How to Fairly Allocate Easy and Difficult Chores

We study the problem of fair allocation of items / goods / resources to m agents. The problem appears in two broad variants: (a) fair allocation of a divisible item wherein one item is to be divided amongst m agents; the problem is popularly known as cake-cutting problem, (b) fair allocation of indivisible items wherein n indivisible or non-shareable items are to be allocated amongst m agents. Further, the item/items can be homogeneous or heterogeneous. Both the variants have several real-time applications such as land-division amongst heirs is an example of allocation of a divisible item, whereas in divorce settlement, division of assets such as house, car, artifacts, etc. is an example of allocation of indivisible items. There are many notions of fairness and social welfare that are important and have been studied: envy-freeness, proportionality, Pareto optimality, maximin share guarantee, Borda maximin share guarantee, utilitarian social welfare, egalitarian social welfare and Nash social welfare. In this chapter, we discuss about the fair-allocation problem and their related results.

Pareto optimization for the two-agent scheduling problems with linear non-increasing deterioration based on Internet of Things

Efficient and fair allocation of multiple types of resources is a crucial objective in a cloud/distributed computing cluster. Users may have diverse resource needs. Furthermore, diversity in server properties/capabilities may mean that only a subset of servers may be usable by a given user. In platforms with such heterogeneity, we identify important limitations in existing multi-resource fair allocation mechanisms, notably Dominant Resource Fairness and its follow-up work. To overcome such limitations, we propose a new server-based approach ; each server allocates resources by maximizing a per-server utility function . We propose a specific class of utility functions which, when appropriately parameterized, adjusts the trade-off between efficiency and fairness, and captures a variety of fairness measures (such as our recently proposed Per-Server Dominant Share Fairness ). We establish conditions for the proposed mechanism to satisfy certain properties that are generally deemed desirable, e.g., envy-freeness, sharing incentive, bottleneck fairness, and Pareto optimality. To implement our resource allocation mechanism, we develop an iterative algorithm which is shown to be globally convergent. Subsequently, we show how the proposed mechanism could be implemented in a distributed fashion. Finally, we carry out extensive trace-driven simulations to show the enhanced performance of our proposed mechanism over the existing ones.

We consider the problem of fairly allocating a set of indivisible items under the criteria of the maximin share guarantee. Specifically, we study approximation of maximin share allocations under hereditary set system valuations, in which each valuation function is based on the independent sets of an underlying hereditary set system. Using a lone divider approach, we show the existence of 1/2-approximate MMS allocations, improving on the 11/30 guarantee of Li and Vetta [ 37 ]. Moreover, we prove that (2/3 + ε)-approximate MMS allocations do not always exist in this model for every ε > 0, an improvement from the recent 3/4 + ε result of Li and Deng [ 36 ]. Our existence proof is constructive but does not directly yield a polynomial-time approximation algorithm. However, we show that a 2/5-approximate MMS allocation can be found in polynomial time, given valuation oracles. Finally, we show that our existence and approximation results transfer to a variety of problems within constrained fair allocation, improving on existing results in some of these settings.

We consider the problem of fair allocation of indivisible items among $n$ agents with additive valuations, when agents have equal entitlements to the goods, and there are no transfers. Best-of-Both-Worlds (BoBW) fairness mechanisms aim to give all agents both an ex-ante guarantee (such as getting the proportional share in expectation) and an ex-post guarantee. Prior BoBW results have focused on ex-post guarantees that are based on the to one paradigm, such as envy-free up to one item (EF1). In this work we attempt to give every agent a high value ex-post, and specifically, a constant fraction of his maximin share (MMS). The up to one item paradigm fails to give such a guarantee, and it is not difficult to present examples in which previous BoBW mechanisms give agents only a $\frac{1}{n}$ fraction of their MMS. Our main result is a deterministic polynomial time algorithm that computes a distribution over allocations that is ex-ante proportional, and ex-post, every allocation gives every agent at least his proportional share up to one item, and more importantly, at least half of his MMS. Moreover, this last ex-post guarantee holds even with respect to a more demanding notion of a share, introduced in this paper, that we refer to as the truncated proportional share (TPS). Our guarantees are nearly best possible, in the sense that one cannot guarantee agents more than their proportional share ex-ante, and one cannot guarantee agents more than a $\frac{n}{2n-1}$ fraction of their TPS ex-post.

With the rapid growth of Internet, plenty of applications has been implemented for users to exchange resources with each other over networks. As a result, it has been a critical challenge to efficiently and fairly allocate resource among participating agents. Motivated by the famous BitTorrent system, a BD Mechanism for resource exchange allocation is proposed, which is based on a combinatorial structure called bottleneck decomposition. The mechanism has been shown to lead to market equilibrium, a state optimizing each individual's utility, and to be truthful, that is robust against agents' strategic behaviors. However, the crux on how to compute a bottleneck decomposition of any graph remains untouched. In this paper, we focus on the computation of bottleneck decomposition to fill the blanks and prove that the bottleneck decomposition of a network can be computed in O(n^6\log(nU)) time, where n and U are the number of agents and the maximal amount of resource any agent has in the network. Based on the bottleneck decomposition, a fair allocation in a resource exchange network can be obtained in polynomial time. We further carefully discuss the fairness in resource exchange scenario, showing the derived allocation satisfies several common fairness notions simultaneously.

We consider the problem of allocating indivisible goods fairly among n agents who have additive and submodular valuations for the goods. Our fairness guarantees are in terms of the maximin share, which is defined to be the maximum value that an agent can ensure for herself, if she were to partition the goods into n bundles, and then receive a minimum valued bundle. Since maximin fair allocations (i.e., allocations in which each agent gets at least her maximin share) do not always exist, prior work has focused on approximation results that aim to find allocations in which the value of the bundle allocated to each agent is (multiplicatively) as close to her maximin share as possible. In particular, Procaccia and Wang (2014) along with Amanatidis et al. (2015) have shown that under additive valuations, a 2/3-approximate maximin fair allocation always exists and can be found in polynomial time. We complement these results by developing a simple and efficient algorithm that achieves the same approximation guarantee. Furthermore, we initiate the study of approximate maximin fair division under submodular valuations. Specifically, we show that when the valuations of the agents are nonnegative, monotone, and submodular, then a 0.21-approximate maximin fair allocation is guaranteed to exist. In fact, we show that such an allocation can be efficiently found by using a simple round-robin algorithm. A technical contribution of the article is to analyze the performance of this combinatorial algorithm by employing the concept of multilinear extensions.

Suppose that two players, P1 and P2, must divide a set of indivisible items that each strictly ranks from best to worst. Assuming that the number of items is even, suppose also that the players desire that the allocations be balanced (each player gets half the items), item-wise envy-free (EF), and Pareto-optimal (PO). Meeting this ideal is frequently impossible. If so, we find a balanced maximal partial allocation of items to the players that is EF, though it may not be PO. Then, we show how to augment it so that it becomes a complete allocation (all items are allocated) that is EF for one player (Pi) and almost-EF for the other player (Pj) in the sense that it is EF for Pj except for one item — it would be EF for Pj if a specific item assigned to Pi were removed. Moreover, we show how low-ranked (for Pj) that exceptional item may be, thereby finding an almost-EF allocation that is as close as possible to EF — as well as complete, balanced, and PO. We provide algorithms to find such almost-EF allocations, adapted from algorithms that apply when complete balanced EF-PO allocations are possible.

Secure communication with low computational resources is a critical issue in the Internet-of-Things (IoT) implementations. It is more challenging in the presence of hybrid adversary enabled with full-duplex (FD) capability to perform eavesdropping and jamming simultaneously. In this work we aim to address this issue through optimal subcarrier allocation towards combating the FD hybrid attacker. We begin with secrecy performance analysis in a multi-user IoT system considering statistical channel state information only of all attacker links. Novel analytical expression for the exact intercept probability is derived and a closed-form approximation is also provided. We further propose an optimisation framework for fair subcarrier allocation with a novel objective of minimising maximum intercept probability among multiple users. Considering the proposed optimisation framework as a non-convex combinatorial, we propose a low-complexity sub-optimal solution by leveraging the integer linear program (ILP) structure of the problem. To reduce the complexity further, the original problem is mapped to the assignment model and solved by exploiting its special structure with graph theory tools providing an optimal solution in polynomial time. Comprehensive investigations, conducted to verify the analysis and quantify the secrecy performance, demonstrate that proposed optimal solutions yield significant enhancement in secrecy performance over relevant schemes.

In this article, we present new results on the fair and efficient allocation of indivisible goods to agents whose preferences correspond to matroid rank functions . This is a versatile valuation class with several desirable properties (such as monotonicity and submodularity), which naturally lends itself to a number of real-world domains. We use these properties to our advantage; first, we show that when agent valuations are matroid rank functions, a socially optimal (i.e., utilitarian social welfare-maximizing) allocation that achieves envy-freeness up to one item (EF1) exists and is computationally tractable. We also prove that the Nash welfare-maximizing and the leximin allocations both exhibit this fairness/efficiency combination by showing that they can be achieved by minimizing any symmetric strictly convex function over utilitarian optimal outcomes. To the best of our knowledge, this is the first valuation function class not subsumed by additive valuations for which it has been established that an allocation maximizing Nash welfare is EF1. Moreover, for a subclass of these valuation functions based on maximum (unweighted) bipartite matching, we show that a leximin allocation can be computed in polynomial time. Additionally, we explore possible extensions of our results to fairness criteria other than EF1 as well as to generalizations of the above valuation classes.

Graphical games have been proposed as a game-theoretic model of large-scale distributed networks of non-cooperative agents. When the number of players is large, and the underlying graph has low degree, they provide a concise way to represent the players' payoffs. It has recently been shown that the problem of finding Nash equilibria in a general degree-3 graphical game with two actions per player is complete for the complexity class PPAD, indicating that it is unlikely that there is any polynomial-time algorithm for this problem. In this paper, we study the complexity of graphical games with two actions per player on bounded-degree trees. This setting was first considered by Kearns, Littman and Singh, who proposed a dynamic programming-based algorithm that computes all Nash equilibria of such games. The running time of their algorithm is exponential, though approximate equilibria can be computed efficiently. Later, Littman, Kearns and Singh proposed a modification to this algorithm that can find a single Nash equilibrium in polynomial time. We show that this modified algorithm is incorrect-the output is not always a Nash equilibrium. We then propose a new algorithm that is based on the ideas of Kearns et al. and computes all Nash equilibria in quadratic time if the input graph is a path, and in polynomial time if it is an arbitrary graph of maximum degree 2. Moreover, our algorithm can be used to compute Nash equilibria of graphical games on arbitrary trees, but the running time can be exponential, even when the tree has bounded degree. We show that this is inevitable -- any algorithm of this type will take exponential time, even on bounded-degree trees with pathwidth 2. It is an open question whether our algorithm runs in polynomial time on graphs with pathwidth 1, but we show that finding a Nash equilibrium for a 2-action graphical game in which the underlying graph has maximum degree 3 and constant pathwidth is PPAD-complete (so is unlikely to be tractable).

We consider the rescheduling on a single machine with release dates to minimize the makespan and total sequence disruption simultaneously. In the literature, a polynomial-time algorithm was presented for minimizing the makespan under a limit on the total sequence disruption. But the algorithm is not strongly polynomial. We present a strongly polynomial-time algorithm for finding all Pareto optimal points of the Pareto optimization problem. Consequently, the rescheduling to minimize the makespan under a limit on the total sequence disruption can be solved in a strongly polynomial time.

The problem of fair division of indivisible goods is a fundamental problem of resource allocation in multi-agent systems, also studied extensively in social choice. Recently, the problem was generalized to the case when goods form a graph and the goal is to allocate goods to agents so that each agent’s bundle forms a connected subgraph. For the maximin share fairness criterion, researchers proved that if goods form a tree, an allocation offering each agent a bundle of at least her maximin share value always exists. Moreover, it can be found in polynomial time. In this paper we consider the problem of maximin share allocations of goods on a cycle. Despite the simplicity of the graph, the problem turns out to be significantly harder than its tree version. We present cases when maximin share allocations of goods on cycles exist and provide in this case results on allocations guaranteeing each agent a certain fraction of her maximin share. We also study algorithms for computing maximin share allocations of goods on cycles.

We study the problem of computing maximin share allocations, a recently introduced fairness notion. Given a set of n agents and a set of goods, the maximin share of an agent is the best she can guarantee to herself, if she is allowed to partition the goods in any way she prefers, into n bundles, and then receive her least desirable bundle. The objective then is to find a partition, where each agent is guaranteed her maximin share. Such allocations do not always exist, hence we resort to approximation algorithms. Our main result is a 2/3-approximation that runs in polynomial time for any number of agents and goods. This improves upon the algorithm of Procaccia and Wang (2014), which is also a 2/3-approximation but runs in polynomial time only for a constant number of agents. To achieve this, we redesign certain parts of the algorithm in Procaccia and Wang (2014), exploiting the construction of carefully selected matchings in a bipartite graph representation of the problem. Furthermore, motivated by the apparent difficulty in establishing lower bounds, we undertake a probabilistic analysis. We prove that in randomly generated instances, maximin share allocations exist with high probability. This can be seen as a justification of previously reported experimental evidence. Finally, we provide further positive results for two special cases arising from previous works. The first is the intriguing case of three agents, where we provide an improved 7/8-approximation. The second case is when all item values belong to {0, 1, 2}, where we obtain an exact algorithm.

We consider the bin packing problem with d different item sizes s i and item multiplicities a i , where all numbers are given in binary encoding. This problem formulation is also known as the one-dimensional cutting stock problem . In this work, we provide an algorithm that, for constant d , solves bin packing in polynomial time. This was an open problem for all d\ge 3 . In fact, for constant d our algorithm solves the following problem in polynomial time: Given two d -dimensional polytopes P and Q , find the smallest number of integer points in P whose sum lies in Q . Our approach also applies to high multiplicity scheduling problems in which the number of copies of each job type is given in binary encoding and each type comes with certain parameters such as release dates, processing times, and deadlines. We show that a variety of high multiplicity scheduling problems can be solved in polynomial time if the number of job types is constant.