Routing Symmetric Demands in Directed Minor-Free Graphs with Constant Congestion

The problem of routing in graphs using node-disjoint paths has received a lot of attention and a polylogarithmic approximation algorithm with constant congestion is known for undirected graphs [Chuzhoy and Li 2016] and [Chekuri and Ene 2013]. However, the problem is hard to approximate within polynomial factors on directed graphs, for any constant congestion [Chuzhoy, Kim and Li 2016]. Recently, [Chekuri, Ene and Pilipczuk 2016] have obtained a polylogarithmic approximation with constant congestion on directed planar graphs, for the special case of symmetric demands. We extend their result by obtaining a polylogarithmic approximation with constant congestion on arbitrary directed minor-free graphs, for the case of symmetric demands

Testing first-order properties for subclasses of sparse graphs

We present a linear-time algorithm for deciding first-order (FO) properties in classes of graphs with bounded expansion, a notion recently introduced by Nesetril and Ossona de Mendez. This generalizes several results from the literature, because many natural classes of graphs have bounded expansion: graphs of bounded tree-width, all proper minor-closed classes of graphs, graphs of bounded degree, graphs with no subgraph isomorphic to a subdivision of a fixed graph, and graphs that can be drawn in a fixed surface in such a way that each edge crosses at most a constant number of other edges. We deduce that there is an almost linear-time algorithm for deciding FO properties in classes of graphs with locally bounded expansion. More generally, we design a dynamic data structure for graphs belonging to a fixed class of graphs of bounded expansion. After a linear-time initialization the data structure allows us to test an FO property in constant time, and the data structure can be updated in constant time after addition/deletion of an edge, provided the list of possible edges to be added is known in advance and their simultaneous addition results in a graph in the class. All our results also hold for relational structures and are based on the seminal result of Nesetril and Ossona de Mendez on the existence of low tree-depth colorings

Testing first-order properties for subclasses of sparse graphs

We present a linear-time algorithm for deciding first-order (FO) properties in classes of graphs with bounded expansion, a notion recently introduced by Nešetřil and Ossona de Mendez. This generalizes several results from the literature, because many natural classes of graphs have bounded expansion: graphs of bounded tree-width, all proper minor-closed classes of graphs, graphs of bounded degree, graphs with no subgraph isomorphic to a subdivision of a fixed graph, and graphs that can be drawn in a fixed surface in such a way that each edge crosses at most a constant number of other edges. We deduce that there is an almost linear-time algorithm for deciding FO properties in classes of graphs with locally bounded expansion. More generally, we design a dynamic data structure for graphs belonging to a fixed class of graphs of bounded expansion. After a linear-time initialization the data structure allows us to test an FO property in constant time, and the data structure can be updated in constant time after addition/deletion of an edge, provided the list of possible edges to be added is known in advance and their simultaneous addition results in a graph in the class. All our results also hold for relational structures and are based on the seminal result of Nešetřil and Ossona de Mendez on the existence of low tree-depth colorings

Spatial mixing and approximation algorithms for graphs with bounded connective constant

The hard core model in statistical physics is a probability distribution on independent sets in a graph in which the weight of any independent set I is proportional to lambda^(|I|), where lambda > 0 is the vertex activity. We show that there is an intimate connection between the connective constant of a graph and the phenomenon of strong spatial mixing (decay of correlations) for the hard core model; specifically, we prove that the hard core model with vertex activity lambda < lambda_c(Delta + 1) exhibits strong spatial mixing on any graph of connective constant Delta, irrespective of its maximum degree, and hence derive an FPTAS for the partition function of the hard core model on such graphs. Here lambda_c(d) := d^d/(d-1)^(d+1) is the critical activity for the uniqueness of the Gibbs measure of the hard core model on the infinite d-ary tree. As an application, we show that the partition function can be efficiently approximated with high probability on graphs drawn from the random graph model G(n,d/n) for all lambda < e/d, even though the maximum degree of such graphs is unbounded with high probability. We also improve upon Weitz's bounds for strong spatial mixing on bounded degree graphs (Weitz, 2006) by providing a computationally simple method which uses known estimates of the connective constant of a lattice to obtain bounds on the vertex activities lambda for which the hard core model on the lattice exhibits strong spatial mixing. Using this framework, we improve upon these bounds for several lattices including the Cartesian lattice in dimensions 3 and higher. Our techniques also allow us to relate the threshold for the uniqueness of the Gibbs measure on a general tree to its branching factor (Lyons, 1989).Comment: 26 pages. In October 2014, this paper was superseded by arxiv:1410.2595. Before that, an extended abstract of this paper appeared in Proc. IEEE Symposium on the Foundations of Computer Science (FOCS), 2013, pp. 300-30

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 $G$ can be embedded into the line with distortion $k$, then $G$ admits a Robertson-Seymour's path-decomposition with bags of diameter at most $k$ in $G$; - 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