140,512 research outputs found
Non-Local Probes Do Not Help with Graph Problems
This work bridges the gap between distributed and centralised models of
computing in the context of sublinear-time graph algorithms. A priori, typical
centralised models of computing (e.g., parallel decision trees or centralised
local algorithms) seem to be much more powerful than distributed
message-passing algorithms: centralised algorithms can directly probe any part
of the input, while in distributed algorithms nodes can only communicate with
their immediate neighbours. We show that for a large class of graph problems,
this extra freedom does not help centralised algorithms at all: for example,
efficient stateless deterministic centralised local algorithms can be simulated
with efficient distributed message-passing algorithms. In particular, this
enables us to transfer existing lower bound results from distributed algorithms
to centralised local algorithms
Minimum Sum Edge Colorings of Multicycles
In the minimum sum edge coloring problem, we aim to assign natural numbers to
edges of a graph, so that adjacent edges receive different numbers, and the sum
of the numbers assigned to the edges is minimum. The {\em chromatic edge
strength} of a graph is the minimum number of colors required in a minimum sum
edge coloring of this graph. We study the case of multicycles, defined as
cycles with parallel edges, and give a closed-form expression for the chromatic
edge strength of a multicycle, thereby extending a theorem due to Berge. It is
shown that the minimum sum can be achieved with a number of colors equal to the
chromatic index. We also propose simple algorithms for finding a minimum sum
edge coloring of a multicycle. Finally, these results are generalized to a
large family of minimum cost coloring problems
Improved approximation for 3-dimensional matching via bounded pathwidth local search
One of the most natural optimization problems is the k-Set Packing problem,
where given a family of sets of size at most k one should select a maximum size
subfamily of pairwise disjoint sets. A special case of 3-Set Packing is the
well known 3-Dimensional Matching problem. Both problems belong to the Karp`s
list of 21 NP-complete problems. The best known polynomial time approximation
ratio for k-Set Packing is (k + eps)/2 and goes back to the work of Hurkens and
Schrijver [SIDMA`89], which gives (1.5 + eps)-approximation for 3-Dimensional
Matching. Those results are obtained by a simple local search algorithm, that
uses constant size swaps.
The main result of the paper is a new approach to local search for k-Set
Packing where only a special type of swaps is considered, which we call swaps
of bounded pathwidth. We show that for a fixed value of k one can search the
space of r-size swaps of constant pathwidth in c^r poly(|F|) time. Moreover we
present an analysis proving that a local search maximum with respect to O(log
|F|)-size swaps of constant pathwidth yields a polynomial time (k + 1 +
eps)/3-approximation algorithm, improving the best known approximation ratio
for k-Set Packing. In particular we improve the approximation ratio for
3-Dimensional Matching from 3/2 + eps to 4/3 + eps.Comment: To appear in proceedings of FOCS 201
Fast Parallel Fixed-Parameter Algorithms via Color Coding
Fixed-parameter algorithms have been successfully applied to solve numerous
difficult problems within acceptable time bounds on large inputs. However, most
fixed-parameter algorithms are inherently \emph{sequential} and, thus, make no
use of the parallel hardware present in modern computers. We show that parallel
fixed-parameter algorithms do not only exist for numerous parameterized
problems from the literature -- including vertex cover, packing problems,
cluster editing, cutting vertices, finding embeddings, or finding matchings --
but that there are parallel algorithms working in \emph{constant} time or at
least in time \emph{depending only on the parameter} (and not on the size of
the input) for these problems. Phrased in terms of complexity classes, we place
numerous natural parameterized problems in parameterized versions of AC. On
a more technical level, we show how the \emph{color coding} method can be
implemented in constant time and apply it to embedding problems for graphs of
bounded tree-width or tree-depth and to model checking first-order formulas in
graphs of bounded degree
Coresets Meet EDCS: Algorithms for Matching and Vertex Cover on Massive Graphs
As massive graphs become more prevalent, there is a rapidly growing need for
scalable algorithms that solve classical graph problems, such as maximum
matching and minimum vertex cover, on large datasets. For massive inputs,
several different computational models have been introduced, including the
streaming model, the distributed communication model, and the massively
parallel computation (MPC) model that is a common abstraction of
MapReduce-style computation. In each model, algorithms are analyzed in terms of
resources such as space used or rounds of communication needed, in addition to
the more traditional approximation ratio.
In this paper, we give a single unified approach that yields better
approximation algorithms for matching and vertex cover in all these models. The
highlights include:
* The first one pass, significantly-better-than-2-approximation for matching
in random arrival streams that uses subquadratic space, namely a
-approximation streaming algorithm that uses space
for constant .
* The first 2-round, better-than-2-approximation for matching in the MPC
model that uses subquadratic space per machine, namely a
-approximation algorithm with memory per
machine for constant .
By building on our unified approach, we further develop parallel algorithms
in the MPC model that give a -approximation to matching and an
-approximation to vertex cover in only MPC rounds and
memory per machine. These results settle multiple open
questions posed in the recent paper of Czumaj~et.al. [STOC 2018]
Dynamic Algorithms for the Massively Parallel Computation Model
The Massive Parallel Computing (MPC) model gained popularity during the last
decade and it is now seen as the standard model for processing large scale
data. One significant shortcoming of the model is that it assumes to work on
static datasets while, in practice, real-world datasets evolve continuously. To
overcome this issue, in this paper we initiate the study of dynamic algorithms
in the MPC model.
We first discuss the main requirements for a dynamic parallel model and we
show how to adapt the classic MPC model to capture them. Then we analyze the
connection between classic dynamic algorithms and dynamic algorithms in the MPC
model. Finally, we provide new efficient dynamic MPC algorithms for a variety
of fundamental graph problems, including connectivity, minimum spanning tree
and matching.Comment: Accepted to the 31st ACM Symposium on Parallelism in Algorithms and
Architectures (SPAA 2019
Best of Two Local Models: Local Centralized and Local Distributed Algorithms
We consider two models of computation: centralized local algorithms and local
distributed algorithms. Algorithms in one model are adapted to the other model
to obtain improved algorithms.
Distributed vertex coloring is employed to design improved centralized local
algorithms for: maximal independent set, maximal matching, and an approximation
scheme for maximum (weighted) matching over bounded degree graphs. The
improvement is threefold: the algorithms are deterministic, stateless, and the
number of probes grows polynomially in , where is the number of
vertices of the input graph.
The recursive centralized local improvement technique by Nguyen and
Onak~\cite{onak2008} is employed to obtain an improved distributed
approximation scheme for maximum (weighted) matching. The improvement is
twofold: we reduce the number of rounds from to for a
wide range of instances and, our algorithms are deterministic rather than
randomized
Towards Work-Efficient Parallel Parameterized Algorithms
Parallel parameterized complexity theory studies how fixed-parameter
tractable (fpt) problems can be solved in parallel. Previous theoretical work
focused on parallel algorithms that are very fast in principle, but did not
take into account that when we only have a small number of processors (between
2 and, say, 1024), it is more important that the parallel algorithms are
work-efficient. In the present paper we investigate how work-efficient fpt
algorithms can be designed. We review standard methods from fpt theory, like
kernelization, search trees, and interleaving, and prove trade-offs for them
between work efficiency and runtime improvements. This results in a toolbox for
developing work-efficient parallel fpt algorithms.Comment: Prior full version of the paper that will appear in Proceedings of
the 13th International Conference and Workshops on Algorithms and Computation
(WALCOM 2019), February 27 - March 02, 2019, Guwahati, India. The final
authenticated version is available online at
https://doi.org/10.1007/978-3-030-10564-8_2
Parallel Sort-Based Matching for Data Distribution Management on Shared-Memory Multiprocessors
In this paper we consider the problem of identifying intersections between
two sets of d-dimensional axis-parallel rectangles. This is a common problem
that arises in many agent-based simulation studies, and is of central
importance in the context of High Level Architecture (HLA), where it is at the
core of the Data Distribution Management (DDM) service. Several realizations of
the DDM service have been proposed; however, many of them are either
inefficient or inherently sequential. These are serious limitations since
multicore processors are now ubiquitous, and DDM algorithms -- being
CPU-intensive -- could benefit from additional computing power. We propose a
parallel version of the Sort-Based Matching algorithm for shared-memory
multiprocessors. Sort-Based Matching is one of the most efficient serial
algorithms for the DDM problem, but is quite difficult to parallelize due to
data dependencies. We describe the algorithm and compute its asymptotic running
time; we complete the analysis by assessing its performance and scalability
through extensive experiments on two commodity multicore systems based on a
dual socket Intel Xeon processor, and a single socket Intel Core i7 processor.Comment: Proceedings of the 21-th ACM/IEEE International Symposium on
Distributed Simulation and Real Time Applications (DS-RT 2017). Best Paper
Award @DS-RT 201
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