracted with queries to either a deterministic or
a randomized set inclusion oracle with one-sided probability of error.
Furthermore, we demonstrate through implementation and experiments that the
algebra-based FPT algorithms are practical, in particular in the setting of the
-path problem. Also discussed are engineering issues such as optimizing
finite field arithmetic.Comment: Journal version, 16 pages. Extended abstract presented at ESA'1
- …