The Parameterized Complexity of the Unique Coverage Problem

We prove semilogarithmic inapproximability for a maximization problem called unique coverage: given a collection of sets, find a subcollection that maximizes the number of elements covered exactly once. Specifically, assuming that $\mathrm{NP}\not\subseteq\operatorname{BPTIME}(2^{n^\varepsilon})$ for an arbitrary $\varepsilon>0$, we prove $O(1/\log^{\sigma}n)$ inapproximability for some constant $\sigma=\sigma(\varepsilon)$. We also prove $O(1/\log^{1/3-\varepsilon}n)$ inapproximability for any $\varepsilon>0$, assuming that refuting random instances of 3SAT is hard on average; and we prove $O(1/\log n)$ inapproximability under a plausible hypothesis concerning the hardness of another problem, balanced bipartite independent set. We establish an $\Omega(1/\log n)$-approximation algorithm, even for a more general (budgeted) setting, and obtain an $\Omega(1/\log B)$-approximation algorithm when every set has at most B elements. We also show that our inapproximability results extend to envy-free pricing, an important problem in computational economics. We describe how the (budgeted) unique coverage problem, motivated by real-world applications, has close connections to other theoretical problems, including max cut, maximum coverage, and radio broadcasting.

We prove semi-logarithmic inapproximability for a maximization problem called unique coverage: given a collection of sets, find a subcollection that maximizes the number of elements covered exactly once. Specifically, we prove O(1/ logσ(e)n) inapproximability assuming that NP n BPTIME(2ne) for some e > 0. We also prove O(1/log1/3-en) inapproximability, for any e > 0, assuming that refuting random instances of 3SAT is hard on average; and prove O(1/log n) inapproximability under a plausible hypothesis concerning the hardness of another problem, balanced bipartite independent set. We establish matching upper bounds up to exponents, even for a more general (budgeted) setting, giving an Ω(1/log n)-approximation algorithm as well as an Ω(1/log B)-approximation algorithm when every set has at most B elements. We also show that our inapproximability results extend to envy-free pricing, an important problem in computational economics. We describe how the (budgeted) unique coverage problem, motivated by real-world applications, has close connections to other theoretical problems including max cut, maximum coverage, and radio broad-casting.

Fixed-parameter tractability (FPT) techniques have recently been successful in solving NP-complete problem instances of practical importance which were too large to be solved with previous methods. In this paper we show how to enhance this approach through the addition of parallelism, thereby allowing even larger problem instances to be solved in practice. More precisely, we demonstrate the potential of parallelism when applied to the bounded-tree search phase of FPT algorithms. We apply our methodology to the k-VERTEX COVER problem which has important applications, e.g., in multiple sequence alignments for computational biochemistry. We have implemented our parallel FPT method for the k-VERTEX COVER problem using C and the MPI communication library, and tested it on a PC cluster. This is the first experimental examination of parallel FPT techniques. We have tested our parallel k-VERTEX COVER method on protein sequences obtained from the National Center for Biotechnology Information. As part of our experiments, we solved larger instances of k-VERTEX COVER than in any previously reported implementations. For example, our code can solve problem instances with k /spl ges/ 400 in less than 1.5 hours. Since our parallel FPT algorithm requires only very little communication between processors, we expect our method to also perform well on Grids.

For fixed integers r,ell geq 0, a graph G is called an (r,ell)-graph if the vertex set V(G) can be partitioned into r independent sets and ell cliques. This brings us to the following natural parameterized questions: Vertex (r,ell)-Partization and Edge (r,ell)-Partization. An input to these problems consist of a graph G and a positive integer k and the objective is to decide whether there exists a set S subseteq V(G) (S subseteq E(G)) such that the deletion of S from G results in an (r,ell)-graph. These problems generalize well studied problems such as Odd Cycle Transversal, Edge Odd Cycle Transversal, Split Vertex Deletion and Split Edge Deletion. We do not hope to get parameterized algorithms for either Vertex (r, ell)-Partization or Edge (r,ell)-Partization when either of r or ell is at least 3 as the recognition problem itself is NP-complete. This leaves the case of r,ell in {1,2}. We almost complete the parameterized complexity dichotomy for these problems by obtaining the following results: - We show that Vertex (r,ell)-Partization is fixed parameter tractable (FPT) for r,ell in {1,2}. Then we design an Oh(sqrt log n)-factor approximation algorithms for these problems. These approximation algorithms are then utilized to design polynomial sized randomized Turing kernels for these problems. - Edge (r,ell)-Partization is FPT when (r,ell)in{(1,2),(2,1)}. However, the parameterized complexity of Edge (2,2)-Partization remains open. For our approximation algorithms and thus for Turing kernels we use an interesting finite forbidden induced graph characterization, for a class of graphs known as (r,ell)-split graphs, properly containing the class of (r,ell)-graphs. This approach to obtain approximation algorithms could be of an independent interest.

We present the first study on the approximability of geometric versions of the unique coverage problem and the minimum membership set cover problem. In the former problem, one is given a family of sets of elements from some universe and aims to select sets that maximize the number of elements contained in precisely one selected set. Unique Coverage has important applications in wireless networks and it is thus natural to consider it in a geometric setting. We use the well-known (unit) disk model for wireless networks and show that Unique Coverage remains NP-hard in this case and present a polynomial-time 1/18-approximation algorithm. This algorithm is extended to the budgeted low-coverage problem, where covering can element multiple times yields less profit and we have a fixed budget to 'buy' sets. We give an asymptotic FPTAS in case the disks have arbitrary size, but bounded ply. For the case that the geometric objects are arbitrary fat objects, we show that these problems are as hard to approximate as in the general case. In the minimum membership set cover problem, the goal is to cover all elements while minimizing the maximum number of sets in which any element is contained. For unit squares and unit disks, we show that the problem remains NP-hard and does not admit a polynomial-time approximation algorithm with ratio smaller than 2 unless P=NP. For unit squares, we give a 5-approximation algorithm for instances where the optimum objective value is bounded by a constant.

This paper presents the study of the hyper-Erlang distribution model and its applications in wireless networks and mobile computing systems. We demonstrate that the hyper-Erlang model provides a very general model for users' mobility and may provide a viable approximation to fat-tailed distribution which leads to the self-similar traffic. The significant difference from the traditional approach in the self-similarity study is that we want to provide an approximation model which preserves the Markovian property of the resulting queueing systems. We also illustrate that the hyper-Erlang distribution is a natural model for the characterization of the systems with mixed types of traffics. As an application, we apply the hyper-Erlang distribution to model the cell residence time (for users' mobility) and demonstrate the effect on channel holding time. This research may open a new avenue for traffic modeling and performance evaluation for future wireless networks and mobile computing systems, over which multiple types of services (voice, data or multimedia) will be supported.

A polynomial-time approximation scheme for the geometric unique coverage problem on unit squares

A 4.31-approximation for the geometric unique coverage problem on unit disks

Algorithms are designed, analyzed and implemented for the maximum clique enumeration (MCE) problem, which asks that we identify all maximum cliques in a finite, simple graph. MCE is closely related to two other well-known and widely-studied problems: the maximum clique optimization problem, which asks us to determine the size of a largest clique, and the maximal clique enumeration problem, which asks that we compile a listing of all maximal cliques. Naturally, these three problems are NP-hard, given that they subsume the classic version of the NP-complete clique decision problem. MCE can be solved in principle with standard enumeration methods due to Bron, Kerbosch, Kose and others. Unfortunately, these techniques are ill-suited to graphs encountered in our applications. We must solve MCE on instances deeply seeded in data mining and computational biology, where high-throughput data capture often creates graphs of extreme size and density. MCE can also be solved in principle using more modern algorithms based in part on vertex cover and the theory of fixed-parameter tractability (FPT). While FPT is an improvement, these algorithms too can fail to scale sufficiently well as the sizes and densities of our datasets grow. An extensive testbed of benchmark MCE instances is devised, based on applications in transcriptomic data analysis. Empirical testing reveals crucial but latent features of such high-throughput biological data. In turn, it is shown that these features distinguish real data from random data intended to reproduce salient topological features. In particular, with real data there tends to be an unusually high degree of maximum clique overlap. Armed with this knowledge, novel decomposition strategies are tuned to the data and coupled with the best FPT MCE implementations. It is demonstrated that the resultant run times are frequently reduced by several orders of magnitude, and that instances once prohibitively time-consuming to solve are now often brought into the domain of realistic feasibility.

In this paper, we consider a multimedia data transmission system over a wireless channel, where packets are queued at the transmitter. Multimedia data transmission over wireless networks often suffers from delay, jitter and packet loss. The main problem to implement a wireless network is the high and variable bit error rate in the radio link (fading, shadowing etc). Among the most important performance measures for real time applications are the packet loss probability. In order to improve the radio link, one often retransmits packets that have not been well received (using the Automatic Retransmission reQuest). This however may lead to queuing phenomena and increased delay due to retransmissions, and to losses of packets due to buffer overflow. In this paper, we allow that the number of retransmission is finite. We model the system state by a three-dimensional Markov chain which represents the evolution the radio link state, the number of packets in the buffer and the number of transmission of a packet in the service. We use an advanced approach based on the theory of singular perturbation in order to compute some performance of interest.

Fractional Link scheduling is one of the most fundamental problems in wireless networks. The prevailing approach for shortest fractional link scheduling is based on a reduction to the maximum-weighted independent set problem, which itself may not admit efficient approximation algorithms. In addition, except for the wireless networks under the protocol interference model, none of the existing scheduling algorithms can produce a link schedule with explicit upper bounds on its length in terms of the link demands. As the result, the polynomial approximate capacity regions in these networks remain blank. This paper develops a purely combinatorial paradigm for fractional link scheduling in wireless networks. In addition to the superior efficiency, it is able to provide explicit upper bounds on the lengths of the produced link schedule. By exploiting these upper bounds, polynomial approximate capacity regions are derived. The effectiveness of this new paradigm is demonstrated by its applications in wireless networks under the physical interference model and wireless MIMO networks under the protocol interference model.

Blockchain technology is finding applications in wireless networks, in particular vehicular networks, for the purposes of establishing trust. A certain set of network resources in terms of communication and computation requirements is also necessary for successful blockchain deployment. Moreover, heterogeneity in wireless networks and changing radio conditions at the physical layer, make it even more challenging to guarantee steady block generation latency. In this work, we study the possible denial of blockchain service attacks where malicious nodes threaten to slow down the block generation latency. We propose a novel Proof-of-Communication-Capability (PoCC) authentication framework that acts as a defense against communication capability spoofing over wireless networks. Our PoCC authentication framework utilizes physical properties such as distance between nodes, channel state information (CSI), and communication puzzle latency to establish the communication capabilities of nodes in the wireless network. Results from a simulated AV network under three different variations of the proposed PoCC framework are encouraging and demonstrate that such attacks can be effectively mitigated.

The communication network indicators are analyzed and cryptographic analysis is considered to ensure information security of wireless communication networks using the application. Exploring fundamental principles and key algorithms like Advanced Encryption Standard and Rivest-Shamir-Adleman, this article equips readers with practical insights. This paper examines the challenges of researching cryptographic analysis to ensure the security of wireless communication networks using the application By understanding cryptographic algorithms and their application in wireless networks, readers can enhance the security their digital communications. A wireless network is used to connect various wired organizational structures and provide connectivity within the organization for employees to move freely by avoiding the hurdle of a physical network. Subsequent sections offer theoretical insights and performance analysis of cryptographic algorithms using the Java programming language.

Channel assignment and its applications in wireless networks by Yanyan Yang thesis 2010 x, 90 p. : ill. ; 30 cm Wireless technologies grow rapidly and benefit almost every aspect of our daily lives. To fulfill various application requirements, researchers…Read more ›

The theoretical analysis of a cross-layer mechanism for improving the quality of service of real-time applications in wireless networks is presented. The mechanism coordinates adaptations of the modulation order at the Physical layer and the media encoding mode at the Application layer, to improve packet loss rate, throughput and mean delay. With the use of Continuous Flow Modeling, the system is considered as a "fluid" queue with inflow and outflow rates representing its traffic generation and service rates, respectively. Each data source is modeled as a Markov chain, from the steady-state of which the optimal adaptation thresholds of the cross-layer mechanism are derived. Extensive performance evaluation results show that the optimized operation of the mechanism attains a significant performance improvement compared to both the sub-optimal case, and a legacy system, which adjusts the modulation order and encoding mode separately and independently of each other.