In the recently introduced model of fair partitioning of friends, there is a set of agents located on the vertices of an underlying graph that indicates the friendships between the agents. The task is to partition the graph into k balanced-sized groups, keeping in mind that the value of an agent for a group is equal to the number of edges they have in that group. The goal is to construct partitions that are "fair", i.e., no agent would like to replace an agent in a different group. We generalize the standard model by considering utilities for the agents that are beyond binary and additive. Having this as our foundation, our contribution is threefold: (a) we adapt several fairness notions that have been developed in the fair division literature to our setting; (b) we give several existence guarantees supported by polynomial-time algorithms; (c) we initiate the study of the computational (and parameterized) complexity of the model and provide an almost complete landscape of the (in)tractability frontier for our fairness concepts.

A team of identical and oblivious ant-like agents – a(ge)nts – leaving pheromone traces, are programmed to jointly patrol an area modeled as a graph. They perform this task using simple local interactions, while also achieving the important byproduct of partitioning the graph into roughly equal-sized disjoint sub-graphs. Each a(ge)nt begins to operate at an arbitrary initial location, and throughout its work does not acquire any information on either the shape or size of the graph, or the number or whereabouts of other a(ge)nts. Graph partitioning occurs spontaneously, as each of the a(ge)nts patrols and expands its own pheromone-marked sub-graph, or region. This graph partitioning algorithm is inspired by molecules hitting the borders of air filled elastic balloons: an a(ge)nt that hits a border edge from the interior of its region more frequently than an external a(ge)nt hits the same edge from an adjacent vertex in the neighboring region, may conquer that adjacent vertex, expanding its region at the expense of the neighbor. Since the rule of patrolling a region ensures that each vertex is visited with a frequency inversely proportional to the size of the region, in terms of vertex count, a smaller region will effectively exert higher “pressure” at its borders, and conquer adjacent vertices from a larger region, thereby increasing the smaller region and shrinking the larger. The algorithm, therefore, tends to equalize the sizes of the regions patrolled, resembling a set of perfectly elastic physical balloons, confined to a closed volume and filled with an equal amount of air. The pheromone based local interactions of agents eventually cause the system to evolve into a partition that is close to balanced rather quickly, and if the graph and the number of a(ge)nts remain unchanged, it is guaranteed that the system settles into a stable and balanced partition.

A convex fair partition of a convex polygonal region is defined as a partition on which all regions are convex and have equal area and equal perimeter. The existence of such a partition for any number of regions remains an open question. In this paper, we address this issue by developing an algorithm to find such a convex fair partition without restrictions on the number of regions. Our approach utilizes the normal flow algorithm (a generalization of Newton’s method) to find a zero for the excess areas and perimeters of the convex hulls of the regions. The initial partition is generated by applying Lloyd’s algorithm to a randomly selected set of points within the polygon, after appropriate scaling. We performed extensive experimentation, and our algorithm can find a convex fair partition for 100% of the tested problem. Our findings support the conjecture that a convex fair partition always exists.

A convex fair partition of a convex polygonal region is defined as a partition on which all regions are convex and have equal area and equal perimeter. In this article we describe an algorithm that finds such fair partition.•The Fair Partitions method finds a fair partition for any given convex polygon and any given number of regions.•Our method relies on two well-known methods: Lloyd's algorithm and the Normal Flow Algorithm.•The method proposed in this article can be used in various contexts and many real-world applications.

This paper is part of an ongoing endeavor to bring the theory of fair division closer to practice by handling requirements from real-life applications. We focus on two requirements originating from the division of land estates: (1) each agent should receive a plot of a usable geometric shape, and (2) plots of different agents must be physically separated. With these requirements, the classic fairness notion of proportionality is impractical, since it may be impossible to attain any multiplicative approximation of it. In contrast, the ordinal maximin share approximation, introduced by Budish in 2011, provides meaningful fairness guarantees. We prove upper and lower bounds on achievable maximin share guarantees when the usable shapes are squares, fat rectangles, or arbitrary axis-aligned rectangles, and explore the algorithmic and query complexity of finding fair partitions in this setting. Our work makes use of tools and concepts from computational geometry such as independent sets of rectangles and guillotine partitions.

We model the societal task of redistricting political districts as a partitioning problem: Given a set of n points in the plane, each belonging to one of two parties, and a parameter k, our goal is to compute a partition P of the plane into regions so that each region contains roughly s = n/k points. P should satisfy a notion of "local" fairness, which is related to the notion of core, a well-studied concept in cooperative game theory. A region is associated with the majority party in that region, and a point is unhappy in P if it belongs to the minority party. A group D of roughly s contiguous points is called a deviating group with respect to P if majority of points in D are unhappy in P. The partition P is locally fair if there is no deviating group with respect to P. This paper focuses on a restricted case when points lie in 1D. The problem is non-trivial even in this case. We consider both adversarial and "beyond worst-case" settings for this problem. For the former, we characterize the input parameters for which a locally fair partition always exists; we also show that a locally fair partition may not exist for certain parameters. We then consider input models where there are "runs" of red and blue points. For such clustered inputs, we show that a locally fair partition may not exist for certain values of s, but an approximate locally fair partition exists if we allow some regions to have smaller sizes. We finally present a polynomial-time algorithm for computing a locally fair partition if one exists.

The increase of mobile data users has created traffic congestion in current cellular networks. Due to this, mobile network providers have been facing difficulty in delivering the best services for customers. Since, detecting community in mobile social network is a valuable technique to leverage the downlink traffic congestion by enhancing local communications within the community, it attracts the attention of many researchers. Therefore, developing an algorithm, which detects community, plays a key role in mobile social network. In this paper, first, we proposed external density metrics to detect mobile social network. External density is defined as the ratio of outgoing links to total links of the community. Second, method to find the best group for common node is proposed. Therefore, an external density algorithm, makes a fair partition by grouping common nodes to a community with relatively higher external density. As a result, the overall modularity value of the network has increased. Third, the proposed algorithm is evaluated. Hence, the evaluation results confirm that our proposed approach has demonstrated good performance improvements than traditional methods.

Cells going through mitosis undergo precisely timed changes in cell shape and organisation, which serve to ensure the fair partitioning of cellular components into the two daughter cells. These structural changes are driven by changes in actin filament and microtubule dynamics and organisation. While most evidence suggests that the two cytoskeletal systems are remodelled in parallel during mitosis, recent work in interphase cells has implicated the centrosome in both microtubule and actin nucleation, suggesting the potential for regulatory crosstalk between the two systems. Here, by using both in vitro and in vivo assays to study centrosomal actin nucleation as cells pass through mitosis, we show that mitotic exit is accompanied by a burst in cytoplasmic actin filament formation that depends on WASH and the Arp2/3 complex. This leads to the accumulation of actin around centrosomes as cells enter anaphase and to a corresponding reduction in the density of centrosomal microtubules. Taken together, these data suggest that the mitotic regulation of centrosomal WASH and the Arp2/3 complex controls local actin nucleation, which may function to tune the levels of centrosomal microtubules during passage through mitosis.

Over the past few decades, increased demand of highly sophisticated real-time applications with complex functionalities has directly led to exponentially increased power consumption and significantly elevated system temperatures. These elevated temperature and thermal variations present formidable challenges towards system reliability, performance, cooling cost and leakages. This article explores the thermal management strength of two fairness based algorithms, namely Proportional Fair (PFair) and Deadline Partitioning Fair (DP-Fair). In related literature, the introduction of fairness is often considered as a tool to achieve optimality in multiprocessor scheduling algorithms. This work shows that these algorithms bring about better thermal profile when compared with the commonly used Earliest Deadline First (EDF) algorithm in similar conditions both in uniprocessor and multiprocessor environments. A simulation is conducted for periodic task set model. The obtained results are encouraging and show that use of fairness based algorithms reduces the operating temperature, peak temperature, and thermal variations.

An efficient aqueous two phase systems using dual inorganic electrolytes to separate 1,3-propanediol from the fermented broth

Algorithms for fair partitioning of convex polygons

Anonymous authentication enables any user to be authenticated without being identified. (t,n)-threshold ring signatures, introduced by Bresson et. al., are ring signature schemes that allow a group of t members to jointly sign a message anonymously in a ring of n members. Threshold ring signature schemes provide a nice tradeoff between anonymity and creditability since it allows multiple ring members to sign a message jointly. The complexity in both signature generation and signature verification of the threshold ring signature scheme proposed by Bresson et. al. is O(n <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ). They also proposed an efficient threshold ring signature scheme based on an (n,t)-complete fair partition, with complexity O(n log n). In this paper, a new efficient (t,n)-threshold ring signature scheme is proposed. This scheme is constructed through a system of t linear equations and n variables, where t is generally a fixed number that is much smaller than n. The proposed threshold ring signature scheme can provide unconditional signer ambiguity, threshold unforgeability and provable security in the random oracle model. The complexity of signature generation and signature verification of the proposed threshold ring signature scheme are O(t log <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> t) and O(n), respectively. Furthermore, the length of the threshold ring signature is the same as the regular ring signature introduced by Rivest et. al., which is 2n+2, while the length of the threshold ring signature scheme proposed by Bresson et. al. is 3n-t+3.

We address the question: Given a positive integer $N$, can any 2D convex polygonal region be partitioned into $N$ convex pieces such that all pieces have the same area and same perimeter? The answer to this question is easily `yes' for $N$=2. We prove the answer to be `yes' for $N$=4 and also discuss higher powers of 2.

Orthogonal Frequency Division Multiple Access (OFD-MA) is a leading air interface candidate for future generation cellular networks. However, if deployed in a multi-user multi-tier cellular system, it is important to fairly share radio resources such as transmission power and sub-carriers among co-tier and cross-tier users. This paper focuses on a mathematical formulation of cell inner-zone/outer-zone radio resource partitioning variables and considers the case of an FFR-based macrocell underlaid with femtocell. By applying an exhaustive search procedure on the developed formulation, we determine the optimal radio resource partitioning parameter values from the perspectives of macrocell user fairness and femtocell throughput maximization.

Trade fairs and exhibitions have been around for centuries. They provide an important meeting point for sellers and potential buyers, be it for cars or clothes. Some fair organizers partition the exhibition space into predefined units, which may limit the flexibility. A free layout of the space allows for any shape or size of the stands and gives the exhibitor a better chance at creating something spectacular that will attract attention. Providing full flexibility, and thus a free layout, means that every new fair will be different. In addition to each individual stand having a varying size, the shape is not constrained to be rectangular, but might be, e.g., triangular or elliptic. All of this puts high demands on the production of the fairs. For example, the location and outline of every stand has to be determined and marked on the floor before the construction phase. This article gives a real world example of how robot technology can be used to automate the marking process for exhibitions and fairs in a way that both reduces the marking time and improves the working conditions. A mobile robot system equipped with a laser scanner and an ink jet printer is presented. The focus of the article is the description of the marking process and the system design. In addition, the requirements for such a system are outlined as well as results from over three years of operation and an analysis for commercialization.

Estimating clam yield potential in the Sacca di Goro lagoon (Italy) by using a two-part conditional model

A GIS-based habitat suitability model for commercial yield estimation of Tapes philippinarum in a Mediterranean coastal lagoon (Sacca di Goro, Italy)