On Random Greedy Triangle Packing

10.37236/1296 ◽  
1997 ◽  
Vol 4 (1) ◽  
Author(s):  
David A. Grable
Keyword(s):  
Greedy Algorithm ◽  
General Problem ◽  
Upper Bound ◽  
Triple System ◽  
Final Number ◽  
Edge Disjoint ◽  
Partial Triple System ◽  
Maximal Packing

The behaviour of the random greedy algorithm for constructing a maximal packing of edge-disjoint triangles on $n$ points (a maximal partial triple system) is analysed with particular emphasis on the final number of unused edges. It is shown that this number is at most $n^{7/4+o(1)}$, "halfway" from the previous best-known upper bound $o(n^2)$ to the conjectured value $n^{3/2+o(1)}$. The more general problem of random greedy packing in hypergraphs is also considered.

scholarly journals Revisiting Modified Greedy Algorithm for Monotone Submodular Maximization with a Knapsack Constraint

2021 ◽  
Vol 5 (1) ◽  
pp. 1-22
Author(s):  
Jing Tang ◽  
Xueyan Tang ◽  
Andrew Lim ◽  
Kai Han ◽  
Chongshou Li ◽  
...  
Keyword(s):  
Approximation Algorithms ◽  
Greedy Algorithm ◽  
Branch And Bound ◽  
Upper Bound ◽  
Optimization Problem ◽  
Approximation Factor ◽  
Real World Application ◽  
Efficiency Of Algorithms ◽  
Knapsack Constraint ◽  
Submodular Maximization

Monotone submodular maximization with a knapsack constraint is NP-hard. Various approximation algorithms have been devised to address this optimization problem. In this paper, we revisit the widely known modified greedy algorithm. First, we show that this algorithm can achieve an approximation factor of 0.405, which significantly improves the known factors of 0.357 given by Wolsey and (1-1/e)/2\approx 0.316 given by Khuller et al. More importantly, our analysis closes a gap in Khuller et al.'s proof for the extensively mentioned approximation factor of (1-1/\sqrte )\approx 0.393 in the literature to clarify a long-standing misconception on this issue. Second, we enhance the modified greedy algorithm to derive a data-dependent upper bound on the optimum. We empirically demonstrate the tightness of our upper bound with a real-world application. The bound enables us to obtain a data-dependent ratio typically much higher than 0.405 between the solution value of the modified greedy algorithm and the optimum. It can also be used to significantly improve the efficiency of algorithms such as branch and bound.

scholarly journals Triple systems with a fixed number of repeated triples

1986 ◽  
Vol 41 (2) ◽  
pp. 180-187 ◽  
Author(s):  
C. A. Rodger
Keyword(s):  
Triple System ◽  
Fixed Number ◽  
Triple Systems ◽  
Partial Triple System

AbstractIn this paper, linear embeddings of partial designs into designs are found where no repeated blocks are introduced in the embedding process. Triple systems, pure cyclic triple systems, cyclic and directed triple systems are considered. In particular, a partial triple system with no repeated triples is embedded linearly in a triple system with no repeated triples.

scholarly journals Large triangle packings and Tuza’s conjecture in sparse random graphs

2020 ◽  
Vol 29 (5) ◽  
pp. 757-779 ◽  
Author(s):  
Patrick Bennett ◽  
Andrzej Dudek ◽  
Shira Zerbib
Keyword(s):  
Greedy Algorithm ◽  
Random Graph ◽  
Random Graphs ◽  
Maximum Size ◽  
Packing Number ◽  
Edge Disjoint ◽  
Large Triangle

AbstractThe triangle packing number v(G) of a graph G is the maximum size of a set of edge-disjoint triangles in G. Tuza conjectured that in any graph G there exists a set of at most 2v(G) edges intersecting every triangle in G. We show that Tuza’s conjecture holds in the random graph G = G(n, m), when m ⩽ 0.2403n3/2 or m ⩾ 2.1243n3/2. This is done by analysing a greedy algorithm for finding large triangle packings in random graphs.

The (3, l)-Rainbow Edge-Index of Cartesian Product Graphs

2017 ◽  
Vol 17 (03n04) ◽  
pp. 1741009
Author(s):  
YUEFANG SUN
Keyword(s):  
Upper Bound ◽  
Cartesian Product ◽  
Edge Coloring ◽  
Index Formula ◽  
Colored Graph ◽  
Common Edge ◽  
Edge Connectivity ◽  
Edge Disjoint ◽  
Product Graphs ◽  
Cartesian Product Graphs

For a graph G and a vertex subset [Formula: see text] of at least two vertices, an S-tree is a subgraph T of G that is a tree with [Formula: see text]. Two S-trees are said to be edge-disjoint if they have no common edge. Let [Formula: see text] denote the maximum number of edge-disjoint S-trees in G. For an integer K with [Formula: see text], the generalized k-edge-connectivity is defined as [Formula: see text]. An S-tree in an edge-colored graph is rainbow if no two edges of it are assigned the same color. Let [Formula: see text] and l be integers with [Formula: see text], the [Formula: see text]-rainbow edge-index [Formula: see text] of G is the smallest number of colors needed in an edge-coloring of G such that for every set S of k vertices of G, there exist l edge-disjoint rainbow S-trees.In this paper, we study the [Formula: see text]-rainbow edge-index of Cartesian product graphs and get a sharp upper bound for [Formula: see text] , where G and H are connected graphs with orders at least three, and [Formula: see text] denotes the Cartesian product of G and H.

Corrigendum: Greed Works—Online Algorithms for Unrelated Machine Stochastic Scheduling

2021 ◽  
Author(s):  
Varun Gupta ◽  
Benjamin Moseley ◽  
Marc Uetz ◽  
Qiaomin Xie
Keyword(s):  
Greedy Algorithm ◽  
Online Algorithms ◽  
Upper Bound ◽  
Objective Function Value ◽  
Stochastic Scheduling ◽  
Performance Guarantee ◽  
Performance Bound ◽  
Unrelated Machines ◽  
Release Times ◽  
A Performance

This corrigendum fixes an incorrect claim in the paper Gupta et al. [Gupta V, Moseley B, Uetz M, Xie Q (2020) Greed works—online algorithms for unrelated machine stochastic scheduling. Math. Oper. Res. 45(2):497–516.], which led us to claim a performance guarantee of 6 for a greedy algorithm for deterministic online scheduling with release times on unrelated machines. The result is based on an upper bound on the increase of the objective function value when adding an additional job [Formula: see text] to a machine [Formula: see text] (Gupta et al., lemma 6). It was pointed out by Sven Jäger from Technische Universität Berlin that this upper bound may fail to hold. We here present a modified greedy algorithm and analysis, which leads to a performance guarantee of 7.216 instead. Correspondingly, also the claimed performance guarantee of [Formula: see text] in theorem 4 of Gupta et al. for the stochastic online problem has to be corrected. We obtain a performance bound [Formula: see text].

ON DETERMINISTIC FINITE STATE MACHINES IN RANDOM ENVIRONMENTS

2018 ◽  
Vol 33 (4) ◽  
pp. 528-563
Author(s):  
Joel Ratsaby
Keyword(s):  
Markov Chain ◽  
Random Environment ◽  
General Problem ◽  
Upper Bound ◽  
Decision Function ◽  
Random Environments ◽  
State Machines ◽  
Random Input ◽  
Output Sequence ◽  
Finite State

AbstractThe general problem under investigation is to understand how the complexity of a system which has been adapted to its random environment affects the level of randomness of its output (which is a function of its random input). In this paper, we consider a specific instance of this problem in which a deterministic finite-state decision system operates in a random environment that is modeled by a binary Markov chain. The system interacts with it by trying to match states of inactivity (represented by 0). Matching means that the system selects the (t + 1)th bit from the Markov chain whenever it predicts at time t that the environment will take a 0 value. The actual value at time t + 1 may be 0 or 1 thus the selected sequence of bits (which forms the system's output) may have both binary values. To try to predict well, the system's decision function is inferred based on a sample of the random environment.We are interested in assessing how non-random the output sequence may be. To do that, we apply the adapted system on a second random sample of the environment and derive an upper bound on the deviation between the average number of 1 bit in the output sequence and the probability of a 1. The bound shows that the complexity of the system has a direct effect on this deviation and hence on how non-random the output sequence may be. The bound takes the form of $O(\sqrt {(2^k/n} ))$ where 2k is the complexity of the system and n is the length of the second sample.

scholarly journals A new efficient RLF-like algorithm for the vertex coloring problem

2016 ◽  
Vol 26 (4) ◽  
pp. 441-456 ◽  
Author(s):  
Mourchid Adegbindin ◽  
Alain Hertz ◽  
Martine Bellaïche
Keyword(s):  
Greedy Algorithm ◽  
Chromatic Number ◽  
Upper Bound ◽  
Computational Experiments ◽  
Vertex Coloring ◽  
Greedy Heuristics ◽  
Selection Rules ◽  
Coloring Problem ◽  
Color Class ◽  
Vertex Coloring Problem

The Recursive Largest First (RLF) algorithm is one of the most popular greedy heuristics for the vertex coloring problem. It sequentially builds color classes on the basis of greedy choices. In particular, the first vertex placed in a color class C is one with a maximum number of uncolored neighbors, and the next vertices placed in C are chosen so that they have as many uncolored neighbors which cannot be placed in C. These greedy choices can have a significant impact on the performance of the algorithm, which explains why we propose alternative selection rules. Computational experiments on 63 difficult DIMACS instances show that the resulting new RLF-like algorithm, when compared with the standard RLF, allows to obtain a reduction of more than 50% of the gap between the number of colors used and the best known upper bound on the chromatic number. The new greedy algorithm even competes with basic metaheuristics for the vertex coloring problem.

Reliability Assessment of Some Regular Networks

2019 ◽  
Author(s):  
Shu-Li Zhao ◽  
Rong-Xia Hao ◽  
Sheng-Lung Peng
Keyword(s):  
Regular Graph ◽  
Upper Bound ◽  
Reliability Assessment ◽  
Alternating Group ◽  
Star Graph ◽  
Edge Disjoint ◽  
Regular Networks ◽  
Alternating Group Network

Abstract The generalized $k$-connectivity of a graph $G$ is a parameter that can measure the reliability of a network $G$ to connect any $k$ vertices in $G$, which is a generalization of traditional connectivity. Let $S\subseteq V(G)$ and $\kappa _{G}(S)$ denote the maximum number $r$ of edge-disjoint trees $T_{1}, T_{2}, \cdots , T_{r}$ in $G$ such that $V(T_{i})\bigcap V(T_{j})=S$ for any $i, j \in \{1, 2, \cdots , r\}$ and $i\neq j$. For an integer $k$ with $2\leq k\leq n$, the generalized $k$-connectivity of a graph $G$ is defined as $\kappa _{k}(G)= min\{\kappa _{G}(S)|S\subseteq V(G)$ and $|S|=k\}$. In this paper, we introduce a family of regular graph $G_{n}$ that can be constructed recursively and each vertex with exactly one outside neighbor. The generalized $3$-connectivity of the regular graph $G_{n}$ is studied, which attains a previously proven upper bound on $\kappa _{3}(G)$. As applications of the main result, the generalized $3$-connectivity of some important networks including some known results such as the alternating group network $AN_{n}$, the star graph $S_{n}$ and the pancake graphs $P_{n}$ can be obtained directly.

Influence of Magnetic Fields on Solar Hydrodynamics: Experimental Results

1977 ◽  
Vol 36 ◽  
pp. 143-180 ◽  
Author(s):  
J.O. Stenflo
Keyword(s):  
Solar Activity ◽  
Energy Balance ◽  
Magnetic Fields ◽  
Solar Atmosphere ◽  
General Problem ◽  
Solar Rotation ◽  
Active Regions ◽  
Physical Conditions ◽  
Dynamic Phenomena

It is well-known that solar activity is basically caused by the Interaction of magnetic fields with convection and solar rotation, resulting in a great variety of dynamic phenomena, like flares, surges, sunspots, prominences, etc. Many conferences have been devoted to solar activity, including the role of magnetic fields. Similar attention has not been paid to the role of magnetic fields for the overall dynamics and energy balance of the solar atmosphere, related to the general problem of chromospheric and coronal heating. To penetrate this problem we have to focus our attention more on the physical conditions in the ‘quiet’ regions than on the conspicuous phenomena in active regions.

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