Comparing approximate relaxations of envy-freeness

In fair division problems with indivisible goods it is well known that one cannot have any guarantees for the classic fairness notions of envy-freeness and proportionality. As a result, several relaxations have been introduced, most of which in quite recent works. We focus on four such notions, namely envy-freeness up to one good (EF1), envy-freeness up to any good (EFX), maximin share fairness (MMS), and pairwise maximin share fairness (PMMS). Since obtaining these relaxations also turns out to be problematic in several scenarios, approximate versions of them have been considered. In this work, we investigate further the connections between the four notions mentioned above and their approximate versions. We establish several tight, or almost tight, results concerning the approximation quality that any of these notions guarantees for the others, providing an almost complete picture of this landscape. Some of our findings reveal interesting and surprising consequences regarding the power of these notions, e.g., PMMS and EFX provide the same worst-case guarantee for MMS, despite PMMS being a strictly stronger notion than EFX. We believe such implications provide further insight on the quality of approximately fair solutions

Rapid mixing of the switch Markov chain for strongly stable degree sequences and 2-class joint degree matrices

The switch Markov chain has been extensively studied as the most natural Markov Chain Monte Carlo approach for sampling graphs with prescribed degree sequences. We use comparison arguments with other, less natural but simpler to analyze, Markov chains, to show that the switch chain mixes rapidly in two different settings. We first study the classic problem of uniformly sampling simple undirected, as well as bipartite, graphs with a given degree sequence. We apply an embedding argument, involving a Markov chain defined by Jerrum and Sinclair (TCS, 1990) for sampling graphs that almost have a given degree sequence, to show rapid mixing for degree sequences satisfying strong stability, a notion closely related to P-stability. This results in a much shorter proof that unifies the currently known rapid mixing results of the switch chain and extends them up to sharp characterizations of P-stability. In particular, our work resolves an open problem posed by Greenhill (SODA, 2015).Secondly, in order to illustrate the power of our approach, we study the problem of uniformly sampling graphs for which, in addition to the degree sequence, a joint degree distribution is given. Although the problem was formalized over a decade ago, and despite its practical significance in generating synthetic network topologies, small progress has been made on the random sampling of such graphs. The case of a single degree class reduces to sampling of regular graphs, but beyond this almost nothing is known. We fully resolve the case of two degree classes, by showing that the switch Markov chain is always rapidly mixing. Again, we first analyze an auxiliary chain for strongly stable instances on an augmented state space and then use an embedding argument.</p

Approximation Algorithms for Computing Maximin Share Allocations

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 algorith

An Overview of the SOLVE-THESEO 2000 Campaign

Between November 1999 and April 2000, two major field experiments, the SAGE III Ozone Loss and Validation Experiment (SOLVE) and the Third European Stratospheric Experiment on Ozone (THESEO 2000), collaborated to form the largest field campaign yet mounted to study Arctic ozone loss. This international campaign involved more than 500 scientists from over 20 countries spread across the high and mid-latitudes of the northern hemisphere. The main scientific aims of SOLVE-THESEO 2000 were to study (a) the processes leading to ozone loss in the Arctic vortex and (b) the effect on ozone amounts over northern mid-latitudes. The campaign included satellites, heavy lift balloon launches, 6 different aircraft, ground stations, and scores of ozone-sonde. Campaign activities were principally conducted in 3 intensive measurement phases centered on early December 1999, late January 2000, and early March 2000. Observations made during the campaign showed that temperatures were unusually cold in the polar lower stratosphere over the course of the 1999-2000 winter. These cold temperatures resulted in the formation of extensive polar stratospheric clouds (PSCs) across the Arctic. Heterogeneous chemical reactions on the surfaces of the PSC particles produced high levels of reactive chlorine within the polar vortex by early January. This reactive chlorine catalytically destroyed about 60% of the ozone in a layer near 20 km between late January and mid-March 2000

An overview of the SOLVE/THESEO 2000 campaign

International audienceBetween November 1999 and April 2000, two major field experiments, the Stratospheric Aerosol and Gas Experiment (SAGE) III Ozone Loss and Validation Experiment (SOLVE) and the Third European Stratospheric Experiment on Ozone (THESEO 2000), collaborated to form the largest field campaign yet mounted to study Arctic ozone loss. This international campaign involved more than 500 scientists from over 20 countries. These scientists made measurements across the high and middle latitudes of the Northern Hemisphere. The main scientific aims of SOLVE/THESEO 2000 were to study (1) the processes leading to ozone loss in the Arctic vortex and (2) the effect on ozone amounts over northern midlatitudes. The campaign included satellites, research balloons, six aircraft, ground stations, and scores of ozonesondes. Campaign activities were principally conducted in three intensive measurement phases centered on early December 1999, late January 2000, and early March 2000. Observations made during the campaign showed that temperatures were below normal in the polar lower stratosphere over the course of the 1999-2000 winter. Because of these low temperatures, extensive polar stratospheric clouds (PSC) formed across the Arctic. Large particles containing nitric acid trihydrate were observed for the first time, showing that denitrification can occur without the formation of ice particles. Heterogeneous chemical reactions on the surfaces of the PSC particles produced high levels of reactive chlorine within the polar vortex by early January. This reactive chlorine catalytically destroyed about 60% of the ozone in a layer near 20 km between late January and mid-March 2000, with good agreement being found between a number of empirical and modeling studies. The measurements made during SOLVE/THESEO 2000 have improved our understanding of key photochemical parameters and the evolution of ozone-destroying forms of chlorine