146,726 research outputs found
The Metric Nearness Problem
Metric nearness refers to the problem of optimally restoring metric properties to
distance measurements that happen to be nonmetric due to measurement errors or otherwise. Metric
data can be important in various settings, for example, in clustering, classification, metric-based
indexing, query processing, and graph theoretic approximation algorithms. This paper formulates
and solves the metric nearness problem: Given a set of pairwise dissimilarities, find a “nearest” set
of distances that satisfy the properties of a metric—principally the triangle inequality. For solving
this problem, the paper develops efficient triangle fixing algorithms that are based on an iterative
projection method. An intriguing aspect of the metric nearness problem is that a special case turns
out to be equivalent to the all pairs shortest paths problem. The paper exploits this equivalence and
develops a new algorithm for the latter problem using a primal-dual method. Applications to graph
clustering are provided as an illustration. We include experiments that demonstrate the computational
superiority of triangle fixing over general purpose convex programming software. Finally, we
conclude by suggesting various useful extensions and generalizations to metric nearness
Efficient Construction of Probabilistic Tree Embeddings
In this paper we describe an algorithm that embeds a graph metric
on an undirected weighted graph into a distribution of tree metrics
such that for every pair , and
. Such embeddings have
proved highly useful in designing fast approximation algorithms, as many hard
problems on graphs are easy to solve on tree instances. For a graph with
vertices and edges, our algorithm runs in time with high
probability, which improves the previous upper bound of shown by
Mendel et al.\,in 2009.
The key component of our algorithm is a new approximate single-source
shortest-path algorithm, which implements the priority queue with a new data
structure, the "bucket-tree structure". The algorithm has three properties: it
only requires linear time in the number of edges in the input graph; the
computed distances have a distance preserving property; and when computing the
shortest-paths to the -nearest vertices from the source, it only requires to
visit these vertices and their edge lists. These properties are essential to
guarantee the correctness and the stated time bound.
Using this shortest-path algorithm, we show how to generate an intermediate
structure, the approximate dominance sequences of the input graph, in time, and further propose a simple yet efficient algorithm to converted
this sequence to a tree embedding in time, both with high
probability. Combining the three subroutines gives the stated time bound of the
algorithm.
Then we show that this efficient construction can facilitate some
applications. We proved that FRT trees (the generated tree embedding) are
Ramsey partitions with asymptotically tight bound, so the construction of a
series of distance oracles can be accelerated
Average Sensitivity of Graph Algorithms
In modern applications of graphs algorithms, where the graphs of interest are
large and dynamic, it is unrealistic to assume that an input representation
contains the full information of a graph being studied. Hence, it is desirable
to use algorithms that, even when only a (large) subgraph is available, output
solutions that are close to the solutions output when the whole graph is
available. We formalize this idea by introducing the notion of average
sensitivity of graph algorithms, which is the average earth mover's distance
between the output distributions of an algorithm on a graph and its subgraph
obtained by removing an edge, where the average is over the edges removed and
the distance between two outputs is the Hamming distance.
In this work, we initiate a systematic study of average sensitivity. After
deriving basic properties of average sensitivity such as composition, we
provide efficient approximation algorithms with low average sensitivities for
concrete graph problems, including the minimum spanning forest problem, the
global minimum cut problem, the minimum - cut problem, and the maximum
matching problem. In addition, we prove that the average sensitivity of our
global minimum cut algorithm is almost optimal, by showing a nearly matching
lower bound. We also show that every algorithm for the 2-coloring problem has
average sensitivity linear in the number of vertices. One of the main ideas
involved in designing our algorithms with low average sensitivity is the
following fact; if the presence of a vertex or an edge in the solution output
by an algorithm can be decided locally, then the algorithm has a low average
sensitivity, allowing us to reuse the analyses of known sublinear-time
algorithms and local computation algorithms (LCAs). Using this connection, we
show that every LCA for 2-coloring has linear query complexity, thereby
answering an open question.Comment: 39 pages, 1 figur
The Power of Dynamic Distance Oracles: Efficient Dynamic Algorithms for the Steiner Tree
In this paper we study the Steiner tree problem over a dynamic set of
terminals. We consider the model where we are given an -vertex graph
with positive real edge weights, and our goal is to maintain a tree
which is a good approximation of the minimum Steiner tree spanning a terminal
set , which changes over time. The changes applied to the
terminal set are either terminal additions (incremental scenario), terminal
removals (decremental scenario), or both (fully dynamic scenario). Our task
here is twofold. We want to support updates in sublinear time, and keep
the approximation factor of the algorithm as small as possible. We show that we
can maintain a -approximate Steiner tree of a general graph in
time per terminal addition or removal. Here,
denotes the stretch of the metric induced by . For planar graphs we achieve
the same running time and the approximation ratio of .
Moreover, we show faster algorithms for incremental and decremental scenarios.
Finally, we show that if we allow higher approximation ratio, even more
efficient algorithms are possible. In particular we show a polylogarithmic time
-approximate algorithm for planar graphs.
One of the main building blocks of our algorithms are dynamic distance
oracles for vertex-labeled graphs, which are of independent interest. We also
improve and use the online algorithms for the Steiner tree problem.Comment: Full version of the paper accepted to STOC'1
Analyzing the Effect of Objective Correlation on the Efficient Set of MNK-Landscapes
In multiobjective combinatorial optimization, there exists two main classes
of metaheuristics, based either on multiple aggregations, or on a dominance
relation. As in the single objective case, the structure of the search space
can explain the difficulty for multiobjective metaheuristics, and guide the
design of such methods. In this work we analyze the properties of
multiobjective combinatorial search spaces. In particular, we focus on the
features related the efficient set, and we pay a particular attention to the
correlation between objectives. Few benchmark takes such objective correlation
into account. Here, we define a general method to design multiobjective
problems with correlation. As an example, we extend the well-known
multiobjective NK-landscapes. By measuring different properties of the search
space, we show the importance of considering the objective correlation on the
design of metaheuristics.Comment: Learning and Intelligent OptimizatioN Conference (LION 5), Rome :
Italy (2011
Line-distortion, Bandwidth and Path-length of a graph
We investigate the minimum line-distortion and the minimum bandwidth problems
on unweighted graphs and their relations with the minimum length of a
Robertson-Seymour's path-decomposition. The length of a path-decomposition of a
graph is the largest diameter of a bag in the decomposition. The path-length of
a graph is the minimum length over all its path-decompositions. In particular,
we show:
- if a graph can be embedded into the line with distortion , then
admits a Robertson-Seymour's path-decomposition with bags of diameter at most
in ;
- for every class of graphs with path-length bounded by a constant, there
exist an efficient constant-factor approximation algorithm for the minimum
line-distortion problem and an efficient constant-factor approximation
algorithm for the minimum bandwidth problem;
- there is an efficient 2-approximation algorithm for computing the
path-length of an arbitrary graph;
- AT-free graphs and some intersection families of graphs have path-length at
most 2;
- for AT-free graphs, there exist a linear time 8-approximation algorithm for
the minimum line-distortion problem and a linear time 4-approximation algorithm
for the minimum bandwidth problem
NetLSD: Hearing the Shape of a Graph
Comparison among graphs is ubiquitous in graph analytics. However, it is a
hard task in terms of the expressiveness of the employed similarity measure and
the efficiency of its computation. Ideally, graph comparison should be
invariant to the order of nodes and the sizes of compared graphs, adaptive to
the scale of graph patterns, and scalable. Unfortunately, these properties have
not been addressed together. Graph comparisons still rely on direct approaches,
graph kernels, or representation-based methods, which are all inefficient and
impractical for large graph collections.
In this paper, we propose the Network Laplacian Spectral Descriptor (NetLSD):
the first, to our knowledge, permutation- and size-invariant, scale-adaptive,
and efficiently computable graph representation method that allows for
straightforward comparisons of large graphs. NetLSD extracts a compact
signature that inherits the formal properties of the Laplacian spectrum,
specifically its heat or wave kernel; thus, it hears the shape of a graph. Our
evaluation on a variety of real-world graphs demonstrates that it outperforms
previous works in both expressiveness and efficiency.Comment: KDD '18: The 24th ACM SIGKDD International Conference on Knowledge
Discovery & Data Mining, August 19--23, 2018, London, United Kingdo
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