Maximal independent set

Résumé

In graph theory, a maximal independent set (MIS) or maximal stable set is an independent set that is not a subset of any other independent set. In other words, there is no vertex outside the independent set that may join it because it is maximal with respect to the independent set property. For example, in the graph P3, a path with three vertices a, b, and c, and two edges {{overline|ab}} and {{overline|bc}}, the sets {b} and {a, c} are both maximally independent. The set {a} is independent, but is not maximal independent, because it is a subset of the larger independent set {a, c}. In this same graph, the maximal cliques are the sets {a, b} and {b, c}. A MIS is also a dominating set in the graph, and every dominating set that is independent must be maximal independent, so MISs are also called independent dominating sets. A graph may have many MISs of widely varying sizes; the largest, or possibly

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https://en.wikipedia.org/wiki/Maximal_independent_set

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We develop a sophisticated framework for solving problems in discrete mathematics through the use of randomness (i.e., coin flipping). This includes constructing mathematical structures with unexpected (and sometimes paradoxical) properties for which no other methods of construction are known.

A first graduate course in algorithms, this course assumes minimal background, but moves rapidly. The objective is to learn the main techniques of algorithm analysis and design, while building a repertory of basic algorithmic solutions to problems in many domains.

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Improved Massively Parallel Computation Algorithms for MIS, Matching, and Vertex Cover

Mohsen Ghaffari, Slobodan Mitrovic

We present O(log log n)-round algorithms in the Massively Parallel Computation (MPC) model, with a(n) memory per machine, that compute a maximal independent set, a 1 + epsilon approximation of maximum matching, and a 2 + epsilon approximation of minimum vertex cover, for any n-vertex graph and any constant epsilon > 0. These improve the state of the art as follows: Our MIS algorithm leads to a simple O(log log A)-round MIS algorithm in the CONGESTED-CLIQUE model of distributed computing, which improves on the (O) over tilde(root log Delta)-round algorithm of Ghaffari [PODC'17]. Our O(log log(2) n)-round (1 + epsilon)-approximate maximum matching algorithm simplifies or improves on the following prior work: O(log2 log n)-round (1 + epsilon)-approximation algorithm of Czumaj et al. [STOC'18] and O(log log n)-round (1 + epsilon) approximation algorithm of Assadi et al. [arXiv'17]. Our O(log log n)-round (2 + epsilon)-approximate minimum vertex cover algorithm improves on an O(log log n)-round O(1)-approximation of Assadi et al. [arXiv'17].

ASSOC COMPUTING MACHINERY2018

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ASSOC COMPUTING MACHINERY

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Many modern services need to routinely perform tasks on a large scale. This prompts us to consider the following question: How can we design efficient algorithms for large-scale computation? In this thesis, we focus on devising a general strategy to address the above question. Our approaches use tools from graph theory and convex optimization, and prove to be very effective on a number of problems that exhibit locality. A recurring theme in our work is to use randomization to obtain simple and practical algorithms. The techniques we developed enabled us to make progress on the following questions:

Parallel Computation of Approximately Maximum Matchings. We put forth a new approach to computing $O(1)$ -approximate maximum matchings in the Massively Parallel Computation (MPC) model. In the regime in which the memory per machine is $\Theta(n)$ , i.e., linear in the size of the vertex-set, our algorithm requires only $O((\log \log{n})^2)$ rounds of computations. This is an almost exponential improvement over the barrier of $\Omega(\log {n})$ rounds that all the previous results required in this regime.
Parallel Computation of Maximal Independent Sets. We propose a simple randomized algorithm that constructs maximal independent sets in the MPC model. If the memory per machine is $\Theta(n)$ our algorithm runs in $O(\log \log{n})$ MPC-rounds. In the same regime, all the previously known algorithms required $O(\log{n})$ rounds of computation.
Network Routing under Link Failures. We design a new protocol for stateless message-routing in $k$ -connected graphs. Our routing scheme has two important features: (1) each router performs the routing decisions based only on the local information available to it; and, (2) a message is delivered successfully even if arbitrary $k-1$ links have failed. This significantly improves upon the previous work of which the routing schemes tolerate only up to $k/2 - 1$ failed links in $k$ -connected graphs.
Streaming Submodular Maximization under Element Removals. We study the problem of maximizing submodular functions subject to cardinality constraint $k$ , in the context of streaming algorithms. In a regime in which up to $m$ elements can be removed from the stream, we design an algorithm that provides a constant-factor approximation for this problem. At the same time, the algorithm stores only $O(k \log^2{k} + m \log^3{k})$ elements. Our algorithm improves quadratically upon the prior work, that requires storing $O(k \cdot m)$ many elements to solve the same problem.
Fast Recovery for the Separated Sparsity Model. In the context of compressed sensing, we put forth two recovery algorithms of nearly-linear time for the separated sparsity signals (that naturally model neural spikes). This improves upon the previous algorithm that had a quadratic running time. We also derive a refined version of the natural dynamic programming (DP) approach to the recovery of the separated sparsity signals. This DP approach leads to a recovery algorithm that runs in linear time for an important class of separated sparsity signals. Finally, we consider a generalization of these signals into two dimensions, and we show that computing an exact projection for the two-dimensional model is NP-hard.

EPFL2018

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Parallel Computation of Approximately Maximum Matchings. We put forth a new approach to computing $O(1)$ -approximate maximum matchings in the Massively Parallel Computation (MPC) model. In the regime in which the memory per machine is $\Theta(n)$ , i.e., linear in the size of the vertex-set, our algorithm requires only $O((\log \log{n})^2)$ rounds of computations. This is an almost exponential improvement over the barrier of $\Omega(\log {n})$ rounds that all the previous results required in this regime.
Parallel Computation of Maximal Independent Sets. We propose a simple randomized algorithm that constructs maximal independent sets in the MPC model. If the memory per machine is $\Theta(n)$ our algorithm runs in $O(\log \log{n})$ MPC-rounds. In the same regime, all the previously known algorithms required $O(\log{n})$ rounds of computation.
Network Routing under Link Failures. We design a new protocol for stateless message-routing in $k$ -connected graphs. Our routing scheme has two important features: (1) each router performs the routing decisions based only on the local information available to it; and, (2) a message is delivered successfully even if arbitrary $k-1$ links have failed. This significantly improves upon the previous work of which the routing schemes tolerate only up to $k/2 - 1$ failed links in $k$ -connected graphs.
Streaming Submodular Maximization under Element Removals. We study the problem of maximizing submodular functions subject to cardinality constraint $k$ , in the context of streaming algorithms. In a regime in which up to $m$ elements can be removed from the stream, we design an algorithm that provides a constant-factor approximation for this problem. At the same time, the algorithm stores only $O(k \log^2{k} + m \log^3{k})$ elements. Our algorithm improves quadratically upon the prior work, that requires storing $O(k \cdot m)$ many elements to solve the same problem.
Fast Recovery for the Separated Sparsity Model. In the context of compressed sensing, we put forth two recovery algorithms of nearly-linear time for the separated sparsity signals (that naturally model neural spikes). This improves upon the previous algorithm that had a quadratic running time. We also derive a refined version of the natural dynamic programming (DP) approach to the recovery of the separated sparsity signals. This DP approach leads to a recovery algorithm that runs in linear time for an important class of separated sparsity signals. Finally, we consider a generalization of these signals into two dimensions, and we show that computing an exact projection for the two-dimensional model is NP-hard.

EPFL2018