58 research outputs found
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 -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 -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.Comment: Accepted to SODA 201
Inequity Aversion Pricing over Social Networks: Approximation Algorithms and Hardness Results
We study a revenue maximization problem in the context of social networks. Namely, we consider a model introduced by Alon, Mansour, and Tennenholtz (EC 2013) that captures inequity aversion, i.e., prices offered to neighboring vertices should not be significantly different. We first provide approximation algorithms for a natural class of instances, referred to as the class of single-value revenue functions. Our results improve on the current state of the art, especially when the number of distinct prices is small. This applies, for example, to settings where the seller will only consider a fixed number of discount types or special offers. We then resolve one of the open questions posed in Alon et al., by establishing APX-hardness for the problem. Surprisingly, we further show that the problem is NP-complete even when the price differences are allowed to be relatively large. Finally, we also provide some extensions of the model of Alon et al., regarding the allowed set of prices
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
Multiple Birds with One Stone: Beating for EFX and GMMS via Envy Cycle Elimination
Several relaxations of envy-freeness, tailored to fair division in settings
with indivisible goods, have been introduced within the last decade. Due to the
lack of general existence results for most of these concepts, great attention
has been paid to establishing approximation guarantees. In this work, we
propose a simple algorithm that is universally fair in the sense that it
returns allocations that have good approximation guarantees with respect to
four such fairness notions at once. In particular, this is the first algorithm
achieving a -approximation of envy-freeness up to any good (EFX) and
a -approximation of groupwise maximin share fairness (GMMS),
where is the golden ratio (). The best known
approximation factor for either one of these fairness notions prior to this
work was . Moreover, the returned allocation achieves envy-freeness up to
one good (EF1) and a -approximation of pairwise maximin share fairness
(PMMS). While EFX is our primary focus, we also exhibit how to fine-tune our
algorithm and improve the guarantees for GMMS or PMMS. Finally, we show that
GMMS -- and thus PMMS and EFX -- allocations always exist when the number of
goods does not exceed the number of agents by more than two
Don’t Roll the Dice, Ask Twice: The Two-Query Distortion of Matching Problems and Beyond
In most social choice settings, the participating agents are typically
required to express their preferences over the different alternatives in the
form of linear orderings. While this simplifies preference elicitation, it
inevitably leads to high distortion when aiming to optimize a cardinal
objective such as the social welfare, since the values of the agents remain
virtually unknown. A recent array of works put forward the agenda of designing
mechanisms that can learn the values of the agents for a small number of
alternatives via queries, and use this extra information to make a
better-informed decision, thus improving distortion. Following this agenda, in
this work we focus on a class of combinatorial problems that includes most
well-known matching problems and several of their generalizations, such as
One-Sided Matching, Two-Sided Matching, General Graph Matching, and
-Constrained Resource Allocation. We design two-query mechanisms that
achieve the best-possible worst-case distortion in terms of social welfare, and
outperform the best-possible expected distortion that can be achieved by
randomized ordinal mechanisms
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