We consider a variant of the game of Cops and Robbers, called Lazy Cops and Robbers, where at most one cop can move in any round. The lazy cop number is the analogue of the usual cop number for this game. Lazy Cops and Robbers was recently introduced by Offner and Ojakian, who provided asymptotic upper and lower bounds on the analogue of the cop number of the hypercube. By investigating expansi...

AbstrAct Organizations have the potential to achieve advantage through communities of practice (COPs) initiatives. However, establishing and sustaining COPs is a challenging task. Facilitation is needed to help COPs BLOCKINovercome BLOCKINdifficulties BLOCKINthroughout BLOCKINtheir BLOCKINlife BLOCKINstages. BLOCKINFacilitators BLOCKINtake BLOCKINon BLOCKINleadership BLOCKINroles BLOCKINwithin ...

A dynamic covalent approach towards rigid aryleneethynylene covalent organic polyhedrons (COPs) was explored. Our study on the relationship of the COP structures and the geometry of their building blocks reveals that the topology of aryleneethynylene COPs strongly depends on the size of the building blocks. A tetramer (D2h symmetric), dimer, or interlocked complex can be formed from monomers wi...

2 Background 2 2.1 NP-hard Combinatorial (Optimization) Problems . . . . . . . . . . . . . . . 2 2.2 Algorithms for Attacking COPs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2.1 Exact Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2.2 Non-Exact Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2.3 Comparison of the Tw...

This article is an introduction to Cluster Optimized Proximity Scaling (COPS) aimed at practitioners, as well a tutorial on the usage of corresponding R package cops. COPS variant multidimensional scaling (MDS) that aims providing clustered configuration while still representing multivariate dissimilarities faithfully. It subsumes most popular MDS versions special cases. We illustrate ideas, us...

Solving constrained optimization problems (COPs) has been gathering attention from many researchers. In this paper, we defined the best fitness value among feasible solutions in current population as gbest. Then, we converted the original COPs to multi-objective optimization problems (MOPs) with one constraint. The constraint set the function value f(x) should be less than or equal to gbest; th...

In recent years, the constrained optimization problems have become a hot topic among the interest of scholars. In this paper, a new improved artificial immune algorithm is proposed and then used for solving constrained optimizations problems (COPs). This algorithm will treat these COPs as multi-objective optimization problems, and it is based on the concept of Pareto optimization to solve COPs....

Vertex pursuit games are widely studied by both graph theorists and computer scientists. Cops and Robbers is a vertex pursuit game played on a graph, where some set of agents (or Cops) attempts to capture a robber. The cop number is the minimum number of cops needed to win. While cop number of a graph has been studied for over 25 years, it is not well understood, and has few connections with ex...

In the classical game of Cops and Robbers on graphs, the capture time is defined by the least number of moves needed to catch all robbers with the smallest amount of cops that suffice. While the case of one cop and one robber is well understood, it is an open question how long it takes for multiple cops to catch multiple robbers. We show that capturing ` ∈ O (n) robbers can take Ω (` · n) time,...