scholarly journals Regarding Two Conjectures on Clique and Biclique Partitions

10.37236/9564 ◽  
2021 ◽  
Vol 28 (4) ◽  
Author(s):  
Dhruv Rohatgi ◽  
John C. Urschel ◽  
Jake Wellens
Keyword(s):  
Upper Bounds ◽  
The Other ◽  
Lower And Upper Bounds ◽  
Minimum Number ◽  
Other Regarding

For a graph $G$, let $cp(G)$ denote the minimum number of cliques of $G$ needed to cover the edges of $G$ exactly once. Similarly, let $bp_k(G)$ denote the minimum number of bicliques (i.e. complete bipartite subgraphs of $G$) needed to cover each edge of $G$ exactly $k$ times. We consider two conjectures – one regarding the maximum possible value of $cp(G) + cp(\overline{G})$ (due to de Caen, Erdős, Pullman and Wormald) and the other regarding $bp_k(K_n)$ (due to de Caen, Gregory and Pritikin). We disprove the first, obtaining improved lower and upper bounds on $\max_G cp(G) + cp(\overline{G})$, and we prove an asymptotic version of the second, showing that $bp_k(K_n) = (1+o(1))n$.

Conflict-Free Colourings of Uniform Hypergraphs With Few Edges

2012 ◽  
Vol 21 (4) ◽  
pp. 611-622 ◽  
Author(s):  
A. KOSTOCHKA ◽  
M. KUMBHAT ◽  
T. ŁUCZAK
Keyword(s):  
Chromatic Number ◽  
Upper Bound ◽  
Upper Bounds ◽  
Uniform Hypergraph ◽  
Lower And Upper Bounds ◽  
Uniform Hypergraphs ◽  
Minimum Number

A colouring of the vertices of a hypergraph is called conflict-free if each edge e of contains a vertex whose colour does not repeat in e. The smallest number of colours required for such a colouring is called the conflict-free chromatic number of , and is denoted by χCF(). Pach and Tardos proved that for an (2r − 1)-uniform hypergraph with m edges, χCF() is at most of the order of rm1/r log m, for fixed r and large m. They also raised the question whether a similar upper bound holds for r-uniform hypergraphs. In this paper we show that this is not necessarily the case. Furthermore, we provide lower and upper bounds on the minimum number of edges of an r-uniform simple hypergraph that is not conflict-free k-colourable.

SETS OF K-INDEPENDENT STRINGS

2010 ◽  
Vol 21 (03) ◽  
pp. 321-327 ◽  
Author(s):  
YEN-WU TI ◽  
CHING-LUEH CHANG ◽  
YUH-DAUH LYUU ◽  
ALEXANDER SHEN
Keyword(s):  
Information Theory ◽  
Kolmogorov Complexity ◽  
Upper Bounds ◽  
The Other ◽  
Lower And Upper Bounds ◽  
Algorithmic Information Theory ◽  
Algorithmic Information ◽  
Conditional Complexity

A bit string is random (in the sense of algorithmic information theory) if it is incompressible, i.e., its Kolmogorov complexity is close to its length. Two random strings are independent if knowing one of them does not simplify the description of the other, i.e., the conditional complexity of each string (using the other as a condition) is close to its length. We may define independence of a k-tuple of strings in the same way. In this paper we address the following question: what is that maximal cardinality of a set of n-bit strings if any k elements of this set are independent (up to a certain constant)? Lower and upper bounds that match each other (with logarithmic precision) are provided.

scholarly journals Obtaining a Proportional Allocation by Deleting Items

Algorithmica ◽  
2021 ◽  
Author(s):  
Britta Dorn ◽  
Ronald de Haan ◽  
Ildikó Schlotter
Keyword(s):  
Control Problem ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Upper Bounds ◽  
Fair Allocation ◽  
Indivisible Goods ◽  
Lower And Upper Bounds ◽  
Proportional Allocation ◽  
Minimum Number

AbstractWe consider the following control problem on fair allocation of indivisible goods. Given a set I of items and a set of agents, each having strict linear preferences over the items, we ask for a minimum subset of the items whose deletion guarantees the existence of a proportional allocation in the remaining instance; we call this problem Proportionality by Item Deletion (PID). Our main result is a polynomial-time algorithm that solves PID for three agents. By contrast, we prove that PID is computationally intractable when the number of agents is unbounded, even if the number k of item deletions allowed is small—we show that the problem is $${\mathsf {W}}[3]$$ W [ 3 ] -hard with respect to the parameter k. Additionally, we provide some tight lower and upper bounds on the complexity of PID when regarded as a function of |I| and k. Considering the possibilities for approximation, we prove a strong inapproximability result for PID. Finally, we also study a variant of the problem where we are given an allocation $$\pi $$ π in advance as part of the input, and our aim is to delete a minimum number of items such that $$\pi $$ π is proportional in the remainder; this variant turns out to be $${{\mathsf {N}}}{{\mathsf {P}}}$$ N P -hard for six agents, but polynomial-time solvable for two agents, and we show that it is $$\mathsf {W[2]}$$ W [ 2 ] -hard when parameterized by the number k of

scholarly journals On the Information Ratio of Graphs with Many Leaves

2019 ◽  
Vol 73 (1) ◽  
pp. 97-108
Author(s):  
Máté Gyarmati ◽  
Péter Ligeti
Keyword(s):  
Lower Bound ◽  
Secret Sharing ◽  
Complete Graph ◽  
Upper Bounds ◽  
Entropy Method ◽  
The Other ◽  
Secret Sharing Schemes ◽  
Lower And Upper Bounds ◽  
Information Ratio ◽  
Recursive Constructions

Abstract We investigate the information ratio of graph-based secret sharing schemes. This ratio characterizes the efficiency of a scheme measured by the amount of information the participants must remember for each bits in the secret. We examine the information ratio of several systems based on graphs with many leaves, by proving non-trivial lower and upper bounds for the ratio. On one hand, we apply the so-called entropy method for proving that the lower bound for the information ratio of n-sunlet graphs composed of a 1-factor between the vertices of a cycle Cn and n independent vertices is 2. On the other hand, some symmetric and recursive constructions are given that yield the upper bounds. In particular, we show that the information ratio of every graph composed of a 1-factor between a complete graph Kn and at most 4 independent vertices is smaller than 2.

scholarly journals Drawing a Graph in a Hypercube

10.37236/1099 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
David R. Wood
Keyword(s):  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Line Segments ◽  
Sidon Sets ◽  
Minimum Number

A $d$-dimensional hypercube drawing of a graph represents the vertices by distinct points in $\{0,1\}^d$, such that the line-segments representing the edges do not cross. We study lower and upper bounds on the minimum number of dimensions in hypercube drawing of a given graph. This parameter turns out to be related to Sidon sets and antimagic injections.

scholarly journals On the geodetic hull number for complementary prisms II

2020 ◽  
Author(s):  
Diane Castonguay ◽  
Erika Morais Martins Coelho ◽  
Hebert Coelho ◽  
Julliano Nascimento
Keyword(s):  
Convex Hull ◽  
Shortest Path ◽  
Disjoint Union ◽  
Perfect Matching ◽  
Convex Set ◽  
Upper Bounds ◽  
The Other ◽  
Lower And Upper Bounds ◽  
Split Graph ◽  
Complementary Prism

In the geodetic convexity, a set of vertices $S$ of a graph $G$ is \textit{convex} if all vertices belonging to any shortest path between two vertices of $S$ lie in $S$. The \textit{convex hull} $H(S)$ of $S$ is the smallest convex set containing $S$. If $H(S) = V(G)$, then $S$ is a \textit{hull set}. The cardinality $h(G)$ of a minimum hull set of $G$ is the \textit{hull number} of $G$. The \textit{complementary prism} $G\overline{G}$ of a graph $G$ arises from the disjoint union of the graph $G$ and $\overline{G}$ by adding the edges of a perfect matching between the corresponding vertices of $G$ and $\overline{G}$. A graph $G$ is \textit{autoconnected} if both $G$ and $\overline{G}$ are connected. Motivated by previous work, we study the hull number for complementary prisms of autoconnected graphs. When $G$ is a split graph, we present lower and upper bounds showing that the hull number is unlimited. In the other case, when $G$ is a non-split graph, it is limited by $3$.

Are You Sure That This Happened? Assessing the Factuality Degree of Events in Text

2012 ◽  
Vol 38 (2) ◽  
pp. 261-299 ◽  
Author(s):  
Roser Saurí ◽  
James Pustejovsky
Keyword(s):  
Computational Model ◽  
Upper Bounds ◽  
Epistemic Modality ◽  
The Other ◽  
Proof Of Concept ◽  
Lower And Upper Bounds ◽  
Information Level ◽  
Two Measures ◽  
The One ◽  
Event Factuality

Identifying the veracity, or factuality, of event mentions in text is fundamental for reasoning about eventualities in discourse. Inferences derived from events judged as not having happened, or as being only possible, are different from those derived from events evaluated as factual. Event factuality involves two separate levels of information. On the one hand, it deals with polarity, which distinguishes between positive and negative instantiations of events. On the other, it has to do with degrees of certainty (e.g., possible, probable), an information level generally subsumed under the category of epistemic modality. This article aims at contributing to a better understanding of how event factuality is articulated in natural language. For that purpose, we put forward a linguistic-oriented computational model which has at its core an algorithm articulating the effect of factuality relations across levels of syntactic embedding. As a proof of concept, this model has been implemented in De Facto, a factuality profiler for eventualities mentioned in text, and tested against a corpus built specifically for the task, yielding an F1 of 0.70 (macro-averaging) and 0.80 (micro-averaging). These two measures mutually compensate for an over-emphasis present in the other (either on the lesser or greater populated categories), and can therefore be interpreted as the lower and upper bounds of the De Facto's performance.

scholarly journals BOUNDING RIGHT-ARM ROTATION DISTANCES

2007 ◽  
Vol 17 (02) ◽  
pp. 369-399 ◽  
Author(s):  
SEAN CLEARY ◽  
JENNIFER TABACK
Keyword(s):  
Word Length ◽  
Upper Bounds ◽  
Fixed Number ◽  
The Other ◽  
Binary Trees ◽  
Distance Measures ◽  
Generating Sets ◽  
Minimum Number ◽  
Thompson’S Group ◽  
The Difference

Rotation distance measures the difference in shape between binary trees of the same size by counting the minimum number of rotations needed to transform one tree to the other. We describe several types of rotation distance where restrictions are put on the locations where rotations are permitted, and provide upper bounds on distances between trees with a fixed number of nodes with respect to several families of these restrictions. These bounds are sharp in a certain asymptotic sense and are obtained by relating each restricted rotation distance to the word length of elements of Thompson's group F with respect to different generating sets, including both finite and infinite generating sets.

Algorithms for four-machine flowshop scheduling problem with uncertain processing times to minimize makespan

2020 ◽  
Vol 54 (2) ◽  
pp. 529-553 ◽  
Author(s):  
Muberra Allahverdi ◽  
Ali Allahverdi
Keyword(s):  
Upper Bounds ◽  
The Other ◽  
Computational Experiments ◽  
Scheduling Problem ◽  
Flowshop Scheduling ◽  
Lower And Upper Bounds ◽  
Significance Level ◽  
Dominance Relations ◽  
Processing Times ◽  
Flowshop Scheduling Problem

We consider the four-machine flowshop scheduling problem to minimize makespan where processing times are uncertain. The processing times are within some intervals, where the only available information is the lower and upper bounds of job processing times. Some dominance relations are developed, and twelve algorithms are proposed. The proposed algorithms first convert the four-machine problem into two stages, then, use the well-known Johnson’s algorithm, known to yield the optimal solution for the two-stage problem. The algorithms also use the developed dominance relations. The proposed algorithms are extensively evaluated through randomly generated data for different numbers of jobs and different gaps between the lower and upper bounds of processing times. Computational experiments indicate that the proposed algorithms perform well. Moreover, the computational experiments reveal that one of the proposed algorithms, Algorithm A7, performs significantly better than the other eleven algorithms for all possible combinations of the number of jobs and the gaps between the lower and upper bounds. More specifically, error percentages of the other eleven algorithms range from 2.3 to 27.7 times that of Algorithm A7. The results have been confirmed by constructing 99% confidence intervals and tests of hypotheses using a significance level of 0.01.

Sign in / Sign up

Export Citation Format

Share Document