25,487 research outputs found
Properties of graphs with large girth
This thesis is devoted to the analysis of a class of
iterative probabilistic algorithms in regular graphs, called
locally greedy algorithms, which will provide bounds for
graph functions in regular graphs with large girth. This class is
useful because, by conveniently setting the parameters associated
with it, we may derive algorithms for some well-known graph
problems, such as algorithms to find a large independent set, a
large induced forest, or even a small dominating set in an input
graph G. The name ``locally greedy" comes from the fact that, in
an algorithm of this class, the probability associated with the
random selection of a vertex v is determined by the current
state of the vertices within some fixed distance of v.
Given r > 2 and an r-regular graph G, we determine the
expected performance of a locally greedy algorithm in G,
depending on the girth g of the input and on the degree r of
its vertices. When the girth of the graph is sufficiently large,
this analysis leads to new lower bounds on the independence number
of G and on the maximum number of vertices in an induced forest
in G, which, in both cases, improve the bounds previously known.
It also implies bounds on the same functions in graphs with large
girth and maximum degree r and in random regular graphs. As a
matter of fact, the asymptotic lower bounds on the cardinality of
a maximum induced forest in a random regular graph improve earlier
bounds, while, for independent sets, our bounds coincide with
asymptotic lower bounds first obtained by Wormald. Our result
provides an alternative proof of these bounds which avoids sharp
concentration arguments.
The main contribution of this work lies in the method presented
rather than in these particular new bounds. This method allows us,
in some sense, to directly analyse prioritised algorithms in
regular graphs, so that the class of locally greedy algorithms, or
slight modifications thereof, may be applied to a wider range of
problems in regular graphs with large girth
Greed is Good for Deterministic Scale-Free Networks
Large real-world networks typically follow a power-law degree distribution. To study such networks, numerous random graph models have been proposed. However, real-world networks are not drawn at random. In fact, the behavior of real-world networks and random graph models can be the complete opposite of one another, depending on the considered property. Brach, Cygan, Lacki, and Sankowski [SODA 2016] introduced two natural deterministic conditions: (1) a power-law upper bound on the degree distribution (PLB-U) and (2) power-law neighborhoods, that is, the degree distribution of neighbors of each vertex is also upper bounded by a power law (PLB-N). They showed that many real-world networks satisfy both deterministic properties and exploit them to design faster algorithms for a number of classical graph problems like transitive closure, maximum matching, determinant, PageRank, matrix inverse, counting triangles and maximum clique.
We complement the work of Brach et al. by showing that some well-studied random graph models exhibit both the mentioned PLB properties and additionally also a power-law lower bound on the degree distribution (PLB-L). All three properties hold with high probability for Chung-Lu Random Graphs and Geometric Inhomogeneous Random Graphs and almost surely for Hyperbolic Random Graphs. As a consequence, all results of Brach et al. also hold with high probability for Chung-Lu Random Graphs and Geometric Inhomogeneous Random Graphs and almost surely for Hyperbolic Random Graphs.
In the second part of this work we study three classical NP-hard combinatorial optimization problems on PLB networks. It is known that on general graphs, a greedy algorithm, which chooses nodes in the order of their degree, only achieves an approximation factor of asymptotically at least logarithmic in the maximum degree for Minimum Vertex Cover and Minimum Dominating Set, and an approximation factor of asymptotically at least the maximum degree for Maximum Independent Set. We prove that the PLB-U property suffices such that the greedy approach achieves a constant-factor approximation for all three problems. We also show that all three combinatorial optimization problems are APX-complete, even if all PLB-properties hold. Hence, a PTAS cannot be expected, unless P=NP
Recommendation Subgraphs for Web Discovery
Recommendations are central to the utility of many websites including
YouTube, Quora as well as popular e-commerce stores. Such sites typically
contain a set of recommendations on every product page that enables visitors to
easily navigate the website. Choosing an appropriate set of recommendations at
each page is one of the key features of backend engines that have been deployed
at several e-commerce sites.
Specifically at BloomReach, an engine consisting of several independent
components analyzes and optimizes its clients' websites. This paper focuses on
the structure optimizer component which improves the website navigation
experience that enables the discovery of novel content.
We begin by formalizing the concept of recommendations used for discovery. We
formulate this as a natural graph optimization problem which in its simplest
case, reduces to a bipartite matching problem. In practice, solving these
matching problems requires superlinear time and is not scalable. Also,
implementing simple algorithms is critical in practice because they are
significantly easier to maintain in production. This motivated us to analyze
three methods for solving the problem in increasing order of sophistication: a
sampling algorithm, a greedy algorithm and a more involved partitioning based
algorithm.
We first theoretically analyze the performance of these three methods on
random graph models characterizing when each method will yield a solution of
sufficient quality and the parameter ranges when more sophistication is needed.
We complement this by providing an empirical analysis of these algorithms on
simulated and real-world production data. Our results confirm that it is not
always necessary to implement complicated algorithms in the real-world and that
very good practical results can be obtained by using heuristics that are backed
by the confidence of concrete theoretical guarantees
A More Reliable Greedy Heuristic for Maximum Matchings in Sparse Random Graphs
We propose a new greedy algorithm for the maximum cardinality matching
problem. We give experimental evidence that this algorithm is likely to find a
maximum matching in random graphs with constant expected degree c>0,
independent of the value of c. This is contrary to the behavior of commonly
used greedy matching heuristics which are known to have some range of c where
they probably fail to compute a maximum matching
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