Matching algorithms have been classically studied with the goal of finding stable solutions. However, in many important societal problems, the degree of fairness in the matching assumes crucial importance, for instance when we have to match COVID-19 patients to care units. We study the problem of finding a stable many-to-one matching while satisfying fairness among all the agentswith cardinal utilities. We consider various fairness definitions from fair allocation literature, such as envy-freeness (EF) and leximin optimal fairness. We find that EF and its weaker versions are incompatible with stability, even under a restricted setting with isometric utilities. We focus on leximin optimal fairness and show that finding such a matching is NP-Hard, even under isometric utilities. Next, we narrow our focus onto ranked isometric utilities and provide a characterisation for the space of stable matchings. We present a novel and efficient algorithm that finds the leximin optimal stable matching under ranked isometric utilities . To the best of our knowledge, we are the first to address the problem of finding a leximin optimally fair and stable matching .
The past few years have seen a surge of work on fairness in social choice literature. A major portion of this work has focused on allocation settings where items must be allocated fairly to agents with diverse preferences. In comparison, fairness in two-sided settings where agents have to be matched to other agents, with preferences on both sides, has received much less attention. In this paper, we explore the problem of finding a stable many-to-one matching while satisfying fairness among the agents, having cardinal utilities. In particular, we focus on leximin optimal fairness and seek leximin optimal many-to-one stable matchings. In the special case of ranked utilities where all agents on each side have the same rankings over the agents on the other side (but not necessarily the same utilities), we provide a characterisation for the space of stable matchings. This leads to a novel algorithm, called FaSt, that finds the leximin optimal stable matching under ranked isometric utilities (where the utilities from matching are the same across an edge). The running time of FaSt is linear in the number of edges. Additionally, we provide an efficient algorithm, called FaSt-Gen, that finds the leximin optimal stable matching for ranked but otherwise unconstrained utilities. The running time of FaSt-Gen is quadratic in the number of edges. We then establish that, in the absence of rankings, finding a leximin optimal stable matching is NP-Hard, even under isometric utilities. In fact, when additivity is the only assumption on the utilities, we show that, unless P=NP, no polynomial time approximation is possible .
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Matching algorithms have been classically studied with the goal of finding stable solutions. However, in many important societal problems, the degree of fairness in the matching assumes crucial importance, for instance when we have to match COVID-19 patients to care units. We study
The past few years have seen a surge of work on fairness in social choice literature. A major portion of this work has focused on allocation settings where items must be allocated fairly to agents with diverse preferences. In comparison, fairness in two-sided settings where agents have to be matched to other agents, with preferences on both sides, has received much less attention. In this paper, we explore
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all the agentswith
the agents, having
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We consider various fairness definitions from fair allocation literature, such as envy-freeness (EF) and leximin optimal fairness. We find that EF and its weaker versions are incompatible with stability, even under a restricted setting with isometric utilities. We
In particular, we
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show that finding such a matching is NP-Hard, even under isometric utilities. Next, we narrow our focus onto ranked isometric utilities and
seek leximin optimal many-to-one stable matchings. In the special case of ranked utilities where all agents on each side have the same rankings over the agents on the other side (but not necessarily the same utilities), we
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We present a novel and efficient algorithm
This leads to a novel algorithm, called FaSt,
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. To the best of our knowledge, we are the first to address the problem of
(where the utilities from matching are the same across an edge). The running time of FaSt is linear in the number of edges. Additionally, we provide an efficient algorithm, called FaSt-Gen, that finds the leximin optimal stable matching for ranked but otherwise unconstrained utilities. The running time of FaSt-Gen is quadratic in the number of edges. We then establish that, in the absence of rankings,
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optimally fair and stable matching
optimal stable matching is NP-Hard, even under isometric utilities. In fact, when additivity is the only assumption on the utilities, we show that, unless P=NP, no polynomial time approximation is possible