Local Approximation Schemes for Ad Hoc and Sensor Networks

We present two local approaches that yield polynomial-time approximation schemes (PTAS) for the Maximum Independent Set and Minimum Dominating Set problem in unit disk graphs. The algorithms run locally in each node and compute a (1+ε)-approximation to the problems at hand for any given ε > 0. The time complexity of both algorithms is O(TMIS + log*! n/εO(1)), where TMIS is the time required to compute a maximal independent set in the graph, and n denotes the number of nodes. We then extend these results to a more general class of graphs in which the maximum number of pair-wise independent nodes in every r-neighborhood is at most polynomial in r. Such graphs of polynomially bounded growth are introduced as a more realistic model for wireless networks and they generalize existing models, such as unit disk graphs or coverage area graphs

Towards a complexity theory for the congested clique

The congested clique model of distributed computing has been receiving attention as a model for densely connected distributed systems. While there has been significant progress on the side of upper bounds, we have very little in terms of lower bounds for the congested clique; indeed, it is now know that proving explicit congested clique lower bounds is as difficult as proving circuit lower bounds. In this work, we use various more traditional complexity-theoretic tools to build a clearer picture of the complexity landscape of the congested clique: -- Nondeterminism and beyond: We introduce the nondeterministic congested clique model (analogous to NP) and show that there is a natural canonical problem family that captures all problems solvable in constant time with nondeterministic algorithms. We further generalise these notions by introducing the constant-round decision hierarchy (analogous to the polynomial hierarchy). -- Non-constructive lower bounds: We lift the prior non-uniform counting arguments to a general technique for proving non-constructive uniform lower bounds for the congested clique. In particular, we prove a time hierarchy theorem for the congested clique, showing that there are decision problems of essentially all complexities, both in the deterministic and nondeterministic settings. -- Fine-grained complexity: We map out relationships between various natural problems in the congested clique model, arguing that a reduction-based complexity theory currently gives us a fairly good picture of the complexity landscape of the congested clique

Rainbow domination and related problems on some classes of perfect graphs

Let $k \in \mathbb{N}$ and let $G$ be a graph. A function $f: V(G) \rightarrow 2^{[k]}$ is a rainbow function if, for every vertex $x$ with $f(x)=\emptyset$, $f(N(x)) =[k]$. The rainbow domination number $\gamma_{kr}(G)$ is the minimum of $\sum_{x \in V(G)} |f(x)|$ over all rainbow functions. We investigate the rainbow domination problem for some classes of perfect graphs

Fast exact algorithms for some connectivity problems parametrized by clique-width

Given a clique-width $k$-expression of a graph $G$, we provide $2^{O(k)}\cdot n$ time algorithms for connectivity constraints on locally checkable properties such as Node-Weighted Steiner Tree, Connected Dominating Set, or Connected Vertex Cover. We also propose a $2^{O(k)}\cdot n$ time algorithm for Feedback Vertex Set. The best running times for all the considered cases were either $2^{O(k\cdot \log(k))}\cdot n^{O(1)}$ or worse

A Breezing Proof of the KMW Bound

In their seminal paper from 2004, Kuhn, Moscibroda, and Wattenhofer (KMW) proved a hardness result for several fundamental graph problems in the LOCAL model: For any (randomized) algorithm, there are input graphs with $n$ nodes and maximum degree $\Delta$ on which $\Omega(\min\{\sqrt{\log n/\log \log n},\log \Delta/\log \log \Delta\})$ (expected) communication rounds are required to obtain polylogarithmic approximations to a minimum vertex cover, minimum dominating set, or maximum matching. Via reduction, this hardness extends to symmetry breaking tasks like finding maximal independent sets or maximal matchings. Today, more than $15$ years later, there is still no proof of this result that is easy on the reader. Setting out to change this, in this work, we provide a fully self-contained and $\mathit{simple}$ proof of the KMW lower bound. The key argument is algorithmic, and it relies on an invariant that can be readily verified from the generation rules of the lower bound graphs.Comment: 21 pages, 6 figure

GraphCombEx: A Software Tool for Exploration of Combinatorial Optimisation Properties of Large Graphs

We present a prototype of a software tool for exploration of multiple combinatorial optimisation problems in large real-world and synthetic complex networks. Our tool, called GraphCombEx (an acronym of Graph Combinatorial Explorer), provides a unified framework for scalable computation and presentation of high-quality suboptimal solutions and bounds for a number of widely studied combinatorial optimisation problems. Efficient representation and applicability to large-scale graphs and complex networks are particularly considered in its design. The problems currently supported include maximum clique, graph colouring, maximum independent set, minimum vertex clique covering, minimum dominating set, as well as the longest simple cycle problem. Suboptimal solutions and intervals for optimal objective values are estimated using scalable heuristics. The tool is designed with extensibility in mind, with the view of further problems and both new fast and high-performance heuristics to be added in the future. GraphCombEx has already been successfully used as a support tool in a number of recent research studies using combinatorial optimisation to analyse complex networks, indicating its promise as a research software tool

Adding Isolated Vertices Makes some Online Algorithms Optimal

An unexpected difference between online and offline algorithms is observed. The natural greedy algorithms are shown to be worst case online optimal for Online Independent Set and Online Vertex Cover on graphs with 'enough' isolated vertices, Freckle Graphs. For Online Dominating Set, the greedy algorithm is shown to be worst case online optimal on graphs with at least one isolated vertex. These algorithms are not online optimal in general. The online optimality results for these greedy algorithms imply optimality according to various worst case performance measures, such as the competitive ratio. It is also shown that, despite this worst case optimality, there are Freckle graphs where the greedy independent set algorithm is objectively less good than another algorithm. It is shown that it is NP-hard to determine any of the following for a given graph: the online independence number, the online vertex cover number, and the online domination number.Comment: A footnote in the .tex file didn't show up in the last version. This was fixe