1,310 research outputs found
Towards a complexity theory for the congested clique
The congested clique model of distributed computing has been receiving
attention as a model for densely connected distributed systems. While there has
been significant progress on the side of upper bounds, we have very little in
terms of lower bounds for the congested clique; indeed, it is now know that
proving explicit congested clique lower bounds is as difficult as proving
circuit lower bounds.
In this work, we use various more traditional complexity-theoretic tools to
build a clearer picture of the complexity landscape of the congested clique:
-- Nondeterminism and beyond: We introduce the nondeterministic congested
clique model (analogous to NP) and show that there is a natural canonical
problem family that captures all problems solvable in constant time with
nondeterministic algorithms. We further generalise these notions by introducing
the constant-round decision hierarchy (analogous to the polynomial hierarchy).
-- Non-constructive lower bounds: We lift the prior non-uniform counting
arguments to a general technique for proving non-constructive uniform lower
bounds for the congested clique. In particular, we prove a time hierarchy
theorem for the congested clique, showing that there are decision problems of
essentially all complexities, both in the deterministic and nondeterministic
settings.
-- Fine-grained complexity: We map out relationships between various natural
problems in the congested clique model, arguing that a reduction-based
complexity theory currently gives us a fairly good picture of the complexity
landscape of the congested clique
Max flow vitality in general and -planar graphs
The \emph{vitality} of an arc/node of a graph with respect to the maximum
flow between two fixed nodes and is defined as the reduction of the
maximum flow caused by the removal of that arc/node. In this paper we address
the issue of determining the vitality of arcs and/or nodes for the maximum flow
problem. We show how to compute the vitality of all arcs in a general
undirected graph by solving only max flow instances and, In
-planar graphs (directed or undirected) we show how to compute the vitality
of all arcs and all nodes in worst-case time. Moreover, after
determining the vitality of arcs and/or nodes, and given a planar embedding of
the graph, we can determine the vitality of a `contiguous' set of arcs/nodes in
time proportional to the size of the set.Comment: 12 pages, 3 figure
A class of pairwise models for epidemic dynamics on weighted networks
In this paper, we study the (susceptible-infected-susceptible) and
(susceptible-infected-removed) epidemic models on undirected, weighted
networks by deriving pairwise-type approximate models coupled with
individual-based network simulation. Two different types of
theoretical/synthetic weighted network models are considered. Both models start
from non-weighted networks with fixed topology followed by the allocation of
link weights in either (i) random or (ii) fixed/deterministic way. The pairwise
models are formulated for a general discrete distribution of weights, and these
models are then used in conjunction with network simulation to evaluate the
impact of different weight distributions on epidemic threshold and dynamics in
general. For the dynamics, the basic reproductive ratio is
computed, and we show that (i) for both network models is maximised if
all weights are equal, and (ii) when the two models are equally matched, the
networks with a random weight distribution give rise to a higher value.
The models are also used to explore the agreement between the pairwise and
simulation models for different parameter combinations
Cooperation through social influence
We consider a simple and altruistic multiagent system in which the agents are eager to perform a collective task but where their real engagement depends on the willingness to perform the task of other influential agents. We model this scenario by an influence game, a cooperative simple game in which a team (or coalition) of players succeeds if it is able to convince enough agents to participate in the task (to vote in favor of a decision). We take the linear threshold model as the influence model. We show first the expressiveness of influence games showing that they capture the class of simple games. Then we characterize the computational complexity of various problems on influence games, including measures (length and width), values (Shapley-Shubik and Banzhaf) and properties (of teams and players). Finally, we analyze those problems for some particular extremal cases, with respect to the propagation of influence, showing tighter complexity characterizations.Peer ReviewedPostprint (author’s final draft
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