We propose to study unweighted graphs of constant distance VC-dimension as a broad generalization of many graph classes for which we can compute the diameter in truly subquadratic-time. In particular for any fixed @math , the class of @math -minor free graphs has distance VC-dimension at most @math . Our first main result is that on graphs of distance VC-dimension at most @math , for any fixed @math we can either compute the diameter or conclude that it is larger than @math in time @math , where @math only depends on @math . Then as a byproduct of our approach, we get the first truly subquadratic-time algorithm for constant diameter computation on all the nowhere dense graph classes. Finally, we show how to remove the dependency on @math for any graph class that excludes a fixed graph @math as a minor. More generally, our techniques apply to any graph with constant distance VC-dimension and polynomial expansion. As a result for all such graphs one obtains a truly subquadratic-time algorithm for computing their diameter. Our approach is based on the work of Chazelle and Welzl who proved the existence of spanning paths with strongly sublinear stabbing number for every hypergraph of constant VC-dimension. We show how to compute such paths efficiently by combining the best known approximation algorithms for the stabbing number problem with a clever use of @math -nets, region decomposition and other partition techniques.
An early example of linear-time solvable special case for diameter computation is the class of interval graphs @cite_0 . For every interval graph @math and for any integer @math , if we first compute an interval representation for @math in linear-time @cite_31 then we can compute by dynamic programming, for every vertex @math , the contiguous segment of all the vertices at a distance @math from @math in @math . It takes almost linear-time and it implies a straightforward quasi linear-time algorithm for diameter computation. More efficient algorithms for diameter computation on interval graphs and related graph classes were proposed in @cite_23 . Nevertheless we will show in what follows that interval orderings are a powerful tool for diameter computation on more general geometric graph classes.
@cite_0 @cite_31 @cite_23
2091612923 2949393806 2399682168 2952839955
Determining the diameter of a graph is a fundamental graph operation, yet no efficient (i.e. linear or quadratic time) algorithm is known. In this paper, we examine the diameter problem on chordal graphs and AT-free graphs and show that a very simple (linear time) 2-sweep LexBFS algorithm identifies a vertex of maximum eccentricity unless the given graph has a specified induced subgraph (it was previously known that a single LexBFS algorithm is guaranteed to end at a vertex that is within 1 of the diameter for chordal graphs and AT-free graphs). As a consequence of the forbidden induced subgraph result on chordal graphs, our algorithm is guaranteed to work optimally for directed path graphs (it was previously known that a single LexBFS algorithm is guaranteed to work optimally for interval graphs). We show a new way to round vector solutions of semidefinite programming (SDP) hierarchies into integral solutions, based on a connection between these hierarchies and the spectrum of the input graph. We demonstrate the utility of our method by providing a new SDP-hierarchy based algorithm for constraint satisfaction problems with 2-variable constraints (2-CSP's). More concretely, we show for every 2-CSP instance I a rounding algorithm for r rounds of the Lasserre SDP hierarchy for I that obtains an integral solution that is at most worse than the relaxation's value (normalized to lie in [0,1]), as long as r > k ( ) ( ) ;, where k is the alphabet size of I, @math , and @math denotes the number of eigenvalues larger than @math in the normalized adjacency matrix of the constraint graph of @math . In the case that @math is a instance, the threshold @math is only a polynomial in @math , and is independent of the alphabet size. Also in this case, we can give a non-trivial bound on the number of rounds for instance. In particular our result yields an SDP-hierarchy based algorithm that matches the performance of the recent subexponential algorithm of Arora, Barak and Steurer (FOCS 2010) in the worst case, but runs faster on a natural family of instances, thus further restricting the set of possible hard instances for Khot's Unique Games Conjecture. Our algorithm actually requires less than the @math constraints specified by the @math level of the Lasserre hierarchy, and in some cases @math rounds of our program can be evaluated in time @math . We develop a new technique for constructing sparse graphs that allow us to prove near-linear lower bounds on the round complexity of computing distances in the CONGEST model. Specifically, we show an @math lower bound for computing the diameter in sparse networks, which was previously known only for dense networks [, SODA 2012]. In fact, we can even modify our construction to obtain graphs with constant degree, using a simple but powerful degree-reduction technique which we define. Moreover, our technique allows us to show @math lower bounds for computing @math -approximations of the diameter or the radius, and for computing a @math -approximation of all eccentricities. For radius, we are unaware of any previous lower bounds. For diameter, these greatly improve upon previous lower bounds and are tight up to polylogarithmic factors [, SODA 2012], and for eccentricities the improvement is both in the lower bound and in the approximation factor [Holzer and Wattenhofer, PODC 2012]. Interestingly, our technique also allows showing an almost-linear lower bound for the verification of @math -spanners, for @math . Consider an undirected weighted graph G=(V,E) with |V|=n and |E|=m, where each vertex v is assigned a label from a set L of labels. We show how to construct a compact distance oracle that can answer queries of the form: "what is the distance from v to the closest lambda-labeled node" for a given node v in V and label lambda in L. This problem was introduced by Hermelin, Levy, Weimann and Yuster [ICALP 2011] where they present several results for this problem. In the first result, they show how to construct a vertex-label distance oracle of expected size O(kn^ 1+1 k ) with stretch (4k - 5) and query time O(k). In a second result, they show how to reduce the size of the data structure to O(kn ^ 1 k ) at the expense of a huge stretch, the stretch of this construction grows exponentially in k, (2^k-1). In the third result they present a dynamic vertex-label distance oracle that is capable of handling label changes in a sub-linear time. The stretch of this construction is also exponential in k, (2 3^ k-1 +1). We manage to significantly improve the stretch of their constructions, reducing the dependence on k from exponential to polynomial (4k-5), without requiring any tradeoff regarding any of the other variables. In addition, we introduce the notion of vertex-label spanners: subgraphs that preserve distances between every node v and label lambda. We present an efficient construction for vertex-label spanners with stretch-size tradeoff close to optimal.