72,340 research outputs found
On the Complexity of Distributed Splitting Problems
One of the fundamental open problems in the area of distributed graph
algorithms is the question of whether randomization is needed for efficient
symmetry breaking. While there are fast, -time randomized
distributed algorithms for all of the classic symmetry breaking problems, for
many of them, the best deterministic algorithms are almost exponentially
slower. The following basic local splitting problem, which is known as the
\emph{weak splitting} problem takes a central role in this context: Each node
of a graph has to be colored red or blue such that each node of
sufficiently large degree has at least one node of each color among its
neighbors. Ghaffari, Kuhn, and Maus [STOC '17] showed that this seemingly
simple problem is complete w.r.t. the above fundamental open question in the
following sense: If there is an efficient -time determinstic
distributed algorithm for weak splitting, then there is such an algorithm for
all locally checkable graph problems for which an efficient randomized
algorithm exists. In this paper, we investigate the distributed complexity of
weak splitting and some closely related problems. E.g., we obtain efficient
algorithms for special cases of weak splitting, where the graph is nearly
regular. In particular, we show that if and are the minimum
and maximum degrees of and if , weak splitting can
be solved deterministically in time
. Further, if and , there is a
randomized algorithm with time complexity
Digraph Complexity Measures and Applications in Formal Language Theory
We investigate structural complexity measures on digraphs, in particular the
cycle rank. This concept is intimately related to a classical topic in formal
language theory, namely the star height of regular languages. We explore this
connection, and obtain several new algorithmic insights regarding both cycle
rank and star height. Among other results, we show that computing the cycle
rank is NP-complete, even for sparse digraphs of maximum outdegree 2.
Notwithstanding, we provide both a polynomial-time approximation algorithm and
an exponential-time exact algorithm for this problem. The former algorithm
yields an O((log n)^(3/2))- approximation in polynomial time, whereas the
latter yields the optimum solution, and runs in time and space O*(1.9129^n) on
digraphs of maximum outdegree at most two. Regarding the star height problem,
we identify a subclass of the regular languages for which we can precisely
determine the computational complexity of the star height problem. Namely, the
star height problem for bideterministic languages is NP-complete, and this
holds already for binary alphabets. Then we translate the algorithmic results
concerning cycle rank to the bideterministic star height problem, thus giving a
polynomial-time approximation as well as a reasonably fast exact exponential
algorithm for bideterministic star height.Comment: 19 pages, 1 figur
Solving k-Set Agreement with Stable Skeleton Graphs
In this paper we consider the k-set agreement problem in distributed
message-passing systems using a round-based approach: Both synchrony of
communication and failures are captured just by means of the messages that
arrive within a round, resulting in round-by-round communication graphs that
can be characterized by simple communication predicates. We introduce the weak
communication predicate PSources(k) and show that it is tight for k-set
agreement, in the following sense: We (i) prove that there is no algorithm for
solving (k-1)-set agreement in systems characterized by PSources(k), and (ii)
present a novel distributed algorithm that achieves k-set agreement in runs
where PSources(k) holds. Our algorithm uses local approximations of the stable
skeleton graph, which reflects the underlying perpetual synchrony of a run. We
prove that this approximation is correct in all runs, regardless of the
communication predicate, and show that graph-theoretic properties of the stable
skeleton graph can be used to solve k-set agreement if PSources(k) holds.Comment: to appear in 16th IEEE Workshop on Dependable Parallel, Distributed
and Network-Centric System
Algorithmic Aspects of a General Modular Decomposition Theory
A new general decomposition theory inspired from modular graph decomposition
is presented. This helps unifying modular decomposition on different
structures, including (but not restricted to) graphs. Moreover, even in the
case of graphs, the terminology ``module'' not only captures the classical
graph modules but also allows to handle 2-connected components, star-cutsets,
and other vertex subsets. The main result is that most of the nice algorithmic
tools developed for modular decomposition of graphs still apply efficiently on
our generalisation of modules. Besides, when an essential axiom is satisfied,
almost all the important properties can be retrieved. For this case, an
algorithm given by Ehrenfeucht, Gabow, McConnell and Sullivan 1994 is
generalised and yields a very efficient solution to the associated
decomposition problem
Finding weakly reversible realizations of chemical reaction networks using optimization
An algorithm is given in this paper for the computation of dynamically
equivalent weakly reversible realizations with the maximal number of reactions,
for chemical reaction networks (CRNs) with mass action kinetics. The original
problem statement can be traced back at least 30 years ago. The algorithm uses
standard linear and mixed integer linear programming, and it is based on
elementary graph theory and important former results on the dense realizations
of CRNs. The proposed method is also capable of determining if no dynamically
equivalent weakly reversible structure exists for a given reaction network with
a previously fixed complex set.Comment: 18 pages, 9 figure
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