14,329 research outputs found
A proof of the four-colour theorem
The four-colour problem remained unsolved for more than a hundred years has played a role of the utmost importance in the development of graph theory. The four-colour theorem was confirmed in 1976, which is not completely satisfied due to: i) part of the proof using computers cannot be verified by hand; ii) even the part, supposedly hand-checkable, is extraordinarily complicated and tedious, and as far as we know, no one has entirely verified it. Seeking a hand-checkable proof of the four-colour theorem is one of world-interested problems, which is addressed in this paper. A necessary and sufficient condition for n-colour theorem in a space is: there exists a largest n-complete graph base in the same space. Examples are given to illustrate applications
Influence Maximization Meets Efficiency and Effectiveness: A Hop-Based Approach
Influence Maximization is an extensively-studied problem that targets at
selecting a set of initial seed nodes in the Online Social Networks (OSNs) to
spread the influence as widely as possible. However, it remains an open
challenge to design fast and accurate algorithms to find solutions in
large-scale OSNs. Prior Monte-Carlo-simulation-based methods are slow and not
scalable, while other heuristic algorithms do not have any theoretical
guarantee and they have been shown to produce poor solutions for quite some
cases. In this paper, we propose hop-based algorithms that can easily scale to
millions of nodes and billions of edges. Unlike previous heuristics, our
proposed hop-based approaches can provide certain theoretical guarantees.
Experimental evaluations with real OSN datasets demonstrate the efficiency and
effectiveness of our algorithms.Comment: Extended version of the conference paper at ASONAM 2017, 11 page
Towards Profit Maximization for Online Social Network Providers
Online Social Networks (OSNs) attract billions of users to share information
and communicate where viral marketing has emerged as a new way to promote the
sales of products. An OSN provider is often hired by an advertiser to conduct
viral marketing campaigns. The OSN provider generates revenue from the
commission paid by the advertiser which is determined by the spread of its
product information. Meanwhile, to propagate influence, the activities
performed by users such as viewing video ads normally induce diffusion cost to
the OSN provider. In this paper, we aim to find a seed set to optimize a new
profit metric that combines the benefit of influence spread with the cost of
influence propagation for the OSN provider. Under many diffusion models, our
profit metric is the difference between two submodular functions which is
challenging to optimize as it is neither submodular nor monotone. We design a
general two-phase framework to select seeds for profit maximization and develop
several bounds to measure the quality of the seed set constructed. Experimental
results with real OSN datasets show that our approach can achieve high
approximation guarantees and significantly outperform the baseline algorithms,
including state-of-the-art influence maximization algorithms.Comment: INFOCOM 2018 (Full version), 12 page
Veech surfaces and simple closed curves
We study the SL(2,R)-infimal lengths of simple closed curves on
half-translation surfaces. Our main result is a characterization of Veech
surfaces in terms of these lengths. We also revisit the "no small virtual
triangles" theorem of Smillie and Weiss and establish the following dichotomy:
the virtual triangle area spectrum of a half-translation surface either has a
gap above zero or is dense in a neighborhood of zero. These results make use of
the auxiliary polygon associated to a curve on a half-translation surface, as
introduced by Tang and Webb.Comment: 12 pages. v2: added proof of continuity of infimal length functions
on quadratic differential space; 16 pages, one figure; to appear in Israel J.
Mat
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