14 research outputs found
Large Flocks of Small Birds: on the Minimal Size of Population Protocols
Population protocols are a well established model of distributed computation by mobile finite-state agents with very limited storage. A classical result establishes that population protocols compute exactly predicates definable in Presburger arithmetic. We initiate the study of the minimal amount of memory required to compute a given predicate as a function of its size. We present results on the predicates x >= n for n in N, and more generally on the predicates corresponding to systems of linear inequalities. We show that they can be computed by protocols with O(log n) states (or, more generally, logarithmic in the coefficients of the predicate), and that, surprisingly, some families of predicates can be computed by protocols with O(log log n) states. We give essentially matching lower bounds for the class of 1-aware protocols
(Near) Optimal Adaptivity Gaps for Stochastic Multi-Value Probing
Consider a kidney-exchange application where we want to find a max-matching in a random graph. To find whether an edge e exists, we need to perform an expensive test, in which case the edge e appears independently with a known probability p_e. Given a budget on the total cost of the tests, our goal is to find a testing strategy that maximizes the expected maximum matching size.
The above application is an example of the stochastic probing problem. In general the optimal stochastic probing strategy is difficult to find because it is adaptive - decides on the next edge to probe based on the outcomes of the probed edges. An alternate approach is to show the adaptivity gap is small, i.e., the best non-adaptive strategy always has a value close to the best adaptive strategy. This allows us to focus on designing non-adaptive strategies that are much simpler. Previous works, however, have focused on Bernoulli random variables that can only capture whether an edge appears or not. In this work we introduce a multi-value stochastic probing problem, which can also model situations where the weight of an edge has a probability distribution over multiple values.
Our main technical contribution is to obtain (near) optimal bounds for the (worst-case) adaptivity gaps for multi-value stochastic probing over prefix-closed constraints. For a monotone submodular function, we show the adaptivity gap is at most 2 and provide a matching lower bound. For a weighted rank function of a k-extendible system (a generalization of intersection of k matroids), we show the adaptivity gap is between O(k log k) and k. None of these results were known even in the Bernoulli case where both our upper and lower bounds also apply, thereby resolving an open question of Gupta et al. [Gupta et al., 2017]
Sublinear Time Hypergraph Sparsification via Cut and Edge Sampling Queries
The problem of sparsifying a graph or a hypergraph while approximately
preserving its cut structure has been extensively studied and has many
applications. In a seminal work, Bencz\'ur and Karger (1996) showed that given
any -vertex undirected weighted graph and a parameter , there is a near-linear time algorithm that outputs a weighted subgraph
of of size such that the weight of every
cut in is preserved to within a -factor in . The
graph is referred to as a {\em -approximate cut
sparsifier} of . Subsequent recent work has obtained a similar result for
the more general problem of hypergraph cut sparsifiers. However, all known
sparsification algorithms require time where denotes the
number of vertices and denotes the number of hyperedges in the hypergraph.
Since can be exponentially large in , a natural question is if it is
possible to create a hypergraph cut sparsifier in time polynomial in , {\em
independent of the number of edges}. We resolve this question in the
affirmative, giving the first sublinear time algorithm for this problem, given
appropriate query access to the hypergraph.Comment: ICALP 202
Online Matching on -Uniform Hypergraphs
The online matching problem was introduced by Karp, Vazirani and Vazirani
(STOC 1990) on bipartite graphs with vertex arrivals. It is well-known that the
optimal competitive ratio is for both integral and fractional versions
of the problem. Since then, there has been considerable effort to find optimal
competitive ratios for other related settings. In this work, we go beyond the
graph case and study the online matching problem on -uniform hypergraphs.
For , we provide an optimal primal-dual fractional algorithm, which
achieves a competitive ratio of . As our main
technical contribution, we present a carefully constructed adversarial
instance, which shows that this ratio is in fact optimal. It combines ideas
from known hard instances for bipartite graphs under the edge-arrival and
vertex-arrival models. For , we give a simple integral algorithm which
performs better than greedy when the online nodes have bounded degree. As a
corollary, it achieves the optimal competitive ratio of 1/2 on 3-uniform
hypergraphs when every online node has degree at most 2. This is because the
special case where every online node has degree 1 is equivalent to the
edge-arrival model on graphs, for which an upper bound of 1/2 is known
Polynomial Convergence of Bandit No-Regret Dynamics in Congestion Games
We introduce an online learning algorithm in the bandit feedback model that,
once adopted by all agents of a congestion game, results in game-dynamics that
converge to an -approximate Nash Equilibrium in a polynomial number
of rounds with respect to , the number of players and the number of
available resources. The proposed algorithm also guarantees sublinear regret to
any agent adopting it. As a result, our work answers an open question from
arXiv:2206.01880 and extends the recent results of arXiv:2306.15543 to the
bandit feedback model. We additionally establish that our online learning
algorithm can be implemented in polynomial time for the important special case
of Network Congestion Games on Directed Acyclic Graphs (DAG) by constructing an
exact -barycentric spanner for DAGs
Near-Quadratic Lower Bounds for Two-Pass Graph Streaming Algorithms
We prove that any two-pass graph streaming algorithm for the -
reachability problem in -vertex directed graphs requires near-quadratic
space of bits. As a corollary, we also obtain near-quadratic space
lower bounds for several other fundamental problems including maximum bipartite
matching and (approximate) shortest path in undirected graphs.
Our results collectively imply that a wide range of graph problems admit
essentially no non-trivial streaming algorithm even when two passes over the
input is allowed. Prior to our work, such impossibility results were only known
for single-pass streaming algorithms, and the best two-pass lower bounds only
ruled out space algorithms, leaving open a large gap between
(trivial) upper bounds and lower bounds