210,744 research outputs found
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 Complexity of Scheduling in Wireless Networks
We consider the problem of throughput-optimal scheduling in wireless networks subject to interference constraints. We model the interference using a family of K-hop interference models, under which no two links within a K-hop distance can successfully transmit at the same time. For a given K, we can obtain a throughput-optimal scheduling policy by solving the well-known maximum weighted matching problem. We show that for K > 1, the resulting problems are NP-Hard that cannot be approximated within a factor that grows polynomially with the number of nodes. Interestingly, for geometric unit-disk graphs that can be used to describe a wide range of wireless networks, the problems admit polynomial time approximation schemes within a factor arbitrarily close to 1. In these network settings, we also show that a simple greedy algorithm can provide a 49-approximation, and the maximal matching scheduling policy, which can be easily implemented in a distributed fashion, achieves a guaranteed fraction of the capacity region for "all K." The geometric constraints are crucial to obtain these throughput guarantees. These results are encouraging as they suggest that one can develop low-complexity distributed algorithms to achieve near-optimal throughput for a wide range of wireless networksopen1
Dynamic Control of Tunable Sub-optimal Algorithms for Scheduling of Time-varying Wireless Networks
It is well known that for ergodic channel processes the Generalized
Max-Weight Matching (GMWM) scheduling policy stabilizes the network for any
supportable arrival rate vector within the network capacity region. This
policy, however, often requires the solution of an NP-hard optimization
problem. This has motivated many researchers to develop sub-optimal algorithms
that approximate the GMWM policy in selecting schedule vectors. One implicit
assumption commonly shared in this context is that during the algorithm
runtime, the channel states remain effectively unchanged. This assumption may
not hold as the time needed to select near-optimal schedule vectors usually
increases quickly with the network size. In this paper, we incorporate channel
variations and the time-efficiency of sub-optimal algorithms into the scheduler
design, to dynamically tune the algorithm runtime considering the tradeoff
between algorithm efficiency and its robustness to changing channel states.
Specifically, we propose a Dynamic Control Policy (DCP) that operates on top of
a given sub-optimal algorithm, and dynamically but in a large time-scale
adjusts the time given to the algorithm according to queue backlog and channel
correlations. This policy does not require knowledge of the structure of the
given sub-optimal algorithm, and with low overhead can be implemented in a
distributed manner. Using a novel Lyapunov analysis, we characterize the
throughput stability region induced by DCP and show that our characterization
can be tight. We also show that the throughput stability region of DCP is at
least as large as that of any other static policy. Finally, we provide two case
studies to gain further intuition into the performance of DCP.Comment: Submitted for journal consideration. A shorter version was presented
in IEEE IWQoS 200
Inter-plane satellite matching in dense LEO constellations
Dense constellations of Low Earth Orbit (LEO) small satellites are envisioned
to make extensive use of the inter-satellite link (ISL). Within the same
orbital plane, the inter-satellite distances are preserved and the links are
rather stable. In contrast, the relative motion between planes makes the
inter-plane ISL challenging. In a dense set-up, each spacecraft has several
satellites in its coverage volume, but the time duration of each of these links
is small and the maximum number of active connections is limited by the
hardware. We analyze the matching problem of connecting satellites using the
inter-plane ISL for unicast transmissions. We present and evaluate the
performance of two solutions to the matching problem with any number of orbital
planes and up to two transceivers: a heuristic solution with the aim of
minimizing the total cost; and a Markovian solution to maintain the on-going
connections as long as possible. The Markovian algorithm reduces the time
needed to solve the matching up to 1000x and 10x with respect to the optimal
solution and to the heuristic solution, respectively, without compromising the
total cost. Our model includes power adaptation and optimizes the network
energy consumption as the exemplary cost in the evaluations, but any other
QoS-oriented KPI can be used instead
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
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