Optimality program in segment and string graphs

Planar graphs are known to allow subexponential algorithms running in time $2^{O(\sqrt n)}$ or $2^{O(\sqrt n \log n)}$ for most of the paradigmatic problems, while the brute-force time $2^{\Theta(n)}$ is very likely to be asymptotically best on general graphs. Intrigued by an algorithm packing curves in $2^{O(n^{2/3}\log n)}$ by Fox and Pach [SODA'11], we investigate which problems have subexponential algorithms on the intersection graphs of curves (string graphs) or segments (segment intersection graphs) and which problems have no such algorithms under the ETH (Exponential Time Hypothesis). Among our results, we show that, quite surprisingly, 3-Coloring can also be solved in time $2^{O(n^{2/3}\log^{O(1)}n)}$ on string graphs while an algorithm running in time $2^{o(n)}$ for 4-Coloring even on axis-parallel segments (of unbounded length) would disprove the ETH. For 4-Coloring of unit segments, we show a weaker ETH lower bound of $2^{o(n^{2/3})}$ which exploits the celebrated Erd\H{o}s-Szekeres theorem. The subexponential running time also carries over to Min Feedback Vertex Set but not to Min Dominating Set and Min Independent Dominating Set.Comment: 19 pages, 15 figure

Improved Local Search for Geometric Hitting Set

International audienceOver the past several decades there has been steady progress towards the goal of polynomial-time approximation schemes (PTAS) for fundamental geometric combinatorial optimization problems. A foremost example is the geometric hitting set problem: given a set P of points and a set D of geometric objects, compute the minimum-sized subset of P that hits all objects in D. For the case where D is a set of disks in the plane, the 30-year quest for a PTAS, starting from the seminal work of Hochbaum [19], was finally achieved in 2010. Surprisingly, the algorithm to achieve the PTAS is simple: local-search. Since then, local-search has turned out to be a powerful algorithmic approach towards achieving good approximation ratios for geometric problems (for geometric independent-set problem, for dominating sets, for the terrain guarding problem and several others). Unfortunately all these algorithms have the same limitation: local search is able to give a PTAS, but with hopeless running times; e.g., a 3-approximation for the geometric hitting takes more than n 66 time [15] for the geometric hitting set problem for disks in the plane! That leaves open the question of whether a better understanding – both combinatorial and algorithmic – of local search and the problem can give a better approximation ratio in a more reasonable time. In this paper, we investigate this question for one of the fundamental problems, hitting sets for disks in the plane. We present tight approximation bounds for (3, 2)-local search 1 and give an (8 + algorithm with running time O(n 2.34); the previous-best result achieving a similar approximation ratio gave a 10-approximation in time O(n 15) – that too just for unit disks. The techniques and ideas generalize to (4, 3) local search. Furthermore, as mentioned earlier, local-search has been used for several other geometric optimization problems; for all these problems our results show that (3, 2) local search gives an 8-approximation and no better 2. Similarly (4, 3)-local search gives a 5-approximation for all these problems

Scaling limits of loop-erased random walks and uniform spanning trees

The uniform spanning tree (UST) and the loop-erased random walk (LERW) are related probabilistic processes. We consider the limits of these models on a fine grid in the plane, as the mesh goes to zero. Although the existence of scaling limits is still unproven, subsequential scaling limits can be defined in various ways, and do exist. We establish some basic a.s. properties of the subsequential scaling limits in the plane. It is proved that any LERW subsequential scaling limit is a simple path, and that the trunk of any UST subsequential scaling limit is a topological tree, which is dense in the plane. The scaling limits of these processes are conjectured to be conformally invariant in 2 dimensions. We make a precise statement of the conformal invariance conjecture for the LERW, and show that this conjecture implies an explicit construction of the scaling limit, as follows. Consider the Loewner differential equation ${\partial f\over\partial t} = z {\zeta(t)+z \over \zeta(t)-z} {\partial f\over\partial z}$ with boundary values $f(z,0)=z$, in the range z\in\U=\{w\in\C\st |w|<1\}, $t\le 0$. We choose \zeta(t):= \B(-2t), where \B(t) is Brownian motion on \partial \U starting at a random-uniform point in \partial \U. Assuming the conformal invariance of the LERW scaling limit in the plane, we prove that the scaling limit of LERW from 0 to \partial\U has the same law as that of the path $f(\zeta(t),t)$. We believe that a variation of this process gives the scaling limit of the boundary of macroscopic critical percolation clusters.Comment: (for V2) inserted another figure and two more reference

Fine Gaussian fluctuations on the Poisson space II: rescaled kernels, marked processes and geometric U-statistics

Continuing the analysis initiated in Lachi\'eze-Rey and Peccati (2011), we use contraction operators to study the normal approximation of random variables having the form of a U-statistic written on the points in the support of a random Poisson measure. Applications are provided: to boolean models, and coverage of random networks

Hitting Subgraphs in Sparse Graphs and Geometric Intersection Graphs

We investigate a fundamental vertex-deletion problem called (Induced) Subgraph Hitting: given a graph $G$ and a set $\mathcal{F}$ of forbidden graphs, the goal is to compute a minimum-sized set $S$ of vertices of $G$ such that $G-S$ does not contain any graph in $\mathcal{F}$ as an (induced) subgraph. This is a generic problem that encompasses many well-known problems that were extensively studied on their own, particularly (but not only) from the perspectives of both approximation and parameterization. We focus on the design of efficient approximation schemes, i.e., with running time $f(\varepsilon,\mathcal{F}) \cdot n^{O(1)}$, which are also of significant interest to both communities. Technically, our main contribution is a linear-time approximation-preserving reduction from (Induced) Subgraph Hitting on any graph class $\mathcal{G}$ of bounded expansion to the same problem on bounded degree graphs within $\mathcal{G}$. This yields a novel algorithmic technique to design (efficient) approximation schemes for the problem on very broad graph classes, well beyond the state-of-the-art. Specifically, applying this reduction, we derive approximation schemes with (almost) linear running time for the problem on any graph classes that have strongly sublinear separators and many important classes of geometric intersection graphs (such as fat-object graphs, pseudo-disk graphs, etc.). Our proofs introduce novel concepts and combinatorial observations that may be of independent interest (and, which we believe, will find other uses) for studies of approximation algorithms, parameterized complexity, sparse graph classes, and geometric intersection graphs. As a byproduct, we also obtain the first robust algorithm for $k$-Subgraph Isomorphism on intersection graphs of fat objects and pseudo-disks, with running time $f(k) \cdot n \log n + O(m)$.Comment: 60 pages, abstract shortened to fulfill the length limi

Structuring Unreliable Radio Networks

In this paper we study the problem of building a connected dominating set with constant degree (CCDS) in the dual graph radio network model [4,9,10]. This model includes two types of links: reliable, which always deliver messages, and unreliable, which sometimes fail to deliver messages. Real networks compensate for this differing quality by deploying low-layer detection protocols to filter unreliable from reliable links. With this in mind, we begin by presenting an algorithm that solves the CCDS problem in the dual graph model under the assumption that every process u is provided a local link detector set consisting of every neighbor connected to u by a reliable link. The algorithm solves the CCDS problem in O(Δ\log[superscript 2] n/b + log[superscript 3] n) rounds, with high probability, where Δ is the maximum degree in the reliable link graph, n is the network size, and b is an upper bound in bits on the message size. The algorithm works by first building a Maximal Independent Set (MIS) in log[superscript 3] n time, and then leveraging the local topology knowledge to efficiently connect nearby MIS processes. A natural follow up question is whether the link detector must be perfectly reliable to solve the CCDS problem. With this in mind, we first describe an algorithm that builds a CCDS in O(Δpolylog(n)) time under the assumption of O(1) unreliable links included in each link detector set. We then prove this algorithm to be (almost) tight by showing that the possible inclusion of only a single unreliable link in each process's local link detector set is sufficient to require Ω(Δ) rounds to solve the CCDS problem, regardless of message size. We conclude by discussing how to apply our algorithm in the setting where the topology of reliable and unreliable links can change over time.Simons Foundation. (Postdoctoral Fellows Program)United States. Air Force Office of Scientific Research (Award FA9550-08-1-0159)National Science Foundation (U.S.) (Award CCF-0937274)National Science Foundation (U.S.) (Award CCF-0726514)National Science Foundation (U.S.) (Purdue University) (Science and Technology Center Award 0939370-CCF

Connectivity, Coverage and Placement in Wireless Sensor Networks

Wireless communication between sensors allows the formation of flexible sensor networks, which can be deployed rapidly over wide or inaccessible areas. However, the need to gather data from all sensors in the network imposes constraints on the distances between sensors. This survey describes the state of the art in techniques for determining the minimum density and optimal locations of relay nodes and ordinary sensors to ensure connectivity, subject to various degrees of uncertainty in the locations of the nodes