On Geometric Spanners of Euclidean and Unit Disk Graphs

We consider the problem of constructing bounded-degree planar geometric spanners of Euclidean and unit-disk graphs. It is well known that the Delaunay subgraph is a planar geometric spanner with stretch factor C_{del\approx 2.42; however, its degree may not be bounded. Our first result is a very simple linear time algorithm for constructing a subgraph of the Delaunay graph with stretch factor \rho =1+2\pi(k\cos{\frac{\pi{k)^{-1 and degree bounded by $k$, for any integer parameter $k\geq 14$. This result immediately implies an algorithm for constructing a planar geometric spanner of a Euclidean graph with stretch factor \rho \cdot C_{del and degree bounded by $k$, for any integer parameter $k\geq 14$. Moreover, the resulting spanner contains a Euclidean Minimum Spanning Tree (EMST) as a subgraph. Our second contribution lies in developing the structural results necessary to transfer our analysis and algorithm from Euclidean graphs to unit disk graphs, the usual model for wireless ad-hoc networks. We obtain a very simple distributed, {\em strictly-localized algorithm that, given a unit disk graph embedded in the plane, constructs a geometric spanner with the above stretch factor and degree bound, and also containing an EMST as a subgraph. The obtained results dramatically improve the previous results in all aspects, as shown in the paper

Independent Set on graphs with maximum degree 3

Let G be an undirected graph with maximum degree at most 3 such that G does not contain either of the two graphs shown in Figure 1 as a subgraph. We prove that the independence number of G is at least n(G)/3 + nt(G)/63, where n(G) is the number of vertices in G and nt(G) is the number of nontriangle vertices in G. We show an application of the aforementioned combinatorial result to the area of parameterized complexity. We present a linear-time kernelization algorithm for the independent set problem on graphs with maximum degree at most 3 that computes a kernel of size at most 630k/211 \u3c 3k, where k is the lower bound on the size of the independent set sought

The role of twins in computing planar supports of hypergraphs

A support or realization of a hypergraph $H$ is a graph $G$ on the same vertex as $H$ such that for each hyperedge of $H$ it holds that its vertices induce a connected subgraph of $G$. The NP-hard problem of finding a planar} support has applications in hypergraph drawing and network design. Previous algorithms for the problem assume that twins}---pairs of vertices that are in precisely the same hyperedges---can safely be removed from the input hypergraph. We prove that this assumption is generally wrong, yet that the number of twins necessary for a hypergraph to have a planar support only depends on its number of hyperedges. We give an explicit upper bound on the number of twins necessary for a hypergraph with $m$ hyperedges to have an $r$-outerplanar support, which depends only on $r$ and $m$. Since all additional twins can be safely removed, we obtain a linear-time algorithm for computing $r$-outerplanar supports for hypergraphs with $m$ hyperedges if $m$ and $r$ are constant; in other words, the problem is fixed-parameter linear-time solvable with respect to the parameters $m$ and $r$

Computing Lightweight Spanning Subgraphs Locally

We consider the problem of computing bounded-degree lightweight plane spanning subgraphs of unit disk graphs in the local distributed model of computation. We are motivated by the hypothesis that such subgraphs can provide the underlying network topology for efficient unicasting and/or multicasting in wireless distributed systems. We start by showing that, for any integer $k \geq 2$, there exists a $k$-local distributed algorithm that, given a unit disk graph $U$ embedded in the plane, constructs a plane subgraph of $U$ containing a Euclidean Minimum Spanning Tree (EMST) of $V(U)$, whose degree is at most 6, and whose total weight is at most $(1 + \frac{2}{k-1})$ times the weight of an EMST of $V(U)$. We show that this bound is tight by proving that, for any $\epsilon \u3e 0$, there exists a unit disk graph $U$ such that no $k$-local distributed algorithm can construct a spanning subgraph of $U$ whose total weight is at most $(1 + \frac{2}{k-1} -\epsilon)$ times the weight of an EMST of $V(U)$. We then go further and present the first $k$-local distributed algorithm, where $k$ is a constant, that computes a bounded-degree plane lightweight {\em spanner} of a given unit disk graph. The upper bounds on the number of communication rounds of the algorithm, the degree, the stretch factor, and the weight of the spanner, are very small. For example, our results imply an $18$-local distributed algorithm that computes a plane spanner of a given unit disk graph $U$, whose degree is at most 14, stretch factor at most 8.81, and weight at most $8.81$ times the weight of an EMST of $V(U)$. All the obtained results rely on an elegant structural result that we develop for weighted planar graphs. We show a wider application of this result by giving an $O(n\lg{n})$ time centralized algorithm that constructs bounded-degree plane lightweight spanners of unit disk graphs (which include Euclidean graphs), with the best upper bounds on the spanner degree, stretch factor, and weight

Computing the k-hop neighborhoods in wireless networks locally

A k-local distributed algorithm (k is a natural number) is a distributed algorithm in which the computation at every point/device in the distributed system modeled as a graph depends solely on the initial states of the points that are at most k hops away from the point. A distributed algorithm is local if it is k-local for some fixed natural number k. Local distributed algorithms are very important, especially for applications in ad-hoc sensor and wireless networks, since such algorithms are naturally scalable, robust, and fault tolerant. Clearly, an essential component of any k-local distributed algorithm is computing the k-hop neighborhoods of the points in the graph. In this paper we show that, given a wireless network modeled as a unit ball graph or as a quasi unit ball graph U on n points in the 3-dimensional (3-D) Euclidean space, and a natural number k, there exists a local distributed algorithm that computes the k-hop neighborhoods of the points in U such that the total number of messages sent by all the points in U at the end of the algorithm is O(n), and where each message has length O(lgn) bits. Clearly, this algorithm is optimal. We note that the same results can be projected to unit disk graphs and quasi unit disk graphs in the Euclidean plane. Moreover, our techniques, and hence our results, can be generalized in a straightforward manner to unit ball graphs in l-dimensional spaces, where l \u3e 3 is an integer

Local algorithms for edge colorings in UDGs

AbstractIn this paper, we consider two problems: the edge coloring and the strong edge coloring problems on unit disk graphs (UDGs). Both problems have important applications in wireless sensor networks as they can be used to model link scheduling problems in such networks. It is well known that both problems are NP-complete, and approximation algorithms for them have been extensively studied under the centralized model of computation. Centralized algorithms, however, are not suitable for ad hoc wireless sensor networks whose devices typically have limited resources, and lack the centralized coordination.We develop local distributed approximation algorithms for the edge coloring and the strong edge coloring problems on unit disk graphs. For the edge coloring problem, our local distributed algorithm has approximation ratio 2 and locality 50. For the strong edge coloring problem on UDGs, we present two local distributed algorithms with different tradeoffs between their approximation ratio and locality. The first algorithm has ratio 128 and locality 22, whereas the second algorithm has ratio 10 and locality 180

Tight lower bounds for certain parameterized NP-hard problems

Based on the framework of parameterized complexity theory, we derive tight lower bounds on the computational complexity for a number of well-known NP-hard problems. We start by proving a general result, namely that the parameterized weighted satisfiability problem on depth-t circuits cannot be solved in time no(k) poly(m), where n is the circuit input length, m is the circuit size, and k is the parameter, unless the (t − 1)-st level W [t − 1] of the W-hierarchy collapses to FPT. By refining this technique, we prove that a group of parameterized NP-hard problems, including weighted sat, dominating set, hitting set, set cover, and feature set, cannot be solved in time no(k) poly(m), where n is the size of the universal set from which the k elements are to be selected and m is the instance size, unless the first level W [1] of the W-hierarchy collapses to FPT. We also prove that another group of parameterized problems which includes weighted q-sat (for any fixed q ≥ 2), clique, and independent set, cannot be solved in time no(k) unless all search problems in the syntactic class SNP, introduced by Papadimitriou and Yannakakis, are solvable in subexponential time. Note that all these parameterized problems have trivial algorithms of running time either n k poly(m) or O(n k).

On the Pseudo-Achromatic Number Problem

We study the parameterized complexity of the pseudo-achromatic number problem: Given an undirected graph and a parameter k, determine if the graph can be partitioned into k groups such that every two groups are connected by at least one edge. This problem has been extensively studied in graph theory and combinatorial optimization. We show that the problem has a kernel of at most (k-2)(k+1) vertices that is constructable in time O(m\sqrt{n}), where n and m are the number of vertices and edges, respectively, in the graph, and k is the parameter. This directly implies that the problem is fixed-parameter tractable. We also study generalizations of the problem and show that they are parameterized intractable

Solving Partition Problems Almost Always Requires Pushing Many Vertices Around

A fundamental graph problem is to recognize whether the vertex set of a graph G can be bipartitioned into sets A and B such that G[A] and G[B] satisfy properties Pi_A and Pi_B, respectively. This so-called (Pi_A,Pi_B)-Recognition problem generalizes amongst others the recognition of 3-colorable, bipartite, split, and monopolar graphs. A powerful algorithmic technique that can be used to obtain fixed-parameter algorithms for many cases of (Pi_A,Pi_B)-Recognition, as well as several other problems, is the pushing process. For bipartition problems, the process starts with an "almost correct" bipartition (A\u27,B\u27), and pushes appropriate vertices from A\u27 to B\u27 and vice versa to eventually arrive at a correct bipartition. In this paper, we study whether (Pi_A,Pi_B)-Recognition problems for which the pushing process yields fixed-parameter algorithms also admit polynomial problem kernels. In our study, we focus on the first level above triviality, where Pi_A is the set of P_3-free graphs (disjoint unions of cliques, or cluster graphs), the parameter is the number of clusters in the cluster graph G[A], and Pi_B is characterized by a set H of connected forbidden induced subgraphs. We prove that, under the assumption that NP not subseteq coNP/poly, (Pi_A,Pi_B)-Recognition admits a polynomial kernel if and only if H contains a graph of order at most 2. In both the kernelization and the lower bound results, we make crucial use of the pushing process