1,014 research outputs found
Beyond pairwise clustering
We consider the problem of clustering in domains where the affinity relations are not dyadic (pairwise), but rather triadic, tetradic or higher. The problem is an instance of the hypergraph partitioning problem. We propose a two-step algorithm for solving this problem. In the first step we use a novel scheme to approximate the hypergraph using a weighted graph. In the second step a spectral partitioning algorithm is used to partition the vertices of this graph. The algorithm is capable of handling hyperedges of all orders including order two, thus incorporating information of all orders simultaneously. We present a theoretical analysis that relates our algorithm to an existing hypergraph partitioning algorithm and explain the reasons for its superior performance. We report the performance of our algorithm on a variety of computer vision problems and compare it to several existing hypergraph partitioning algorithms
Algorithms for the minimum sum coloring problem: a review
The Minimum Sum Coloring Problem (MSCP) is a variant of the well-known vertex
coloring problem which has a number of AI related applications. Due to its
theoretical and practical relevance, MSCP attracts increasing attention. The
only existing review on the problem dates back to 2004 and mainly covers the
history of MSCP and theoretical developments on specific graphs. In recent
years, the field has witnessed significant progresses on approximation
algorithms and practical solution algorithms. The purpose of this review is to
provide a comprehensive inspection of the most recent and representative MSCP
algorithms. To be informative, we identify the general framework followed by
practical solution algorithms and the key ingredients that make them
successful. By classifying the main search strategies and putting forward the
critical elements of the reviewed methods, we wish to encourage future
development of more powerful methods and motivate new applications
Constant-degree graph expansions that preserve the treewidth
Many hard algorithmic problems dealing with graphs, circuits, formulas and
constraints admit polynomial-time upper bounds if the underlying graph has
small treewidth. The same problems often encourage reducing the maximal degree
of vertices to simplify theoretical arguments or address practical concerns.
Such degree reduction can be performed through a sequence of splittings of
vertices, resulting in an _expansion_ of the original graph. We observe that
the treewidth of a graph may increase dramatically if the splittings are not
performed carefully. In this context we address the following natural question:
is it possible to reduce the maximum degree to a constant without substantially
increasing the treewidth?
Our work answers the above question affirmatively. We prove that any simple
undirected graph G=(V, E) admits an expansion G'=(V', E') with the maximum
degree <= 3 and treewidth(G') <= treewidth(G)+1. Furthermore, such an expansion
will have no more than 2|E|+|V| vertices and 3|E| edges; it can be computed
efficiently from a tree-decomposition of G. We also construct a family of
examples for which the increase by 1 in treewidth cannot be avoided.Comment: 12 pages, 6 figures, the main result used by quant-ph/051107
Optimal competitiveness for the Rectilinear Steiner Arborescence problem
We present optimal online algorithms for two related known problems involving
Steiner Arborescence, improving both the lower and the upper bounds. One of
them is the well studied continuous problem of the {\em Rectilinear Steiner
Arborescence} (). We improve the lower bound and the upper bound on the
competitive ratio for from and to
, where is the number of Steiner
points. This separates the competitive ratios of and the Symetric-,
two problems for which the bounds of Berman and Coulston is STOC 1997 were
identical. The second problem is one of the Multimedia Content Distribution
problems presented by Papadimitriou et al. in several papers and Charikar et
al. SODA 1998. It can be viewed as the discrete counterparts (or a network
counterpart) of . For this second problem we present tight bounds also in
terms of the network size, in addition to presenting tight bounds in terms of
the number of Steiner points (the latter are similar to those we derived for
)
Linear orderings of random geometric graphs (extended abstract)
In random geometric graphs, vertices are randomly distributed on [0,1]^2 and pairs of vertices are connected by edges
whenever they are sufficiently close together. Layout problems seek a linear ordering of the vertices of a graph such that a
certain measure is minimized. In this paper, we study several layout problems on random geometric graphs: Bandwidth,
Minimum Linear Arrangement, Minimum Cut, Minimum Sum Cut, Vertex Separation and Bisection. We first prove that
some of these problems remain \NP-complete even for geometric graphs. Afterwards, we compute lower bounds that hold
with high probability on random geometric graphs. Finally, we characterize the probabilistic behavior of the lexicographic
ordering for our layout problems on the class of random geometric graphs.Postprint (published version
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