On the Giant Component of Geometric Inhomogeneous Random Graphs

In this paper we study the threshold model of geometric inhomogeneous random graphs (GIRGs); a generative random graph model that is closely related to hyperbolic random graphs (HRGs). These models have been observed to capture complex real-world networks well with respect to the structural and algorithmic properties. Following comprehensive studies regarding their connectivity, i.e., which parts of the graphs are connected, we have a good understanding under which circumstances a giant component (containing a constant fraction of the graph) emerges. While previous results are rather technical and challenging to work with, the goal of this paper is to provide more accessible proofs. At the same time we significantly improve the previously known probabilistic guarantees, showing that GIRGs contain a giant component with probability 1 - exp(-?(n^{(3-?)/2})) for graph size n and a degree distribution with power-law exponent ? ? (2, 3). Based on that we additionally derive insights about the connectivity of certain induced subgraphs of GIRGs

A Primal-Dual Algorithm for Multicommodity Flows and Multicuts in Treewidth-2 Graphs

We study the problem of multicommodity flow and multicut in treewidth-2 graphs and prove bounds on the multiflow-multicut gap. In particular, we give a primal-dual algorithm for computing multicommodity flow and multicut in treewidth-2 graphs and prove the following approximate max-flow min-cut theorem: given a treewidth-2 graph, there exists a multicommodity flow of value f with congestion 4, and a multicut of capacity c such that c ? 20 f. This implies a multiflow-multicut gap of 80 and improves upon the previous best known bounds for such graphs. Our algorithm runs in polynomial time when all the edges have capacity one. Our algorithm is completely combinatorial and builds upon the primal-dual algorithm of Garg, Vazirani and Yannakakis for multicut in trees and the augmenting paths framework of Ford and Fulkerson

Efficient Constructions for the Gy\H{o}ri-Lov\'{a}sz Theorem on Almost Chordal Graphs

In the 1970s, Gy\H{o}ri and Lov\'{a}sz showed that for a $k$-connected $n$-vertex graph, a given set of terminal vertices $t_1, \dots, t_k$ and natural numbers $n_1, \dots, n_k$ satisfying $\sum_{i=1}^{k} n_i = n$, a connected vertex partition $S_1, \dots, S_k$ satisfying $t_i \in S_i$ and $|S_i| = n_i$ exists. However, polynomial algorithms to actually compute such partitions are known so far only for $k \leq 4$. This motivates us to take a new approach and constrain this problem to particular graph classes instead of restricting the values of $k$. More precisely, we consider $k$-connected chordal graphs and a broader class of graphs related to them. For the first, we give an algorithm with $O(n^2)$ running time that solves the problem exactly, and for the second, an algorithm with $O(n^4)$ running time that deviates on at most one vertex from the given required vertex partition sizes

Balanced Crown Decomposition for Connectivity Constraints

We introduce the balanced crown decomposition that captures the structure imposed on graphs by their connected induced subgraphs of a given size. Such subgraphs are a popular modeling tool in various application areas, where the non-local nature of the connectivity condition usually results in very challenging algorithmic tasks. The balanced crown decomposition is a combination of a crown decomposition and a balanced partition which makes it applicable to graph editing as well as graph packing and partitioning problems. We illustrate this by deriving improved approximation algorithms and kernelization for a variety of such problems. In particular, through this structure, we obtain the first constant-factor approximation for the Balanced Connected Partition (BCP) problem, where the task is to partition a vertex-weighted graph into k connected components of approximately equal weight. We derive a 3-approximation for the two most commonly used objectives of maximizing the weight of the lightest component or minimizing the weight of the heaviest component

Connected k-Partition of k-Connected Graphs and c-Claw-Free Graphs

w_k. In particular for the balanced version, i.e. w? = w? == w_k, this gives a partition with 1/3w_i ? w(T_i) ? 3w_i