scholarly journals Laplacian Fractional Revival on Graphs

2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Ada Chan ◽  
Bobae Johnson ◽  
Mengzhen Liu ◽  
Malena Schmidt ◽  
Zhanghan Yin ◽  
...  
Keyword(s):  
Prime Number ◽  
Polynomial Time ◽  
Characterization Theorem ◽  
Quantum Walk ◽  
Laplacian Matrix ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Spectral Characterization ◽  
Cartesian Products

We develop the theory of fractional revival in the quantum walk on a graph using its Laplacian matrix as the Hamiltonian. We first give a spectral characterization of Laplacian fractional revival, which leads to a polynomial time algorithm to check this phenomenon and find the earliest time when it occurs. We then apply the characterization theorem to special families of graphs. In particular, we show that no tree admits Laplacian fractional revival except for the paths on two and three vertices, and the only graphs on a prime number of vertices that admit Laplacian fractional revival are double cones. Finally, we construct, through Cartesian products and joins, several infinite families of graphs that admit Laplacian fractional revival; some of these graphs exhibit polygamous fractional revival.

scholarly journals Liveness in broadcast networks

Computing ◽  
2021 ◽  
Author(s):  
Peter Chini ◽  
Roland Meyer ◽  
Prakash Saivasan
Keyword(s):  
Model Checking ◽  
Polynomial Time ◽  
Message Passing ◽  
Linear Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Final State ◽  
Broadcast Network ◽  
Broadcast Networks

AbstractWe study liveness and model checking problems for broadcast networks, a system model of identical clients communicating via message passing. The first problem that we consider is Liveness Verification. It asks whether there is a computation such that one clients visits a final state infinitely often. The complexity of the problem has been open. It was shown to be $$\texttt {P}$$ P -hard but in $$\texttt {EXPSPACE}$$ EXPSPACE . We close the gap by a polynomial-time algorithm. The latter relies on a characterization of live computations in terms of paths in a suitable graph, combined with a fixed-point iteration to efficiently check the existence of such paths. The second problem is Fair Liveness Verification. It asks for a computation where all participating clients visit a final state infinitely often. We adjust the algorithm to also solve fair liveness in polynomial time. Both problems can be instrumented to answer model checking questions for broadcast networks against linear time temporal logic specifications. The first problem in this context is Fair Model Checking. It demands that for all computations of a broadcast network, all participating clients satisfy the specification. We solve the problem via the Vardi–Wolper construction and a reduction to Liveness Verification. The second problem is Sparse Model Checking. It asks whether each computation has a participating client that satisfies the specification. We reduce the problem to Fair Liveness Verification.

scholarly journals Arc-Completion of 2-Colored Best Match Graphs to Binary-Explainable Best Match Graphs

Algorithms ◽  
2021 ◽  
Vol 14 (4) ◽  
pp. 110
Author(s):  
David Schaller ◽  
Manuela Geiß ◽  
Marc Hellmuth ◽  
Peter F. Stadler
Keyword(s):  
Phylogenetic Tree ◽  
Polynomial Time ◽  
Binary Tree ◽  
A Priori ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Minimum Cardinality ◽  
Mathematical Phylogenetics ◽  
Special Case

Best match graphs (BMGs) are vertex-colored digraphs that naturally arise in mathematical phylogenetics to formalize the notion of evolutionary closest genes w.r.t. an a priori unknown phylogenetic tree. BMGs are explained by unique least resolved trees. We prove that the property of a rooted, leaf-colored tree to be least resolved for some BMG is preserved by the contraction of inner edges. For the special case of two-colored BMGs, this leads to a characterization of the least resolved trees (LRTs) of binary-explainable trees and a simple, polynomial-time algorithm for the minimum cardinality completion of the arc set of a BMG to reach a BMG that can be explained by a binary tree.

Learning Importance of Preferences

10.29007/v68w ◽  
2018 ◽  
Author(s):  
Ying Zhu ◽  
Mirek Truszczynski
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Scoring Rules ◽  
Np Hard ◽  
Individual Preferences

We study the problem of learning the importance of preferences in preference profiles in two important cases: when individual preferences are aggregated by the ranked Pareto rule, and when they are aggregated by positional scoring rules. For the ranked Pareto rule, we provide a polynomial-time algorithm that finds a ranking of preferences such that the ranked profile correctly decides all the examples, whenever such a ranking exists. We also show that the problem to learn a ranking maximizing the number of correctly decided examples (also under the ranked Pareto rule) is NP-hard. We obtain similar results for the case of weighted profiles when positional scoring rules are used for aggregation.

scholarly journals A Polynomial Time Algorithm for Spatio-Temporal Security Games

2017 ◽  
Author(s):  
Soheil Behnezhad ◽  
Mahsa Derakhshan ◽  
MohammadTaghi Hajiaghayi ◽  
Aleksandrs Slivkins
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Security Games ◽  
Spatio Temporal

A Polynomial-Time Algorithm for the Knapsack Problem with Two Variables

1976 ◽  
Vol 23 (1) ◽  
pp. 147-154 ◽  
Author(s):  
D. S. Hirschberg ◽  
C. K. Wong
Keyword(s):  
Polynomial Time ◽  
Knapsack Problem ◽  
Polynomial Time Algorithm ◽  
Time Algorithm

scholarly journals Polynomial Time Algorithm for Min-Ranks of Graphs with Simple Tree Structures

Algorithmica ◽  
2013 ◽  
Vol 71 (1) ◽  
pp. 152-180 ◽  
Author(s):  
Son Hoang Dau ◽  
Yeow Meng Chee
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Tree Structures ◽  
Simple Tree
Sign in / Sign up

Export Citation Format

Share Document