A Lower Bound for the Complexity of Monotone Graph Properties

2013 ◽  
Vol 27 (1) ◽  
pp. 257-265 ◽  
Author(s):  
Robert Scheidweiler ◽  
Eberhard Triesch
Keyword(s):  
Lower Bound ◽  
Graph Properties ◽  
Monotone Graph

Review of Introduction To Property Testing by Oded Goldreich

2021 ◽  
Vol 51 (4) ◽  
pp. 6-10
Author(s):  
Sarvagya Upadhyay
Keyword(s):  
Decision Making ◽  
Lower Bound ◽  
Computer Science ◽  
Property Testing ◽  
Probabilistically Checkable Proofs ◽  
Graph Properties ◽  
Locally Testable Codes ◽  
Entire Object ◽  
Functions And Graphs ◽  
Distribution Testing

The area of property testing is concerned with designing methods to decide whether an input object possesses a certain property or not. Usually the problem is described as a promise problem: either the input object has the property or the input object is far from possessing the property. Here, the meaning of object being far from possessing the property is based on a specified and meaningful notion of distance. The main objective of property testing is accomplishing this decision making by developing a super efficient tester. A tester that reads through the entire object can easily determine whether the property is satisfied or not. However, one wishes the tester to probe the input at very few random locations and determine whether the property is satisfied. As such, randomness is a necessary ingredient for testing and having the tester erring on few instances is a necessary price to pay for designing highly efficient methodologies. Much of the literature on property testing has focused on two types of objects: functions and graphs. Naturally they form the major portion of the book: functions are discussed from Chapters 2 to 6 and graph properties are discussed from Chapters 8 to 10. The final three chapters focus on distribution testing, probabilistically checkable proofs (PCPs) and locally testable codes, and ramifications of property testing on other related topics in Computer Science and Statistics. A separate chapter is devoted to query lower bound techniques.

scholarly journals Shannon Capacity and the Categorical Product

10.37236/9113 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Gábor Simonyi
Keyword(s):  
Lower Bound ◽  
Recent Result ◽  
Chromatic Number ◽  
Upper Bound ◽  
Graph Invariant ◽  
Shannon Capacity ◽  
Graph Homomorphisms ◽  
Complementary Graph ◽  
Product Of Graphs ◽  
Monotone Graph

Shannon OR-capacity $C_{\rm OR}(G)$ of a graph $G$, that is the traditionally more often used Shannon AND-capacity of the complementary graph, is a homomorphism monotone graph parameter therefore $C_{\rm OR}(F\times G)\leqslant\min\{C_{\rm OR}(F),C_{\rm OR}(G)\}$ holds for every pair of graphs, where $F\times G$ is the categorical product of graphs $F$ and $G$. Here we initiate the study of the question when could we expect equality in this inequality. Using a strong recent result of Zuiddam, we show that if this "Hedetniemi-type" equality is not satisfied for some pair of graphs then the analogous equality is also not satisfied for this graph pair by some other graph invariant that has a much "nicer" behavior concerning some different graph operations. In particular, unlike Shannon OR-capacity or the chromatic number, this other invariant is both multiplicative under the OR-product and additive under the join operation, while it is also nondecreasing along graph homomorphisms. We also present a natural lower bound on $C_{\rm OR}(F\times G)$ and elaborate on the question of how to find graph pairs for which it is known to be strictly less than the upper bound $\min\{C_{\rm OR}(F),C_{\rm OR}(G)\}$. We present such graph pairs using the properties of Paley graphs.

An asymptotic bound for the complexity of monotone graph properties

COMBINATORICA ◽  
2010 ◽  
Vol 30 (6) ◽  
pp. 735-743 ◽  
Author(s):  
Torsten Korneffel ◽  
Eberhard Triesch
Keyword(s):  
Asymptotic Bound ◽  
Graph Properties ◽  
Monotone Graph
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