A QPTAS for ɛ-Envy-Free Profit-Maximizing Pricing on Line Graphs

2021 ◽  
Vol 9 (3) ◽  
pp. 1-31
Author(s):  
Khaled Elbassioni
Keyword(s):  
Approximation Scheme ◽  
Line Graph ◽  
Hierarchical Decomposition ◽  
Running Time ◽  
The Best Approximation ◽  
Maximum Price ◽  
Unlimited Supply ◽  
On Line ◽  
Doubling Metrics ◽  
Limited Supply

We consider the problem of pricing edges of a line graph so as to maximize the profit made from selling intervals to single-minded customers. An instance is given by a set E of n edges with a limited supply for each edge, and a set of m clients, where each client specifies one interval of E she is interested in and a budget B j which is the maximum price she is willing to pay for that interval. An envy-free pricing is one in which every customer is allocated an (possibly empty) interval maximizing her utility. Grandoni and Rothvoss (SIAM J. Comput. 2016) proposed a polynomial-time approximation scheme ( PTAS ) for the unlimited supply case with running time ( nm ) O ((1/ɛ) 1/ɛ ) , which was extended to the limited supply case by Grandoni and Wiese (ESA 2019). By utilizing the known hierarchical decomposition of doubling metrics , we give a PTAS with running time ( nm ) O (1/ ɛ 2 ) for the unlimited supply case. We then consider the limited supply case, and the notion of ɛ-envy-free pricing in which a customer gets an allocation maximizing her utility within an additive error of ɛ. For this case, we develop an approximation scheme with running time ( nm ) O (log 5/2 max e H e /ɛ 3 ) , where H e = B max ( e )/ B min ( e ) is the maximum ratio of the budgets of any two customers demanding edge e . This yields a PTAS in the uniform budget case, and a quasi-PTAS for the general case. The best approximation known, in both cases, for the exact envy-free pricing version is O (log c max ), where c max is the maximum item supply. Our method is based on the known hierarchical decomposition of doubling metrics, and can be applied to other problems, such as the maximum feasible subsystem problem with interval matrices.

Uncovering local community structure on line graph through degree centrality and expansion

2021 ◽  
pp. 2150120
Author(s):  
Guishen Wang ◽  
Kaitai Wang ◽  
Hongmei Wang ◽  
Huimin Lu ◽  
Xiaotang Zhou ◽  
...  
Keyword(s):  
Community Structure ◽  
Community Detection ◽  
Local Community ◽  
Line Graph ◽  
Detection Methods ◽  
Degree Centrality ◽  
Original Graph ◽  
Normalized Mutual Information ◽  
Local Community Detection ◽  
On Line

Local community detection algorithms are an important type of overlapping community detection methods. Local community detection methods identify local community structure through searching seeds and expansion process. In this paper, we propose a novel local community detection method on line graph through degree centrality and expansion (LCDDCE). We firstly employ line graph model to transfer edges into nodes of a new graph. Secondly, we evaluate edges relationship through a novel node similarity method on line graph. Thirdly, we introduce local community detection framework to identify local node community structure of line graph, combined with degree centrality and PageRank algorithm. Finally, we transfer them back into original graph. The experimental results on three classical benchmarks show that our LCDDCE method achieves a higher performance on normalized mutual information metric with other typical methods.

On Spanning and Dominating Circuits in Graphs

1977 ◽  
Vol 20 (2) ◽  
pp. 215-220 ◽  
Author(s):  
L. Lesniak-Foster ◽  
James E. Williamson
Keyword(s):  
Line Graph ◽  
Dominating Set ◽  
Line Graphs ◽  
Sufficient Condition ◽  
Vertex Set ◽  
On Line ◽  
One To One

AbstractA set E of edges of a graph G is said to be a dominating set of edges if every edge of G either belongs to E or is adjacent to an edge of E. If the subgraph 〈E〉 induced by E is a trail T, then T is called a dominating trail of G. Dominating circuits are defined analogously. A sufficient condition is given for a graph to possess a spanning (and thus dominating) circuit and a sufficient condition is given for a graph to possess a spanning (and thus dominating) trail between each pair of distinct vertices. The line graph L(G) of a graph G is defined to be that graph whose vertex set can be put in one-to-one correspondence with the edge set of G in such a way that two vertices of L(G) are adjacent if and only if the corresponding edges of G are adjacent. The existence of dominating trails and circuits is employed to present results on line graphs and second iterated line graphs, respectively.

Lower Bounds for On-line Graph Problems with Application to On-line Circuit and Optical Routing

2006 ◽  
Vol 36 (2) ◽  
pp. 354-393 ◽  
Author(s):  
Yair Bartal ◽  
Amos Fiat ◽  
Stefano Leonardi
Keyword(s):  
Lower Bounds ◽  
Line Graph ◽  
Graph Problems ◽  
On Line ◽  
Optical Routing

On-line graph coloring of $${\mathbb{P}_5}$$ -free graphs

2007 ◽  
Vol 45 (2) ◽  
pp. 79-91 ◽  
Author(s):  
Iwona Cieślik
Keyword(s):  
Graph Coloring ◽  
Line Graph ◽  
On Line

Nonexistence of a pair of arc disjoint directed Hamilton cycles on line digraphs of 2-diregular digraphs

2015 ◽  
Vol 07 (03) ◽  
pp. 1550034
Author(s):  
T. Govindan ◽  
A. Muthusamy
Keyword(s):  
Line Graph ◽  
Hamilton Cycle ◽  
Hamilton Cycles ◽  
On Line

Bermond conjectured that if G is Hamilton cycle decomposable, then L(G), the line graph of G is Hamilton cycle decomposable. In this paper, we prove that, for any k > 5, there exists a directed Hamilton cycle decomposable 2-diregular digraph D of order 2k such that L(D) is not directed Hamilton cycle decomposable.

scholarly journals Lower Bounds for On-line Graph Colorings

2014 ◽  
pp. 507-515 ◽  
Author(s):  
Grzegorz Gutowski ◽  
Jakub Kozik ◽  
Piotr Micek ◽  
Xuding Zhu
Keyword(s):  
Lower Bounds ◽  
Line Graph ◽  
Graph Colorings ◽  
On Line

scholarly journals Learning Within an Instance for Designing High-Revenue Combinatorial Auctions

2021 ◽  
Author(s):  
Maria-Florina Balcan ◽  
Siddharth Prasad ◽  
Tuomas Sandholm
Keyword(s):  
Random Sampling ◽  
Combinatorial Auctions ◽  
Complexity Measure ◽  
Revenue Maximization ◽  
Learning Groups ◽  
Winner Determination ◽  
Unlimited Supply ◽  
Intrinsic Complexity ◽  
Limited Supply ◽  
New Framework

We develop a new framework for designing truthful, high-revenue (combinatorial) auctions for limited supply. Our mechanism learns within an instance. It generalizes and improves over previously-studied random-sampling mechanisms. It first samples a participatory group of bidders, then samples several learning groups of bidders from the remaining pool of bidders, learns a high-revenue auction from the learning groups, and finally runs that auction on the participatory group. Previous work on random-sampling mechanisms focused primarily on unlimited supply. Limited supply poses additional significant technical challenges, since allocations of items to bidders must be feasible. We prove guarantees on the performance of our mechanism based on a market-shrinkage term and a new complexity measure we coin partition discrepancy. Partition discrepancy simultaneously measures the intrinsic complexity of the mechanism class and the uniformity of the set of bidders. We then introduce new auction classes that can be parameterized in a way that does not depend on the number of bidders participating, and prove strong guarantees for these classes. We show how our mechanism can be implemented efficiently by leveraging practically-efficient routines for solving winner determination. Finally, we show how to use structural revenue maximization to decide what auction class to use with our framework when there is a constraint on the number of learning groups.

Sign in / Sign up

Export Citation Format

Share Document