A Primal-Dual Bicriteria Distributed Algorithm for Capacitated Vertex Cover

Over the years many interesting and efficient parallel algorithms have been developed to solve a wide variety of problems, but not much attention has been paid to studying the inherent parallelism in sequential algorithms---i.e., understanding the depth of their dependence structure, and how shallow dependence structures might beused to develop efficient parallel implementations. In this talk I will describe recent work on analyzing the dependence depth of iterative sequential algorithms---ones that loop over a collection of elements. Many of these algorithms have deep dependence chains in the worst case, but shallow chains (polylog w.h.p.) if the elements are randomly ordered. Examples include many fundamental algorithms: the Knuth shuffle for random permutations, sorting by insertion into a binary search tree, greedy maximal independent set (MIS), greedy maximal matching, greedy graph-coloring, counting cycles in a permutation, incremental k-dimensional linear programming, and incremental 2d Delaunay triangulation. An advantage of the approach is that it can lead to very simple and efficient parallel algorithms. Our MIS algorithm, for example can be coded in a dozen or so lines, and is significantly faster than Luby's algorithm on modern multicore machines. Also the approach encourages snapping the view that sequential and parallel algorithms are distinct, and instead thinking of algorithms, in general, as collections of instructions with dependences among them.

The Maximal Independent Set problem is a well-studied problem in the distributed community. We study Maximum and Maximal Independent Set problems on two geometric intersection graphs; interval graphs and axis-parallel segment intersection graphs, and present deterministic distributed algorithms in the local communication model. We compute the maximum independent set on interval graphs in O(k) rounds and O(n) messages, where k is the size of the maximum independent set and n is the number of nodes in the graph. We provide a matching lower bound of \(\varOmega (k)\) on the number of rounds whereas \(\varOmega (n)\) is a trivial lower bound on message complexity. Thus our algorithm is both time and message optimal. We also study the maximal independent set problem in bi-interval graphs, a special case of the interval graphs where the intervals have two different lengths. We prove that a maximal independent set can be computed in bi-interval graphs in constant rounds that is \(\frac{1}{6}\)-approximation. For axis-parallel segment intersection graphs, we design an algorithm that finds a maximal independent set in O(D) rounds, where D is the diameter of the graph. We further show that this independent set is a \(\frac{1}{2}\)-approximation. The results in this paper extend the results of Molla et al. [J. Parallel Distrib. Comput. 2019].

Suppose that finding the solution to a problem is P-complete. It is natural to ask if it is any easier to obtain an approximate solution. For decision problems this might mean considering the corresponding combinatorial optimization problem. That is, a problem in which we try to minimize or maximize a given quantity. As one might expect from the theory of NP-completeness, the answer is both yes (for example in the case of Bin Packing, Problem A.4.7) and no (for example in the case of the Lexicographically First Maximal Independent Set Size Problem, see Lemma 10.2.2.). There are several motivations for developing good NC approximation algorithms. First, in all likelihood P-complete problems cannot be solved fast in parallel. Therefore, it may be useful to approximate them quickly in parallel. Second, problems that are P- complete but that can be approximated well seem to be special boundary cases. Perhaps by examining these types of problems more closely we can improve our understanding of parallelism. Third, it is important to build a theoretical foundation for studying and classifying additional approximation problems. Finally, it may be possible to speed up sequential approximation algorithms, of NP-complete problems, using fast parallel approximations. Our goal in this section is to develop the basic theory of parallel approximation algorithms. We begin by showing that certain P-complete problems are not amenable to NC approximation algorithms. Later we present examples of P-complete problems that can be approximated well in parallel. We start by considering the Lexicographically First Maximal Independent Set Problem, introduced in Definition 7.1.1, and proven P-complete in Problem A.2.1. As defined, LFMIS it is not directly amenable to approximation. We can phrase the problem in terms of computing the size of the independent set. Definition 10.2.1 Lexicographically First Maximal Independent Set Size (LFMISsize) Given: An undirected graph G = (V, E) with an ordering on the vertices and an integer k. Problem: Is the size of the lexicographically first maximal independent set of G less than or equal to k ? The following lemma shows that computing just the size of the lexicographically first maximal independent set is P-complete.

Convergence and approximation in potential games

Efficient BSP/CGM Algorithms for the Maximum Subsequence Sum and Related Problems 1

A 2-approximation NC algorithm for connected vertex cover and tree cover

Energy efficient head node selection is the major issue in wireless sensor networks. Due to irregular node distribution energy consumption is not a balanced one in cluster-based wireless sensor networks. The proposed system provides techniques to rectify this problem through: (i) Forming Maximal Independent Set (MIS) (ii) Effective Set Head selection (iii) Multi-hop communication. Non adjacency of nodes has been leveraged for creating MIS. Intra-set energy consumption has been reduced by selecting a node as Set Head (SH), which is geographically nearer to all other nodes. Inter Set efficient energy consumption is achieved by using multi-hop communication between SH to BS. The proposed system EECDC forms more efficient and stable sets, which increases the lifetime of sensor networks i.e., Number of data collection rounds.

We formalize the class of “sequential local and show that these algorithms fail to find satisfying assignments on random instances of the “Not-All-Equal-$K$-SAT” (NAE-$K$-SAT) problem if the number of message passing iterations is bounded by a function moderately growing in the number of variables and if the clause-to-variable ratio is above $(1+o_K(1)){2^{K-1}\over K}\ln^2 K$ for sufficiently large $K$. Sequential local algorithms are those that iteratively set variables based on some local information and/or local randomness and then recurse on the reduced instance. Our model captures some weak abstractions of natural algorithms such as Survey Propagation (SP)-guided as well as Belief Propagation (BP)-guided decimation algorithms---two widely studied message-passing--based algorithms---when the number of message-passing rounds in these algorithms is restricted to be growing only moderately with the number of variables. The approach underlying our paper is based on an intricate geometry of th...

We present a randomized parallel list ranking algorithm for distributed memory multiprocessors, using a BSP type model. We first describe a simple version which requires, with high probability, log(3p)+log ln(n)=O(logp+log logn) communication rounds (h-relations withh=O(n/p)) andO(n/p)) local computation. We then outline an improved version that requires high probability, onlyr⩽(4k+6) log(2/3p)+8=O(k logp) communication rounds wherek=min{i⩾0 |ln(i+1)n⩽(2/3p)2i+1}. Notek<ln*(n) is an extremely small number. Forn andp⩾4, the value ofk is at most 2. Hence, for a given number of processors,p, the number of communication rounds required is, for all practical purposes, independent ofn. Forn⩽1, 500,000 and 4⩽p⩽2048, the number of communication rounds in our algorithm is bounded, with high probability, by 78, but the actual number of communication rounds observed so far is 25 in the worst case. Forn⩽10010100 and 4⩽p⩽2048, the number of communication rounds in our algorithm is bounded, with high probability, by 118; and we conjecture that the actual number of communication rounds required will not exceed 50. Our algorithm has a considerably smaller member of communication rounds than the list ranking algorithm used in Reid-Miller’s empirical study of parallel list ranking on the Cray C-90.(1) To our knowledge, Reid-Miller’s algorithm(1) was the fastest list ranking implementation so far. Therefore, we expect that our result will have considerable practical relevance.

This paper presents a randomized parallel algorithm for the Maximal Independent Set problem. Our algorithm uses a BSP-like computer with p processors and requires that n+m/p=/spl Omega/(p) for a graph with 72 vertices and m edges. Under this scalability assumption, and after a preprocessing phase, it computes a maximal independent set after O(log p) communication rounds, with high probability, each round requiring linear computation time O(n+p/p). The preprocessing phase is deterministic and important in order to ensure that degree computations can be implemented efficiently. For this, we give an optimal parallel BSP/CGM algorithm to the p-quantiles search problem, which runs in O(m log p/p) time and a constant number of communication rounds, and could be of interest in its own right, as shown in the text.

In this book chapter, the authors discuss some important communication issues to obtain a highly scalable computing system. They consider the CGM (Coarse-Grained Multicomputer) model, a realistic computing model to obtain scalable parallel algorithms. The communication cost is modeled by the number of communication rounds and the objective is to design algorithms that require the minimum number of communication rounds. They discuss some important issues and make considerations of practical importance, based on our previous experience in the design and implementation of parallel algorithms. The first issue is the amount of data transmitted in a communication round. For a practical implementation to be successful they should attempt to minimize this amount, even when it is already within the limit allowed by the CGM model. The second issue concerns the trade-off between the number of communication rounds which the CGM attempts to minimize and the overall communication time taken in the communication rounds. Sometimes a larger number of communication rounds may actually reduce the total amount of data transmitted in the communications rounds. These two issues have guided us to present efficient parallel algorithms for the string similarity problem, used as an illustration.

We present improved results for approximating maximum-weight independent set (MaxIS) in the CONGEST and LOCAL models of distributed computing. Given an input graph, let n and Δ be the number of nodes and maximum degree, respectively, and let MIS(n,Δ) be the running time of finding a maximal independent set (MIS) in the CONGEST model. Bar-Yehuda et al. [PODC 2017] showed that there is an algorithm in the CONGEST model that finds a Δ-approximation for MaxIS in O(MIS(n,Δ)log W) rounds, where W is the maximum weight of a node in the graph, which can be as large as poly (n). Whether their algorithm is deterministic or randomized that succeeds with high probability depends on the MIS algorithm that is used as a black-box. Our results: 1) A deterministic O(MIS(n,Δ)/e)-round algorithm that finds a (1+e)Δ-approximation for MaxIS in the CONGEST model. 2) A randomized (poly(log log n)/e)-round algorithm that finds, with high probability, a (1+e)Δ-approximation for MaxIS in the CONGEST model. That is, by sacrificing only a tiny fraction of the approximation guarantee, we achieve an exponential speed-up in the running time over the previous best known result. 3) A randomized O(log n⋅ poly(log log n)/e)-round algorithm that finds, with high probability, a 8(1+e)α-approximation for MaxIS in the CONGEST model, where α is the arboricity of the graph. For graphs of arboricity α < Δ/(8(1+e)), this result improves upon the previous best known result in both the approximation factor and the running time. One may wonder whether it is possible to approximate MaxIS with high probability in fewer than poly(log log n) rounds. Interestingly, a folklore randomized ranking algorithm by Boppana implies a single round algorithm that gives an expected Δ-approximation in the CONGEST model. However, it is unclear how to convert this algorithm to one that succeeds with high probability without sacrificing a large number of rounds. For unweighted graphs of maximum degree Δ ≤ n/log n, we show a new analysis of the randomized ranking algorithm, which we combine with the local-ratio technique, to provide a O(1/e)-round algorithm in the CONGEST model that, with high probability, finds an independent set of size at least n/((1+e)(Δ+1)). This result cannot be extended to very high degree graphs, as we show a lower bound of Ω(log^*n) rounds for any randomized algorithm that with probability at least 1-1/log n finds an independent set of size Ω(n/Δ). This lower bound holds even for the LOCAL model. The hard instances that we use to prove our lower bound are graphs of maximum degree Δ = Ω(n/log^*n).

Parameterized approximation algorithms for weighted vertex cover

Due to the very high symbol rate of terrestrial HDTV systems, up to now there exist no equalization solutions with sufficiently low hardware complexity and satisfactory performance for commercial applications. We present a block sequential least squares decision feedback equalization algorithm with application to over-the-air HDTV channels. The proposed adaptive algorithm is derived on the basis of minimization of the least squares criterion, thereby achieving faster convergent and tracking rate relative to the recommended LMS algorithm. Meanwhile, good numerical stability is guaranteed because it successfully eliminates time updates of the filtering coefficients, which is the main cause of instability of FTF-like algorithms. Also of great significance is its drastic reduction in computational complexity by means of block operation, taking advantage of the slowly time-varying nature of terrestrial broadcasting channels. Simulation results show that the equalizer achieves an almost 3.5 dB SNR improvement at a bit error rate of 3/spl times/10/sup -6/ without significant increase in computational complexity, as compared to the conventional LMS decision feedback equalizer (DFE) when applied to the equalization of over-the-air HDTV channels.

The minimum vertex cover (MVC) problem is a well-studied NP-Complete problem and has various applications. In this paper, a new heuristic approach has been proposed to find the minimum vertex cover of a graph. The proposed algorithm has been tested on random graphs and BHOSLIB instances. The results have shown that the proposed algorithm can yield better solutions especially on dense graphs for solving the minimum vertex cover problem.