Characterizations of Decomposable Dependency Models

Decomposable dependency models possess a number of interesting and useful properties. This paper presents new characterizations of decomposable models in terms of independence relationships, which are obtained by adding a single axiom to the well-known set characterizing dependency models that are isomorphic to undirected graphs. We also briefly discuss a potential application of our results to the problem of learning graphical models from data.Comment: See http://www.jair.org/ for any accompanying file

Vertex elimination orderings for hereditary graph classes

We provide a general method to prove the existence and compute efficiently elimination orderings in graphs. Our method relies on several tools that were known before, but that were not put together so far: the algorithm LexBFS due to Rose, Tarjan and Lueker, one of its properties discovered by Berry and Bordat, and a local decomposition property of graphs discovered by Maffray, Trotignon and Vu\vskovi\'c. We use this method to prove the existence of elimination orderings in several classes of graphs, and to compute them in linear time. Some of the classes have already been studied, namely even-hole-free graphs, square-theta-free Berge graphs, universally signable graphs and wheel-free graphs. Some other classes are new. It turns out that all the classes that we study in this paper can be defined by excluding some of the so-called Truemper configurations. For several classes of graphs, we obtain directly bounds on the chromatic number, or fast algorithms for the maximum clique problem or the coloring problem

Hardness of Graph Pricing through Generalized Max-Dicut

The Graph Pricing problem is among the fundamental problems whose approximability is not well-understood. While there is a simple combinatorial 1/4-approximation algorithm, the best hardness result remains at 1/2 assuming the Unique Games Conjecture (UGC). We show that it is NP-hard to approximate within a factor better than 1/4 under the UGC, so that the simple combinatorial algorithm might be the best possible. We also prove that for any $\epsilon > 0$, there exists $\delta > 0$ such that the integrality gap of $n^{\delta}$-rounds of the Sherali-Adams hierarchy of linear programming for Graph Pricing is at most 1/2 + $\epsilon$. This work is based on the effort to view the Graph Pricing problem as a Constraint Satisfaction Problem (CSP) simpler than the standard and complicated formulation. We propose the problem called Generalized Max-Dicut($T$), which has a domain size $T + 1$ for every $T \geq 1$. Generalized Max-Dicut(1) is well-known Max-Dicut. There is an approximation-preserving reduction from Generalized Max-Dicut on directed acyclic graphs (DAGs) to Graph Pricing, and both our results are achieved through this reduction. Besides its connection to Graph Pricing, the hardness of Generalized Max-Dicut is interesting in its own right since in most arity two CSPs studied in the literature, SDP-based algorithms perform better than LP-based or combinatorial algorithms --- for this arity two CSP, a simple combinatorial algorithm does the best.Comment: 28 page

Markov models for fMRI correlation structure: is brain functional connectivity small world, or decomposable into networks?

Correlations in the signal observed via functional Magnetic Resonance Imaging (fMRI), are expected to reveal the interactions in the underlying neural populations through hemodynamic response. In particular, they highlight distributed set of mutually correlated regions that correspond to brain networks related to different cognitive functions. Yet graph-theoretical studies of neural connections give a different picture: that of a highly integrated system with small-world properties: local clustering but with short pathways across the complete structure. We examine the conditional independence properties of the fMRI signal, i.e. its Markov structure, to find realistic assumptions on the connectivity structure that are required to explain the observed functional connectivity. In particular we seek a decomposition of the Markov structure into segregated functional networks using decomposable graphs: a set of strongly-connected and partially overlapping cliques. We introduce a new method to efficiently extract such cliques on a large, strongly-connected graph. We compare methods learning different graph structures from functional connectivity by testing the goodness of fit of the model they learn on new data. We find that summarizing the structure as strongly-connected networks can give a good description only for very large and overlapping networks. These results highlight that Markov models are good tools to identify the structure of brain connectivity from fMRI signals, but for this purpose they must reflect the small-world properties of the underlying neural systems

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