A PTAS for the minimum dominating set problem in unit disk graphs

We present a polynomial-time approximation scheme (PTAS) for the minimum dominating set problem in unit disk graphs. In contrast to previously known approximation schemes for the minimum dominating set problem on unit disk graphs, our approach does not assume a geometric representation of the vertices (specifying the positions of the disks in the plane) to be given as part of the input. \u

Max-Cut and Max-Bisection are NP-hard on unit disk graphs

We prove that the Max-Cut and Max-Bisection problems are NP-hard on unit disk graphs. We also show that $\lambda$-precision graphs are planar for $\lambda$ > 1 / \sqrt{2}$

Approximating Minimum Independent Dominating Sets in Wireless Networks

We present the first polynomial-time approximation scheme (PTAS) for the Minimum Independent Dominating Set problem in graphs of polynomially bounded growth. Graphs of bounded growth are used to characterize wireless communication networks, and this class of graph includes many models known from the literature, e.g. (Quasi) Unit Disk Graphs. An independent dominating set is a dominating set in a graph that is also independent. It thus combines the advantages of both structures, and there are many applications that rely on these two structures e.g. in the area of wireless ad hoc networks. The presented approach yields a robust algorithm, that is, the algorithm accepts any undirected graph as input, and returns a (1+")- pproximate minimum dominating set, or a certificate showing that the input graph does not reflect a wireless network

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

Level-treewidth property, exact algorithms and approximation schemes

Informally, a class of graphs Q is said to have the level-treewidth property (LT-property) if for every G {element_of} Q there is a layout (breadth first ordering) L{sub G} such that the subgraph induced by the vertices in k-consecutive levels in the layout have treewidth O(f (k)), for some function f. We show that several important and well known classes of graphs including planar and bounded genus graphs, (r, s)-civilized graphs, etc, satisfy the LT-property. Building on the recent work, we present two general types of results for the class of graphs obeying the LT-property. (1) All problems in the classes MPSAT, TMAX and TMIN have polynomial time approximation schemes. (2) The problems considered in Eppstein have efficient polynomial time algorithms. These results can be extended to obtain polynomial time approximation algorithms and approximation schemes for a number of PSPACE-hard combinatorial problems specified using different kinds of succinct specifications studied in. Many of the results can also be extended to {delta}-near genus and {delta}-near civilized graphs, for any fixed {delta}. Our results significantly extend the work in and affirmatively answer recent open questions

On the Complexity of Scheduling in Wireless Networks

We consider the problem of throughput-optimal scheduling in wireless networks subject to interference constraints. We model the interference using a family of K-hop interference models, under which no two links within a K-hop distance can successfully transmit at the same time. For a given K, we can obtain a throughput-optimal scheduling policy by solving the well-known maximum weighted matching problem. We show that for K > 1, the resulting problems are NP-Hard that cannot be approximated within a factor that grows polynomially with the number of nodes. Interestingly, for geometric unit-disk graphs that can be used to describe a wide range of wireless networks, the problems admit polynomial time approximation schemes within a factor arbitrarily close to 1. In these network settings, we also show that a simple greedy algorithm can provide a 49-approximation, and the maximal matching scheduling policy, which can be easily implemented in a distributed fashion, achieves a guaranteed fraction of the capacity region for "all K." The geometric constraints are crucial to obtain these throughput guarantees. These results are encouraging as they suggest that one can develop low-complexity distributed algorithms to achieve near-optimal throughput for a wide range of wireless networksopen1

Algebraic and Combinatorial Methods in Computational Complexity

At its core, much of Computational Complexity is concerned with combinatorial objects and structures. But it has often proven true that the best way to prove things about these combinatorial objects is by establishing a connection (perhaps approximate) to a more well-behaved algebraic setting. Indeed, many of the deepest and most powerful results in Computational Complexity rely on algebraic proof techniques. The PCP characterization of NP and the Agrawal-Kayal-Saxena polynomial-time primality test are two prominent examples. Recently, there have been some works going in the opposite direction, giving alternative combinatorial proofs for results that were originally proved algebraically. These alternative proofs can yield important improvements because they are closer to the underlying problems and avoid the losses in passing to the algebraic setting. A prominent example is Dinur's proof of the PCP Theorem via gap amplification which yielded short PCPs with only a polylogarithmic length blowup (which had been the focus of significant research effort up to that point). We see here (and in a number of recent works) an exciting interplay between algebraic and combinatorial techniques. This seminar aims to capitalize on recent progress and bring together researchers who are using a diverse array of algebraic and combinatorial methods in a variety of settings

The complexity of searching implicit graphs

AbstractThe standard complexity classes of complexity theory do not allow for direct classification of most of the problems solved by heuristic search algorithms. The reason is that, almost always, these are defined in terms of implicit graphs of state or problem reduction spaces, while the standard definitions of all complexity classes are specifically tailored to explicit inputs.To allow for more precise comparisons with standard complexity classes, we introduce here a model for the analysis of algorithms on graphs given by vertex expansion procedures. It is based on previously studied concepts of “succinct representation” techniques, and allows us to prove PSPACE-completeness or EXPTIME-completeness of specific, natural problems on implicit graphs, such as those solved by A∗, AO∗, and other best-first search strategies