851 research outputs found
A Quasi-Polynomial Time Partition Oracle for Graphs with an Excluded Minor
Motivated by the problem of testing planarity and related properties, we
study the problem of designing efficient {\em partition oracles}. A {\em
partition oracle} is a procedure that, given access to the incidence lists
representation of a bounded-degree graph and a parameter \eps,
when queried on a vertex , returns the part (subset of vertices) which
belongs to in a partition of all graph vertices. The partition should be
such that all parts are small, each part is connected, and if the graph has
certain properties, the total number of edges between parts is at most \eps
|V|. In this work we give a partition oracle for graphs with excluded minors
whose query complexity is quasi-polynomial in 1/\eps, thus improving on the
result of Hassidim et al. ({\em Proceedings of FOCS 2009}) who gave a partition
oracle with query complexity exponential in 1/\eps. This improvement implies
corresponding improvements in the complexity of testing planarity and other
properties that are characterized by excluded minors as well as sublinear-time
approximation algorithms that work under the promise that the graph has an
excluded minor.Comment: 13 pages, 1 figur
A Local Algorithm for the Sparse Spanning Graph Problem
Constructing a sparse spanning subgraph is a fundamental primitive in graph
theory. In this paper, we study this problem in the Centralized Local model,
where the goal is to decide whether an edge is part of the spanning subgraph by
examining only a small part of the input; yet, answers must be globally
consistent and independent of prior queries.
Unfortunately, maximally sparse spanning subgraphs, i.e., spanning trees,
cannot be constructed efficiently in this model. Therefore, we settle for a
spanning subgraph containing at most edges (where is the
number of vertices and is a given approximation/sparsity
parameter). We achieve query complexity of
, (-notation hides
polylogarithmic factors in ). where is the maximum degree of the
input graph. Our algorithm is the first to do so on arbitrary bounded degree
graphs. Moreover, we achieve the additional property that our algorithm outputs
a spanner, i.e., distances are approximately preserved. With high probability,
for each deleted edge there is a path of
hops in the output that connects its endpoints
ALLOCATION OF PRIZES IN CONTESTS WITH PARTICIPATION CONSTRAINTS
We study all-pay contests with an exogenous minimal effort constraint where a player can participate in a contest only if his effort (output) is equal to or higher than the minimal effort constraint. Contestants are privately informed about a parameter (ability) that affects their cost of effort. The designer decides about the size and the number of prizes. We analyze the optimal prize allocation for the contest designer who wishes to maximize either the total effort or the highest effort. It is shown that if the minimal effort constraint is relatively high, the winner-take-all contest in which the contestant with the highest effort wins the entire prize sum does not maximize the expected total effort nor the expected highest effort. In that case, the random contest in which the entire prize sum is equally allocated to all the participants yields a higher expected total effort as well as a higher expected highest effort than the winner-take-all contest.Winner-take-all contests, all-pay auctions, participation constraints.
A Local Algorithm for Constructing Spanners in Minor-Free Graphs
Constructing a spanning tree of a graph is one of the most basic tasks in
graph theory. We consider this problem in the setting of local algorithms: one
wants to quickly determine whether a given edge is in a specific spanning
tree, without computing the whole spanning tree, but rather by inspecting the
local neighborhood of . The challenge is to maintain consistency. That is,
to answer queries about different edges according to the same spanning tree.
Since it is known that this problem cannot be solved without essentially
viewing all the graph, we consider the relaxed version of finding a spanning
subgraph with edges (where is the number of vertices and
is a given sparsity parameter). It is known that this relaxed
problem requires inspecting edges in general graphs, which
motivates the study of natural restricted families of graphs. One such family
is the family of graphs with an excluded minor. For this family there is an
algorithm that achieves constant success probability, and inspects
edges (for each edge it is queried
on), where is the maximum degree in the graph and is the size of the
excluded minor. The distances between pairs of vertices in the spanning
subgraph are at most a factor of larger than in
.
In this work, we show that for an input graph that is -minor free for any
of size , this task can be performed by inspecting only edges. The distances between pairs of vertices in the spanning
subgraph are at most a factor of larger
than in . Furthermore, the error probability of the new algorithm is
significantly improved to . This algorithm can also be easily
adapted to yield an efficient algorithm for the distributed setting
Testing bounded arboricity
In this paper we consider the problem of testing whether a graph has bounded
arboricity. The family of graphs with bounded arboricity includes, among
others, bounded-degree graphs, all minor-closed graph classes (e.g. planar
graphs, graphs with bounded treewidth) and randomly generated preferential
attachment graphs. Graphs with bounded arboricity have been studied extensively
in the past, in particular since for many problems they allow for much more
efficient algorithms and/or better approximation ratios.
We present a tolerant tester in the sparse-graphs model. The sparse-graphs
model allows access to degree queries and neighbor queries, and the distance is
defined with respect to the actual number of edges. More specifically, our
algorithm distinguishes between graphs that are -close to having
arboricity and graphs that -far from having
arboricity , where is an absolute small constant. The query
complexity and running time of the algorithm are
where denotes
the number of vertices and denotes the number of edges. In terms of the
dependence on and this bound is optimal up to poly-logarithmic factors
since queries are necessary (and .
We leave it as an open question whether the dependence on can be
improved from quasi-polynomial to polynomial. Our techniques include an
efficient local simulation for approximating the outcome of a global (almost)
forest-decomposition algorithm as well as a tailored procedure of edge
sampling
Faster and Simpler Distributed Algorithms for Testing and Correcting Graph Properties in the CONGEST-Model
In this paper we present distributed testing algorithms of graph properties
in the CONGEST-model [Censor-Hillel et al. 2016]. We present one-sided error
testing algorithms in the general graph model.
We first describe a general procedure for converting -testers with
a number of rounds , where denotes the diameter of the graph, to
rounds, where is the number of
processors of the network. We then apply this procedure to obtain an optimal
tester, in terms of , for testing bipartiteness, whose round complexity is
, which improves over the -round algorithm by Censor-Hillel et al. (DISC 2016). Moreover, for
cycle-freeness, we obtain a \emph{corrector} of the graph that locally corrects
the graph so that the corrected graph is acyclic. Note that, unlike a tester, a
corrector needs to mend the graph in many places in the case that the graph is
far from having the property.
In the second part of the paper we design algorithms for testing whether the
network is -free for any connected of size up to four with round
complexity of . This improves over the
-round algorithms for testing triangle freeness by
Censor-Hillel et al. (DISC 2016) and for testing excluded graphs of size by
Fraigniaud et al. (DISC 2016).
In the last part we generalize the global tester by Iwama and Yoshida (ITCS
2014) of testing -path freeness to testing the exclusion of any tree of
order . We then show how to simulate this algorithm in the CONGEST-model in
rounds
A Sublinear Tester for Outerplanarity (and Other Forbidden Minors) With One-Sided Error
We consider one-sided error property testing of -minor freeness
in bounded-degree graphs for any finite family of graphs that
contains a minor of , the -circus graph, or the -grid
for any . This includes, for instance, testing whether a graph
is outerplanar or a cactus graph. The query complexity of our algorithm in
terms of the number of vertices in the graph, , is . Czumaj et~al.\ showed that cycle-freeness and -minor
freeness can be tested with query complexity by using
random walks, and that testing -minor freeness for any that contains a
cycles requires queries. In contrast to these results, we
analyze the structure of the graph and show that either we can find a subgraph
of sublinear size that includes the forbidden minor , or we can find a pair
of disjoint subsets of vertices whose edge-cut is large, which induces an
-minor.Comment: extended to testing outerplanarity, full version of ICALP pape
Characterizing Efficient Referrals in Social Networks
Users of social networks often focus on specific areas of that network,
leading to the well-known "filter bubble" effect. Connecting people to a new
area of the network in a way that will cause them to become active in that area
could help alleviate this effect and improve social welfare.
Here we present preliminary analysis of network referrals, that is, attempts
by users to connect peers to other areas of the network. We classify these
referrals by their efficiency, i.e., the likelihood that a referral will result
in a user becoming active in the new area of the network. We show that by using
features describing past experience of the referring author and the content of
their messages we are able to predict whether referral will be effective,
reaching an AUC of 0.87 for those users most experienced in writing efficient
referrals. Our results represent a first step towards algorithmically
constructing efficient referrals with the goal of mitigating the "filter
bubble" effect pervasive in on line social networks.Comment: Accepted to the 2018 Web conference (WWW2018
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