Due to works by Bestvina-Mess, Swarup and Bowditch, we now have complete knowledge of how splittings of a word-hyperbolic group G as a graph of groups with finite or two-ended edge groups relate to the cut point structure of its boundary. It is central in the theory that ∂G is a locally connected continuum (a Peano space). Motivated by the structure of tight circle packings, we propose to gener...

Today we continue our discussion of exact recovery under perturbation-stability assumptions. We’ll look at a different cut problem than last lecture, and this will also give us an excuse to touch on two cool and fundamental topics, metric embeddings and semidefinite programming. The Maximum Cut problem is a famous NP -hard problem. The input is an undirected graph G = (V,E), where each edge e ∈...

Abstract Let G be a nontrivial edge-colored connected graph. A rainbow edge-cut is an R of G, and all edges have different colors in G. For two vertices u v u-v-edge-cut separating them. An graph called strong disconnected if for every distinct there exists both minimum such edge-coloring disconnection coloring (srd-coloring short) the number (srd-number denoted by srd(G), required to make disc...

For a connected graph G = (V , E), an edge set S ⊂ E is a k-restricted edge cut if G − S is disconnected and every component of G − S contains at least k vertices. The k-restricted edge connectivity of G, denoted by λk(G), is defined as the cardinality of a minimum krestricted edge cut. For U1,U2 ⊂ V (G), denote the set of edges of Gwith one end in U1 and the other in U2 by [U1,U2]. Define ξk(G...

Graph sparsification was first introduced by Benzcur and Karger [BK96]. Given a graph G = (V,E), they proposed a randomized algorithm that samples a subset of the edges to construct a capacitated graph G′ = (V,E ′) such that E ′ ⊂ E with edge capacities c : E ′ → R+. G′ is a (1± )-cut sparsifier if the value of any cut over G′ is within a (1± ) multiplicative factor of the corresponding cut val...

The maximum cut problem this paper deals with can be formulated as follows. Given an undirected simple graph G = (V,E) where V and E stand for the node and edge sets respectively, and given weights assigned to the edges: (wij)ij∈E , a cut δ(S), with S ⊆ V is de ned as the set of edges in E with exactly one endnode in S, i.e. δ(S) = {ij ∈ E | |S ∩ {i, j}| = 1}. The weight w(S) of the cut δ(S) is...

This work presents an efficient method for image multi-zone segmentation under human supervision. As most of the segmentation methods, our procedure relies on an energy minimization and this energy contains a data-consistent term and a coherence term. In this work, the pixel-by-pixel data-consistent term is based on a Support Vector Machine designed in order to deal with color and textural info...

We give an approximation algorithm for non-uniform sparsest cut with the following guarantee: For any ε, δ ∈ (0, 1), given cost and demand graphs with edge weights C,D : ( V 2 ) → R+ respectively, we can find a set T ⊆ V with C(T,V \T ) D(T,V \T ) at most 1+ε δ times the optimal non-uniform sparsest cut value, in time 2 poly(n) provided λr ≥ Φ∗/(1 − δ). Here λr is the r’th smallest generalized ...

We study the problem of computing approximate minimum edge cuts by distributed algorithms. We present two randomized approximation algorithms that both run in a standard synchronous message passing model where in each round, O(log n) bits can be transmitted over every edge (a.k.a. the CONGEST model). The first algorithm is based on a simple and new approach for analyzing random edge sampling, w...