Skip to content
Free planNo credit card requiredStart writing
AI for Research home
AI for Research home
Dashboard
Sign inGet started

Search 250 million research papers

Read the abstract, see who wrote it and who cites it, and get the citation in APA, MLA, Chicago or Harvard.

About 10,000 papers for “Maximal set”

Maximal Independent Set

n.n. · 2013

In this chapter we present a highlight of this course, a fast maximal independent set (MIS) algorithm. The algorithm is the first randomized algorithm that we study in this class. In distributed computing, randomization is a powerful and therefore omnipresent concept, as it allows for relatively simple yet efficient algorithms. As such the studied algorithm is archetypal. A MIS is a basic building block in distributed computing, some other problems pretty much follow directly from the MIS problem. At the end of this chapter, we will give two examples: matching and vertex coloring (see Chapter 1).

Cited by 0

Distributed Maximal Matching and Maximal Independent Set on Hypergraphs

Balliu, Alkida, Brandt, Sebastian, Kuhn, Fabian et al. · Figshare · 2023

We investigate the distributed complexity of maximal matching and maximal independent set (MIS) in hypergraphs in the LOCAL model. A maximal matching of a hypergraph H=(V_H,E_H) is a maximal disjoint set M ⊆ E_H of hyperedges and an MIS S ⊆ V_H is a maximal set of nodes such that no hyperedge is fully contained in S. Both problems can be solved by a simple sequential greedy algorithm, which can be implemented naively in O(∆r + log* n) rounds, where ∆ is the maximum degree, r is the rank, and n is the number of nodes of the hypergraph. We show that for maximal matching, this naive algorithm is optimal in the following sense. Any deterministic algorithm for solving the problem requires Ω(min{∆r, log_{∆r} n}) rounds, and any randomized one requires Ω(min{∆r, log_{∆r} log n}) rounds. Hence, for any algorithm with a complexity of the form O(f(∆,r) + g(n)), we have f(∆,r) ∈ Ω(∆r) if g(n) is not too large, and in particular if g(n) = log* n (which is the optimal asymptotic dependency on n due to Linial's lower bound [FOCS'87]). Our lower bound proof is based on the round elimination framework, and its structure is inspired by a new round elimination fixed point that we give for the ∆-vertex coloring problem in hypergraphs, where nodes need to be colored such that there are no monochromatic hyperedges. For the MIS problem on hypergraphs, we show that for ∆ ≪ r, there are significant improvements over the naive O(∆r + log* n)-round algorithm. We give two deterministic algorithms for the problem. We show that a hypergraph MIS can be computed in O(∆^2 · log r + ∆ · log r · log* r + log* n) rounds. We further show that at the cost of a much worse dependency on ∆, the dependency on r can be removed almost entirely, by giving an algorithm with round complexity ∆^{O(∆)} · log* r + O(log* n).

Cited by 0Open access

Distributed Maximal Matching and Maximal Independent Set on Hypergraphs

Balliu, Alkida, Brandt, Sebastian, Kuhn, Fabian et al. · CISPA Helmholtz Center · 2023

We investigate the distributed complexity of maximal matching and maximal independent set (MIS) in hypergraphs in the LOCAL model. A maximal matching of a hypergraph H=(V_H,E_H) is a maximal disjoint set M ⊆ E_H of hyperedges and an MIS S ⊆ V_H is a maximal set of nodes such that no hyperedge is fully contained in S. Both problems can be solved by a simple sequential greedy algorithm, which can be implemented naively in O(∆r + log* n) rounds, where ∆ is the maximum degree, r is the rank, and n is the number of nodes of the hypergraph. We show that for maximal matching, this naive algorithm is optimal in the following sense. Any deterministic algorithm for solving the problem requires Ω(min{∆r, log_{∆r} n}) rounds, and any randomized one requires Ω(min{∆r, log_{∆r} log n}) rounds. Hence, for any algorithm with a complexity of the form O(f(∆,r) + g(n)), we have f(∆,r) ∈ Ω(∆r) if g(n) is not too large, and in particular if g(n) = log* n (which is the optimal asymptotic dependency on n due to Linial's lower bound [FOCS'87]). Our lower bound proof is based on the round elimination framework, and its structure is inspired by a new round elimination fixed point that we give for the ∆-vertex coloring problem in hypergraphs, where nodes need to be colored such that there are no monochromatic hyperedges. For the MIS problem on hypergraphs, we show that for ∆ ≪ r, there are significant improvements over the naive O(∆r + log* n)-round algorithm. We give two deterministic algorithms for the problem. We show that a hypergraph MIS can be computed in O(∆^2 · log r + ∆ · log r · log* r + log* n) rounds. We further show that at the cost of a much worse dependency on ∆, the dependency on r can be removed almost entirely, by giving an algorithm with round complexity ∆^{O(∆)} · log* r + O(log* n).

Cited by 0Open access

On maximal weakly separated set-systems

В. И. Данилов, Alexander V. Karzanov, Gleb Koshevoy · arXiv (Cornell University) · 2009

For a permutation $ω\in S_n$, Leclerc and Zelevinsky \cite{LZ} introduced a concept of $ω$-{\em chamber weakly separated collection} of subsets of $\{1,2,...,n\}$ and conjectured that all inclusion-wise maximal collections of this sort have the same cardinality $\ell(ω)+n+1$, where $\ell(ω)$ is the length of $ω$. We answer affirmatively this conjecture and present a generalization and additional results.

Cited by 3Open access

Set Systems Containing Many Maximal Chains

J. Robert Johnson, Imre Leader, Paul A. Russell · Combinatorics Probability Computing · 2014

The purpose of this short problem paper is to raise the following extremal question on set systems: Which set systems of a given size maximise the number of ( n + 1)-element chains in the power set $\mathcal{P}$ (1,2,. . ., n )? We will show that for each fixed α > 0 there is a family of α2 n sets containing (α + o (1)) n ! such chains, and that this is asymptotically best possible. For smaller set systems we conjecture that a ‘tower of cubes’ construction is extremal. We finish by mentioning briefly a connection to an extremal problem on posets and a variant of our question for the grid graph.

Cited by 1Open access

Algorithms for maximal independent set problems

Anil Kumar Yadav, Manisha Manjul · 2019

This study involves the development of Maximal Independent Set (MIS) for an arbitrary input graph G. Here we investigate the polynomial-time algorithm using various techniques like greedy, random, divide and conquer and dynamic programming for maximal independent set of a simple graph. An algorithm is be considered ‘efficient’ if its running time is polynomial: O(nc ) for some constant c, where n is the size of the input. The computation of a maximal independent set of minimum size is NP-hard for general graphs.

Cited by 0

Set Systems Containing Many Maximal Chains

Johnson, J. Robert, Leader, Imre, Russell, Paul A. · arXiv (Cornell University) · 2013

The purpose of this short problem paper is to raise an extremal question on set systems which seems to be natural and appealing. Our question is: which set systems of a given size maximise the number of $(n+1)$-element chains in the power set $\mathcal{P}(\{1,2,\dots,n\})$? We will show that for each fixed $α>0$ there is a family of $α2^n$ sets containing $(α+o(1))n!$ such chains, and that this is asymptotically best possible. For smaller set systems we are unable to answer the question. We conjecture that a `tower of cubes' construction is extremal. We finish by mentioning briefly a connection to an extremal problem on posets and a variant of our question for the grid graph.

Cited by 0Open access

Maximal point set domination in graphs

R. Sumathi, K. Angaleeswari, R. Sundareswaran et al. · Discrete Mathematics Algorithms and Applications · 2024

Let [Formula: see text] be a graph with vertex set [Formula: see text] and edge set [Formula: see text] A subset [Formula: see text] of [Formula: see text] is called a dominating set of [Formula: see text] if for any [Formula: see text] in [Formula: see text] there exists a vertex [Formula: see text] such that [Formula: see text] and [Formula: see text] are adjacent. The minimum cardinality of a dominating set of [Formula: see text] is called the domination number of [Formula: see text] and is denoted by [Formula: see text] Several domination parameters have been introduced. Among them, point set domination was conceived by Sampathkumar and Pushpa Latha [E. Sampathkumar and L. Pushpa Latha, Point-set domination number of a graph, Indian J. Pure Appl. Math. 24(4) (1993) 225–229]. A subset [Formula: see text] of [Formula: see text] is called a point set dominating set of [Formula: see text] if for every subset [Formula: see text] of [Formula: see text] there exists a vertex [Formula: see text] such that [Formula: see text] is connected in [Formula: see text] The minimum cardinality of a point set dominating set of [Formula: see text] is called the point set domination number of [Formula: see text] and is denoted by [Formula: see text] Kulli [V. R. Kulli, The Maximal Domination Number of a Graph, Graph Theory Notes of New York XXXIII (Academy of Sciences, 1997), pp. 11–13] introduced the concept of maximal domination number in a graph. A subset [Formula: see text] of [Formula: see text] is called a maximal dominating set of [Formula: see text] if [Formula: see text] is a dominating set of [Formula: see text] and [Formula: see text] is not a dominating set of [Formula: see text] The minimum cardinality of a maximal dominating set of [Formula: see text] is denoted by [Formula: see text] In this paper, the concept of maximal domination is combined with point set domination and maximal point set domination is studied.

Cited by 0

On maximal weakly separated set-systems

Кошевой Г.А. ЦЭМИ РАН RePEc:rus:doahtw:lvs-51, Danilov V.I. and Karzanov A.V., Contacts: repec:rus:ecoorg:cemi-ras_admin.45009-org1 · Central Economics and Mathematics Institute RAS

Онлайновая научная инфраструктура, научное информационное пространство, научно-образовательная социальная сеть, средства для научного 'ÑÐ°Ð¼Ð¾Ð°Ñ€Ñ Ð¸Ð²Ð¸Ñ€Ð¾Ð²Ð°Ð½Ð¸Ñ', институтские Открытые ÐÑ€Ñ Ð¸Ð²Ñ‹

Cited by 0

Maximal Independent Set via Mobile Agents

Debasish Pattanayak, Subhash Bhagat, Sruti Gan Chaudhuri et al. · 2024

We consider the problem of finding a maximal independent set (MIS) in an unknown graph by mobile agents or mobile robots. Suppose n mobile agents are initially located at arbitrary nodes of an n-node anonymous graph G = (V, E) and the goal is that the agents autonomously relocate themselves to find a subset S ⊂ V of nodes such that S forms an MIS. The objective is to minimize both (i) the time required to find an MIS and (ii) the memory required at each agent. We consider the mobile agents with communicate-compute-move model in the synchronous setting, in which agents can communicate with other agents if they are located at the same node (called local communication).

Cited by 10

MAXIMAL DISCRETE SETS (Set Theory and Infinity)

Schrittesser, David · Kyoto University Research Information Repository (Kyoto University) · 2020

We survey results and open questions as well as give some new results regarding the definability and size of maximal discrete sets in analytic hypergraphs. Our main examples include maximal almost disjoint (or mad) families, L-mad families, maximal eventually different families, and maximal cofinitary groups. We discuss the non-increasing sequence of cardinal characteristics ae, forE<w1 as well as the notions of spectra of characteristics and optimal projective witnesses. We give a streamlined account of Zhang's forcing to add generic cofinitary permutations, and of a version of this forcing with built-in coding.

Cited by 0Open access

Constructing a Maximal Independent Set in Parallel

Mark Goldberg, Thomas H. Spencer · SIAM Journal on Discrete Mathematics · 1989

The problem of constructing in parallel a maximal independent set of a given graph is considered. A new deterministic NC-algorithm implemented in the EREW PRAM model is presented. On graphs with n vertices and m edges, it uses $O((n + m)/\log n)$ processors and runs in $O(\log^3 n)$ time. This reduces by a factor of $\log n$ both the running time and the processor count of the previously fastest deterministic algorithm that solves the problem using a linear number of processors.

Cited by 82