34 research outputs found
Simple Distributed Weighted Matchings
Wattenhofer [WW04] derive a complicated distributed algorithm to compute a
weighted matching of an arbitrary weighted graph, that is at most a factor 5
away from the maximum weighted matching of that graph. We show that a variant
of the obvious sequential greedy algorithm [Pre99], that computes a weighted
matching at most a factor 2 away from the maximum, is easily distributed. This
yields the best known distributed approximation algorithm for this problem so
far
Local algorithms in (weakly) coloured graphs
A local algorithm is a distributed algorithm that completes after a constant
number of synchronous communication rounds. We present local approximation
algorithms for the minimum dominating set problem and the maximum matching
problem in 2-coloured and weakly 2-coloured graphs. In a weakly 2-coloured
graph, both problems admit a local algorithm with the approximation factor
, where is the maximum degree of the graph. We also give
a matching lower bound proving that there is no local algorithm with a better
approximation factor for either of these problems. Furthermore, we show that
the stronger assumption of a 2-colouring does not help in the case of the
dominating set problem, but there is a local approximation scheme for the
maximum matching problem in 2-coloured graphs.Comment: 14 pages, 3 figure
Cross-layer Congestion Control, Routing and Scheduling Design in Ad Hoc Wireless Networks
This paper considers jointly optimal design of crosslayer congestion control, routing and scheduling for ad hoc
wireless networks. We first formulate the rate constraint and scheduling constraint using multicommodity flow variables, and formulate resource allocation in networks with fixed wireless channels (or single-rate wireless devices that can mask channel variations) as a utility maximization problem with these constraints.
By dual decomposition, the resource allocation problem
naturally decomposes into three subproblems: congestion control,
routing and scheduling that interact through congestion price.
The global convergence property of this algorithm is proved. We
next extend the dual algorithm to handle networks with timevarying
channels and adaptive multi-rate devices. The stability
of the resulting system is established, and its performance is
characterized with respect to an ideal reference system which
has the best feasible rate region at link layer.
We then generalize the aforementioned results to a general
model of queueing network served by a set of interdependent
parallel servers with time-varying service capabilities, which
models many design problems in communication networks. We
show that for a general convex optimization problem where a
subset of variables lie in a polytope and the rest in a convex set,
the dual-based algorithm remains stable and optimal when the
constraint set is modulated by an irreducible finite-state Markov
chain. This paper thus presents a step toward a systematic way
to carry out cross-layer design in the framework of “layering as
optimization decomposition” for time-varying channel models
Distributed Approximation of Maximum Independent Set and Maximum Matching
We present a simple distributed -approximation algorithm for maximum
weight independent set (MaxIS) in the model which completes
in rounds, where is the maximum
degree, is the number of rounds needed to compute a maximal
independent set (MIS) on , and is the maximum weight of a node. %Whether
our algorithm is randomized or deterministic depends on the \texttt{MIS}
algorithm used as a black-box.
Plugging in the best known algorithm for MIS gives a randomized solution in
rounds, where is the number of nodes.
We also present a deterministic -round algorithm based
on coloring.
We then show how to use our MaxIS approximation algorithms to compute a
-approximation for maximum weight matching without incurring any additional
round penalty in the model. We use a known reduction for
simulating algorithms on the line graph while incurring congestion, but we show
our algorithm is part of a broad family of \emph{local aggregation algorithms}
for which we describe a mechanism that allows the simulation to run in the
model without an additional overhead.
Next, we show that for maximum weight matching, relaxing the approximation
factor to () allows us to devise a distributed algorithm
requiring rounds for any constant
. For the unweighted case, we can even obtain a
-approximation in this number of rounds. These algorithms are
the first to achieve the provably optimal round complexity with respect to
dependency on
Distributed Maximum Matching Verification in CONGEST
We study the maximum cardinality matching problem in a standard distributed setting, where the nodes V of a given n-node network graph G = (V,E) communicate over the edges E in synchronous rounds. More specifically, we consider the distributed CONGEST model, where in each round, each node of G can send an O(log n)-bit message to each of its neighbors. We show that for every graph G and a matching M of G, there is a randomized CONGEST algorithm to verify M being a maximum matching of G in time O(|M|) and disprove it in time O(D + ?), where D is the diameter of G and ? is the length of a shortest augmenting path. We hope that our algorithm constitutes a significant step towards developing a CONGEST algorithm to compute a maximum matching in time O?(s^*), where s^* is the size of a maximum matching
Distributed CONGEST Approximation of Weighted Vertex Covers and Matchings
We provide CONGEST model algorithms for approximating minimum weighted vertex
cover and the maximum weighted matching. For bipartite graphs, we show that a
-approximate weighted vertex cover can be computed
deterministically in polylogarithmic time. This generalizes a corresponding
result for the unweighted vertex cover problem shown in [Faour, Kuhn; OPODIS
'20]. Moreover, we show that in general weighted graph families that are closed
under taking subgraphs and in which we can compute an independent set of weight
at least a -fraction of the total weight, one can compute a
-approximate weighted vertex cover in
polylogarithmic time in the CONGEST model. Our result in particular implies
that in graphs of arboricity , one can compute a
-approximate weighted vertex cover.
For maximum weighted matchings, we show that a -approximate
solution can be computed deterministically in polylogarithmic CONGEST rounds
(for constant ). We also provide a more efficient randomized
algorithm. Our algorithm generalizes results of [Lotker, Patt-Shamir, Pettie;
SPAA '08] and [Bar-Yehuda, Hillel, Ghaffari, Schwartzman; PODC '17] for the
unweighted case.
Finally, we show that even in the LOCAL model and in bipartite graphs of
degree , if for some constant
, then computing a -approximation for the
unweighted minimum vertex cover problem requires rounds. This generalizes aresult of [G\"o\"os, Suomela;
DISC '12], who showed that computing a -approximation in
such graphs requires rounds
Distributed local approximation algorithms for maximum matching in graphs and hypergraphs
We describe approximation algorithms in Linial's classic LOCAL model of
distributed computing to find maximum-weight matchings in a hypergraph of rank
. Our main result is a deterministic algorithm to generate a matching which
is an -approximation to the maximum weight matching, running in rounds. (Here, the
notations hides and factors).
This is based on a number of new derandomization techniques extending methods
of Ghaffari, Harris & Kuhn (2017).
As a main application, we obtain nearly-optimal algorithms for the
long-studied problem of maximum-weight graph matching. Specifically, we get a
approximation algorithm using randomized time and deterministic time.
The second application is a faster algorithm for hypergraph maximal matching,
a versatile subroutine introduced in Ghaffari et al. (2017) for a variety of
local graph algorithms. This gives an algorithm for -edge-list
coloring in rounds deterministically or
rounds randomly. Another consequence (with
additional optimizations) is an algorithm which generates an edge-orientation
with out-degree at most for a graph of
arboricity ; for fixed this runs in
rounds deterministically or rounds randomly