Vertex Sparsifiers: New Results from Old Techniques

Given a capacitated graph $G = (V,E)$ and a set of terminals $K \subseteq V$, how should we produce a graph $H$ only on the terminals $K$ so that every (multicommodity) flow between the terminals in $G$ could be supported in $H$ with low congestion, and vice versa? (Such a graph $H$ is called a flow-sparsifier for $G$.) What if we want $H$ to be a "simple" graph? What if we allow $H$ to be a convex combination of simple graphs? Improving on results of Moitra [FOCS 2009] and Leighton and Moitra [STOC 2010], we give efficient algorithms for constructing: (a) a flow-sparsifier $H$ that maintains congestion up to a factor of $O(\log k/\log \log k)$, where $k = |K|$, (b) a convex combination of trees over the terminals $K$ that maintains congestion up to a factor of $O(\log k)$, and (c) for a planar graph $G$, a convex combination of planar graphs that maintains congestion up to a constant factor. This requires us to give a new algorithm for the 0-extension problem, the first one in which the preimages of each terminal are connected in $G$. Moreover, this result extends to minor-closed families of graphs. Our improved bounds immediately imply improved approximation guarantees for several terminal-based cut and ordering problems.Comment: An extended abstract appears in the 13th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX), 2010. Final version to appear in SIAM J. Computin

Maximum Weight Disjoint Paths in Outerplanar Graphs via Single-Tree Cut Approximators

Since 1997 there has been a steady stream of advances for the maximum disjoint paths problem. Achieving tractable results has usually required focusing on relaxations such as: (i) to allow some bounded edge congestion in solutions, (ii) to only consider the unit weight (cardinality) setting, (iii) to only require fractional routability of the selected demands (the all-or-nothing flow setting). For the general form (no congestion, general weights, integral routing) of edge-disjoint paths ({\sc edp}) even the case of unit capacity trees which are stars generalizes the maximum matching problem for which Edmonds provided an exact algorithm. For general capacitated trees, Garg, Vazirani, Yannakakis showed the problem is APX-Hard and Chekuri, Mydlarz, Shepherd provided a $4$-approximation. This is essentially the only setting where a constant approximation is known for the general form of \textsc{edp}. We extend their result by giving a constant-factor approximation algorithm for general-form \textsc{edp} in outerplanar graphs. A key component for the algorithm is to find a {\em single-tree} $O(1)$ cut approximator for outerplanar graphs. Previously $O(1)$ cut approximators were only known via distributions on trees and these were based implicitly on the results of Gupta, Newman, Rabinovich and Sinclair for distance tree embeddings combined with results of Anderson and Feige.Comment: 19 pages, 6 figure

A node-capacitated Okamura-Seymour theorem

The classical Okamura-Seymour theorem states that for an edge-capacitated, multi-commodity flow instance in which all terminals lie on a single face of a planar graph, there exists a feasible concurrent flow if and only if the cut conditions are satisfied. Simple examples show that a similar theorem is impossible in the node-capacitated setting. Nevertheless, we prove that an approximate flow/cut theorem does hold: For some universal c > 0, if the node cut conditions are satisfied, then one can simultaneously route a c-fraction of all the demands. This answers an open question of Chekuri and Kawarabayashi. More generally, we show that this holds in the setting of multi-commodity polymatroid networks introduced by Chekuri, et. al. Our approach employs a new type of random metric embedding in order to round the convex programs corresponding to these more general flow problems.Comment: 30 pages, 5 figure

Information Nonanticipative Rate Distortion Function and Its Applications

This paper investigates applications of nonanticipative Rate Distortion Function (RDF) in a) zero-delay Joint Source-Channel Coding (JSCC) design based on average and excess distortion probability, b) in bounding the Optimal Performance Theoretically Attainable (OPTA) by noncausal and causal codes, and computing the Rate Loss (RL) of zero-delay and causal codes with respect to noncausal codes. These applications are described using two running examples, the Binary Symmetric Markov Source with parameter p, (BSMS(p)) and the multidimensional partially observed Gaussian-Markov source. For the multidimensional Gaussian-Markov source with square error distortion, the solution of the nonanticipative RDF is derived, its operational meaning using JSCC design via a noisy coding theorem is shown by providing the optimal encoding-decoding scheme over a vector Gaussian channel, and the RL of causal and zero-delay codes with respect to noncausal codes is computed. For the BSMS(p) with Hamming distortion, the solution of the nonanticipative RDF is derived, the RL of causal codes with respect to noncausal codes is computed, and an uncoded noisy coding theorem based on excess distortion probability is shown. The information nonanticipative RDF is shown to be equivalent to the nonanticipatory epsilon-entropy, which corresponds to the classical RDF with an additional causality or nonanticipative condition imposed on the optimal reproduction conditional distribution.Comment: 34 pages, 12 figures, part of this paper was accepted for publication in IEEE International Symposium on Information Theory (ISIT), 2014 and in book Coordination Control of Distributed Systems of series Lecture Notes in Control and Information Sciences, 201

Graph indexing and retrieval based on graph prototypes

[ANGLÈS] Taking a query from a high number of data stored into a database, as fast as possible, is a recurrent problem in the field of computer sciences practically since its origins. At the existence of this problem, it’s necessary to add, moreover, the fact that actually databases contains data types of more diverse and unexpected character possible. Now we are not talking about originating databases which only contained sets of numbers or characters strings. (...) All that I want to make into the present work and I think that was achieved as far as possible, has been to develop and to present a methodology to carry out this process. The Metric Trees of prototypes are based on a well-known strategy, which is based on grouping the data stored in database at the smartest possible way. But also we has added the concept of a graph prototype. A structure that contains information of a set of instances represented by graphs, used until now for classification and recognition. In this thesis we have used graphs as representatives of elements that have to be queried in databases. Note that graphs have the capacity to represent complex objects, for this reason the number of graph databases is increasing. Due to in the literature appears different ways to build a prototype, the work presented here shows a comparative study between the main methods. Combining these two concepts, the Metric Tree and the graph prototype, we propose the construction of metric trees where the graph prototypes are routing nodes to help to decide the way to explore when we make a search in the tree. We have used Metric Trees to make classification and to find all instances that are lower than a maximum distance. (...)[CATALÀ] El trobar-nos davant una gran quantitat de dades i tenir que fer cerques d’aquestes el més ràpid possible és un problema recurrent en el camp de les ciències de la computació pràcticament des dels seus orígens. A l'existència d'aquest problema, se li ha d’afegir, a més a més, el fet de que actualment les bases de dades emmagatzemen tipus de dades de la naturalesa més diversa i molts cops inesperada possible. Ja no parlem de les bases de dades originaries que únicament contenien números o cadenes caràcters. (...) El que he volgut en aquest treball i penso que en la mesura del que era possible s'ha aconseguit, és desenvolupar i presentar una metodologia per portar a terme aquest procés. Els Metric Trees de prototips, que es basen en la ja coneguda estratègia d'agrupar les dades que anem guardant a una base de dades de la forma més intel·ligent possible per no haver d’explorar totes les instàncies que tenim quan volem fer una cerca, però a més a més s'ha afegit el concepte de prototip. Una estructura, que agrupa la informació d'un conjunt d'instàncies, utilitzada fins ara per a fer classificació i reconeixement. Conjugant aquests dos conceptes, el de Metric Tree i el de prototip, plantejem la construcció d'arbres de cerca on els prototips siguin els nodes intermedis, que ens ajudin a decidir quin camí explorar quan volem fer una cerca sobre l'arbre. I utilitzant, aquests tant per a fer classificació com per a buscar totes les instàncies que estiguin una distància més petita d’una distància máxima. Tot això tenint present, que les dades amb que treballem són grafs, és a dir que la metodologia presentada, té la versatilitat de poder-se aplicar, a qualsevol tipus d'informació que es pugui representar d'aquesta manera. (...

Potential Theory on Trees, Graphs and Ahlfors Regular Metric Spaces

We investigate connections between potential theories on a Ahlfors-regular metric space X, on a graph G associated with X, and on the tree T obtained by removing the "horizontal edges" in G. Applications to the calculation of set capacity are given.Comment: 45 pages; presentation improved based on referee comment