Matchings and Copeland's Method

Given a graph $G = (V,E)$ where every vertex has a weak ranking over its neighbors, we consider the problem of computing an optimal matching as per agent preferences. The classical notion of optimality in this setting is stability. However stable matchings, and more generally, popular matchings need not exist when $G$ is non-bipartite. Unlike popular matchings, Copeland winners always exist in any voting instance -- we study the complexity of computing a matching that is a Copeland winner and show there is no polynomial-time algorithm for this problem unless $\mathsf{P} = \mathsf{NP}$. We introduce a relaxation of both popular matchings and Copeland winners called weak Copeland winners. These are matchings with Copeland score at least $\mu/2$, where $\mu$ is the total number of matchings in $G$; the maximum possible Copeland score is $(\mu-1/2)$. We show a fully polynomial-time randomized approximation scheme to compute a matching with Copeland score at least $\mu/2\cdot(1-\varepsilon)$ for any $\varepsilon > 0$

Application of Wireless Sensor networks for Environmental Monitoring and Development of an Energy Efficient Cluster Based Routing

Wireless Sensor Networks (WSNs) have attracted the attention of many researchers. Wireless Sensor Networks (WSNs) are used for various applications such as habitat monitoring, automation, agriculture, and security. Since numerous sensors are usually deployed on remote and inaccessible places, the deployment and maintenance should be easy and scalable. Wireless sensor network consists of large number of small nodes. The nodes then sense environmental changes and report them to other nodes over flexible network architecture. Sensor nodes are great for deployment in hostile environments or over large geographical areas. The measurement of temperature & light by the use of Crossbow sensor kit in which there are different nodes/motes placed at different locations. These nodes are having different node identification & they will sense the temperature & light of there surrounding location and send it to the base station node which is connected through USB port to the computer by the use of MoteView & MoteConfig environment. The data acquisition board that we have used is MDA100CB (Mote Data Acquisition). The programming of the sensor nodes is done by MoteConfig & live data is viewed through MoteView environment. The nodes that we have used are MicaZ, the MDA100CB board is fixed over these nodes by means of 51 Input/output pins. An energy efficient hierarchical cluster-based routing protocol for continuous stream queries in WSN. We introduce a set of cluster heads, head-set, for cluster-based routing. The head-set members are responsible for control and management of the network. On rotation basis, a head-set member receives data from the neighboring nodes and transmits the aggregated results to the distant base station. For a given number of data collecting sensor nodes, the number of control and management nodes can be systematically adjusted to reduce the energy consumption, which increases the network life. Nodes in a sensor network are severely constrained by energy, storage capacity and computing power. To prolong the lifetime of the sensor nodes, designing efficient routing protocols is critical

On Approximate Envy-Freeness for Indivisible Chores and Mixed Resources

We study the fair allocation of undesirable indivisible items, or chores. While the case of desirable indivisible items (or goods) is extensively studied, with many results known for different notions of fairness, less is known about the fair division of chores. We study envy-free allocation of chores and make three contributions. First, we show that determining the existence of an envy-free allocation is NP-complete even in the simple case when agents have binary additive valuations. Second, we provide a polynomial-time algorithm for computing an allocation that satisfies envy-freeness up to one chore (EF1), correcting a claim in the existing literature. A modification of our algorithm can be used to compute an EF1 allocation for doubly monotone instances (where each agent can partition the set of items into objective goods and objective chores). Our third result applies to a mixed resources model consisting of indivisible items and a divisible, undesirable heterogeneous resource (i.e., a bad cake). We show that there always exists an allocation that satisfies envy-freeness for mixed resources (EFM) in this setting, complementing a recent result of Bei et al. [Bei et al., 2021] for indivisible goods and divisible cake

Hide, Not Seek: Perceived Fairness in Envy-Free Allocations of Indivisible Goods

Fair division provides a rich computational and mathematical framework for the allocation of indivisible goods, which has given rise to numerous fairness concepts and their relaxations. In recent years, much attention has been given to theoretical and computational aspects of various fairness concepts. Nonetheless, the choice of which fairness concept is in practice perceived to be fairer by individuals is not well understood. We consider two conceptually different relaxations of envy-freeness and investigate how individuals perceive the induced allocations as fair. In particular, we examine a well-studied relaxation of envy-freeness up to one good (EF1) which is based on counterfactual thinking that any pairwise envy can be eliminated by the hypothetical removal of a single good from the envied agent's bundle. In contrast, a recently proposed epistemic notion, namely envy-freeness up to $k$ hidden goods (HEF-$k$), provides a relaxation by hiding information about a small subset of $k$ goods. Through various crowdsourcing experiments, we empirically demonstrate that allocations achieved by withholding information are perceived to be fairer compared to two variants of EF1.Comment: 21 pages, 10 figure