14 research outputs found
Twenty-Two New Approximate Proof Labeling Schemes
Introduced by Korman, Kutten, and Peleg (Distributed Computing 2005), a proof labeling scheme (PLS) is a system dedicated to verifying that a given configuration graph satisfies a certain property. It is composed of a centralized prover, whose role is to generate a proof for yes-instances in the form of an assignment of labels to the nodes, and a distributed verifier, whose role is to verify the validity of the proof by local means and accept it if and only if the property is satisfied. To overcome lower bounds on the label size of PLSs for certain graph properties, Censor-Hillel, Paz, and Perry (SIROCCO 2017) introduced the notion of an approximate proof labeling scheme (APLS) that allows the verifier to accept also some no-instances as long as they are not "too far" from satisfying the property.
The goal of the current paper is to advance our understanding of the power and limitations of APLSs. To this end, we formulate the notion of APLSs in terms of distributed graph optimization problems (OptDGPs) and develop two generic methods for the design of APLSs. These methods are then applied to various classic OptDGPs, obtaining twenty-two new APLSs. An appealing characteristic of our APLSs is that they are all sequentially efficient in the sense that both the prover and the verifier are required to run in (sequential) polynomial time. On the negative side, we establish "combinatorial" lower bounds on the label size for some of the aforementioned OptDGPs that demonstrate the optimality of our corresponding APLSs. For other OptDGPs, we establish conditional lower bounds that exploit the sequential efficiency of the verifier alone (under the assumption that NP ? co-NP) or that of both the verifier and the prover (under the assumption that P ? NP, with and without the unique games conjecture)
Distributed distance-r covering problems on sparse high-girth graphs
We prove that the distance-r dominating set, distance-r connected dominating set,
distance-r vertex cover, and distance-r connected vertex cover problems admit constant
factor approximations in the CONGEST model of distributed computing in a constant
number of rounds on classes of sparse high-girth graphs. In this paper, sparse means
bounded expansion, and high-girth means girth at least 4r + 2. Our algorithm is quite
simple; however, the proof of its approximation guarantee is non-trivial. To complement
the algorithmic results, we show tightness of our approximation by providing a loosely
matching lower bound on rings.
Our result is the first to show the existence of constant-factor approximations in a constant
number of rounds in non-trivial classes of graphs for distance-r covering problems
Input-Dynamic Distributed Algorithms for Communication Networks
Consider a distributed task where the communication network is fixed but the
local inputs given to the nodes of the distributed system may change over time.
In this work, we explore the following question: if some of the local inputs
change, can an existing solution be updated efficiently, in a dynamic and
distributed manner?
To address this question, we define the batch dynamic CONGEST model in which
we are given a bandwidth-limited communication network and a dynamic edge
labelling defines the problem input. The task is to maintain a solution to a
graph problem on the labeled graph under batch changes. We investigate, when a
batch of edge label changes arrive,
-- how much time as a function of we need to update an existing
solution, and
-- how much information the nodes have to keep in local memory between
batches in order to update the solution quickly.
Our work lays the foundations for the theory of input-dynamic distributed
network algorithms. We give a general picture of the complexity landscape in
this model, design both universal algorithms and algorithms for concrete
problems, and present a general framework for lower bounds. In particular, we
derive non-trivial upper bounds for two selected, contrasting problems:
maintaining a minimum spanning tree and detecting cliques