Randomly removing g handles at once

It was shown in [Indyk-Sidiropoulos 07] that any orientable graph of genus g can be probabilistically embedded into a graph of genus g-1 with constant distortion. Removing handles one by one gives an embedding into a distribution over planar graphs with distortion 2O(g). By removing all $g$ handles at once, we present a probabilistic embedding with distortion O(g2) for both orientable and non-orientable graphs. Our result is obtained by showing that the minimum-cut graph of [Erickson-HarPeled 04] has low dilation, and then randomly cutting this graph out of the surface using the Peeling Lemma from [Lee-Sidiropoulos 08].

We exhibit the following new upper bounds on the space complexity and the parallel complexity of the Bipartite Perfect Matching (BPM) problem for graphs of small genus: (1) BPM in planar graphs is in UL (improves upon the SPL bound from Datta, Kulkarni, and Roy; (2) BPM in constant genus graphs is in NL (orthogonal to the SPL bound from Datta, Kulkarni, Tewari, and Vinodchandran.; (3) BPM in poly-logarithmic genus graphs is in NC; (extends the NC bound for O(log n) genus graphs from Mahajan and Varadarajan, and Kulkarni, Mahajan, and Varadarajan. For Part (1) we combine the flow technique of Miller and Naor with the double counting technique of Reinhardt and Allender . For Part (2) and (3) we extend Miller and Naor's result to higher genus surfaces in the spirit of Chambers, Erickson and Nayyeri.

The dual of a polyhedron is a polyhedron—or in graph‐theoretical terms: the dual of a 3‐connected plane graph is a 3‐connected plane graph. Astonishingly, except for sufficiently large facewidth, not much is known about the connectivity of the dual on higher surfaces. Are the duals of 3‐connected embedded graphs of higher genus 3‐connected, too? If not, which connectivity guarantees 3‐connectedness of the dual? In this article, we give answers to some of these and related questions. We prove that there is no connectivity that guarantees the 3‐connectedness or 2‐connectedness of the dual for every genus, and give upper bounds for the minimum genus for which (with ) a c‐connected embedded graph with a dual that has a 1‐ or 2‐cut can occur. We prove that already on the torus, we need 6‐connectedness to guarantee 3‐connectedness of the dual and 4‐connectedness to guarantee 2‐connectedness of the dual. In the last section, we answer a related question by Plummer and Zha on orientable embeddings of highly connected noncomplete graphs.

This paper presents some recent results on lower bounds for independence ratios of graphs of positive genus and shows that in a limiting sense these graphs have the same independence ratios as do planar graphs. This last result is obtained by an application of Menger's Theorem to show that every triangulation of a surface of positive genus has a short cycle which does not separate the graph and is non‐contractible on that surface.

We describe efficient deterministic techniques for breaking symmetry in parallel. The techniques work well on rooted trees and graphs of constant degree or genus. Our primary technique allows us to 3-color a rooted tree in Ο(lg*n) time on an EREW PRAM using a linear number of processors. We apply these techniques to construct fast linear processor algorithms for several problems, including (Δ + 1)-coloring constant-degree graphs, 5-coloring planar graphs, and finding depth-first-search trees in planar graphs. We also prove lower bounds for 2-coloring directed lists and for finding maximal independent sets in arbitrary graphs.

Embedding between metric spaces is a very powerful algorithmic tool and has been used for finding good approximation algorithms for several problems. In particular, embedding to an [cursive l]₁ norm has been used as the key step in an approximation algorithm for the sparsest cut problem. The sparsest cut problem, in turn, is the main ingredient of many algorithms that have a divide and conquer nature and are used in various fields. While every metric is embeddable into [cursive l]₁ with distortion O (log n) [13], and the bound is tight [39], for special classes of metrics better bounds exist. Shortest path metrics for trees and outerplanar graphs are isometrically embeddable into [cursive l]₁ [41]. Series-parallel graphs [28] and k-outerplanar graphs [19] (for constant k) are embeddable into[cursive l]₁ with constant distortion planar graphs and bounded tree-width graphs are conjectured to have constant distortion in embedding to [cursive l]₁ . Bounded tree-width graphs are one of most general graph classes on which several hard problems are tractable. We study the embedding of series-parallel graphs (or, more generally, graphs with tree-width two) into [cursive l]₁ and also the embedding between two line metrics. We then move our attention to the generalization of tree-width to digraphs and hypergraphs and study several relevant problems.

The Minimum Dominating Set (MDS) problem is a fundamental and challenging problem in distributed computing. While it is well known that minimum dominating sets cannot be well approximated locally on general graphs, in recent years there has been much progress on computing good local approximations on sparse graphs and in particular on planar graphs. In this article, we study distributed and deterministic MDS approximation algorithms for graph classes beyond planar graphs. In particular, we show that existing approximation bounds for planar graphs can be lifted to bounded genus graphs and more general graphs, which we call locally embeddable graphs, and present (1) a local constant-time, constant-factor MDS approximation algorithm on locally embeddable graphs, and (2) a local O (log * n )-time (1+ϵ)-approximation scheme for any ϵ > 0 on graphs of bounded genus. Our main technical contribution is a new analysis of a slightly modified variant of an existing algorithm by Lenzen et al. [21]. Interestingly, unlike existing proofs for planar graphs, our analysis does not rely on direct topological arguments but on combinatorial density arguments only.

Factorization of the pomeron sector and currents in the dual resonance model

: This paper introduces a class of 1 player games of perfect information, which we call complementing games; the player is allowed moves which complement the value of successive plays. A complementing game is symmetric if all noncomplement moves are reversible (i.e., form a symmetric relation). These games are naturally related to a class of machines we call symmetric complementing machines. Symmetric nondeterministic machines were studied in (Lewis and Papadimitriou, 801): they are identical to our symmetric complementing machines with complement moves allowed only on termination. (A companion paper to appear will describe the computational complexity of symmetric complementing and alternating machines.) Of particular interest is the complexity class Sum *CSYMLOG, which contains the outcome problem of symmetric complementing games with constant complement bound with game positions encoded in log space, and next move relations computable in log space. We show that the decision problem for a restricted quantified Boolean logic Sum *QBF sum is complete in Sum *CSYMLOG. We also show that Sum *CSYMLOG contains many well- known and common combinatiorial problems: minimum spanning forests; k- connectivity and k-connected components, and also the recognition problems for many classes of graphs: planar graphs of valence 3; chordal graphs; comparability graphs; interval graphs; split graphs, and permutation graphs.

We present a new method for upper bounding the second eigenvalue of theLaplacian of graphs. Our approach uses multi-commodity flows to deform the geometry of the graph; we embed the resulting metric into Euclidean space to recover a bound on the Rayleigh quotient. Using this, we show that every n-vertex graph of genus g and maximum degree d satisfies lambda2(G) = O((g+1)3d/n).This recovers the O(d/n) bound of Spielman and Teng for planar graphs, and compares to Kelner's bound of O((g+1)poly(d)/n), but our proof does not make use of conformal mappings or circle packings. We are thus able to extend this to resolve positively a conjecture of Spielman and Teng, by proving that lambda2(G) = O(dh6log h/n) whenever G is Kh-minor free. This shows, in particular, that spectral partitioning can be used to recover O(radicn)-sized separators in bounded degree graphs that exclude a fixed minor. We extend this further by obtaining nearly optimal bounds on lambda2 for graphs which exclude small-depth minors in the sense of Plotkin, Rao, and Smith. Consequently, we show that spectral algorithms find small separators in a general class of geometric graphs. Moreover, while the standard sweep'' algorithm applied to the second eigenvector may fail to find good quotient cuts in graphs of unbounded degree, our approach produces a vector that works for arbitrary graphs. This yields an alternate proof of the result of Alon, Seymour, and Thomas that every excluded-minor family of graphs has O(radicn)-node balanced separators.

Planarity can be verified by an approximate proof labeling scheme in constant-time