Graph pebbling is a problem in which pebbles are distributed across the vertices of a graph and moved according to a specific rule: two pebbles are removed from a vertex to place one on an adjacent vertex. The goal is to determine the minimum number of pebbles required to ensure that any target vertex can be reached, known as the pebbling number. Computing the pebbling number lies beyond NP in the polynomial hierarchy, leading to bounding methods. One of the most prominent techniques for upper bounds is the Weight Function Lemma (WFL), which relies on costly integer linear optimization. To mitigate this cost, an alternative approach is to consider the dual formulation of the problem, which allows solutions to be constructed by hand through the selection of strategies given by subtrees with associated weight functions. To improve the bounds, the weights should be distributed as uniformly as possible among the vertices, balancing their individual contribution. However, despite its simplicity, this approach lacks a formal framework. To fill this gap, we introduce a novel heuristic method that refines the selection of balanced strategies. The method is motivated by our theoretical analysis of the limitations of the dual approach, in which we prove lower bounds on the best bounds achievable. Our theoretical analysis shows that the bottleneck lies in the farthest vertices from the target, forcing surplus weight onto the closer neighborhoods. To minimize surplus weight beyond the theoretical minimum, our proposed heuristic prioritizes weight assignment to the farthest vertices, building the subtrees starting from the shortest paths to them and then filling in the weights for the remaining vertices. Applying our heuristic to Flower snarks and Blanuša snarks, we improve the best-known upper bounds, demonstrating the effectiveness of a structured strategy selection when using WFL.

Abstract Studies on smart grid and quantum mechanics have potential to yield several benefits to society, with the former resulting in economic and environmental benefits and the latter providing perfect security and privacy at affordable costs. Recently, security and privacy have become important issues in electrical power grids. In this paper, we describe two quantum privacy‐enhancing protocols: one of them requires that the parties initially share a certain amount of quantum entangled states, while the other uses only quantum key distribution methods without sharing quantum entangled states. The two proposed protocols are resistant to attacks from quantum computers. This paper also describes some recent advances of classical and quantum privacy‐enhancing technologies. Copyright © 2014 John Wiley & Sons, Ltd.

The “abstract search algorithm” is a well known quantum method to find a marked vertex in a graph. It has been applied with success to searching algorithms for the hypercube and the two-dimensional grid. In this work we provide an example for which that method fails to provide the best algorithm in terms of time complexity. We analyze search algorithms in degree-3 hierarchical networks using quantum walks driven by non-groverian coins. Our conclusions are based on numerical simulations, but the hierarchical structures of the graphs seems to allow analytical results.

The spatial search problem consists of minimising the number of steps required to find a given site in a network under the restriction that only oracle queries or translations to neighbouring sites are allowed. We propose a quantum algorithm for the spatial search problem on a honeycomb lattice withNsites and torus-like boundary conditions. The search algorithm is based on a modified quantum walk on an hexagonal lattice and the general framework proposed by Ambainis, Kempe and Rivosh (Ambainiset al. 2005) is employed to show that the time complexity of this quantum search algorithm is $O(\sqrt{N \log N})$ .