Fair Scheduling for Time-dependent Resources

1 Introduction . With the rapid progress of AI technologies , AI algorithms are widely deployed in many societal settings such as the distribution of job and education opportunities where complex social effects may significantly diminish the performance of the algorithms . To motivate our study , let us consider a problem faced by the Students Affairs Office ( SAO ) . An SAO clerk is assigning multiple part-time jobs to the students who submitted job applications . Each part-time job occupies a consecutive time period within a possibly flexible interval . For example , an one-hour math tutorial needs to be given between 8:00am and 11:00am on June 26th . A feasible assignment requires that the jobs assigned to an applicant can be scheduled without mutual overlap . The students are heterogeneous , i.e. , different students may hold different job preferences . It is important that the students are treated equally in terms of getting job opportunities , and thus the clerk ’ s task is to make the assignment fair . The SAO problem falls under the umbrella of the research on job scheduling , which has been studied in numerous fields , including operations research Gentner et al . [ 2004 ] , machine learning Paleja et al . [ 2020 ] , parallel computing Drozdowski [ 2009 ] , cloud computing Al-Arasi and Saif [ 2020 ] , etc . Following the convention of job scheduling research , each part-time job , or job for short , is associated with release time , deadline , and processing time . The students are modeled as machines , who have different utility gains for completing jobs . Traditionally , the objective of designing scheduling algorithms is solely focused on efficiency or profit . However , motivated by various real-world AI driven deployments where the data points of the algorithms are real human beings who should be ∗Equal contribution †Corresponding author 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . treated unbiasedly , addressing the individual fairness becomes important . Accordingly , the past several years has seen considerable efforts in developing fair AI algorithms Chierichetti et al . [ 2017 ] , where combinatorial structures are incorporated into the design , such as vertex cover Rahmattalabi et al . [ 2019 ] , facility location Chen et al . [ 2019 ] and knapsack Amanatidis et al . [ 2020 ] . It is noted that people have different criteria on evaluating fairness , and in this work , we consider two of the most widely accepted definitions . The first is motivated by the max-min objective , i.e. , maximizing the worst-case utility , which has received observable attention for various learning scenarios Rahmattalabi et al . [ 2019 ] . However , for heterogeneous agents , optimizing the worst case is not enough , as different people have different perspectives and may not agree on the output . Accordingly , one popular research agenda is centered around computing an assignment such that everyone believes that it ( approximately ) maximizes the worst case utility . This criterion is named maximin share ( MMS ) fairness in Budish [ 2010 ] . The second one is envy-freeness ( EF ) , which has been very widely studied in social sciences and economics but arguably less explored in machine learning . Informally , an assignment is called EF if everyone believes she has obtained the best resource compared with any other agent ’ s assignment . We note that , due to the scheduling-feasible constraint , some jobs may not be allocated . Thus EF alone is not able to satisfy the agents as keeping all resources unallocated does not incur any envy among them , but the agents envy the charity where unallocated/disregarded items are assumed to be donated to a charity . To resolve this issue , in this work , we want to understand how we can compute allocations that are simultaneously EF and Pareto efficient ( PO ) , where an allocation is called PO if there does not exist another allocation that makes nobody worse off but somebody strictly better off . Recently , Chiarelli et al . [ 2020 ] and Hummel and Hetland [ 2021 ] studied the fair allocation of conflicting items , where the items are connected via graphs . An edge between two items means they are in conflict and should be allocated to different agents . However , in our model , the conflict among items can not be described as the edges in a graph . For example , two one-hour tutorials between 9:00am and 11:00am can be feasibly scheduled , but three such tutorials are not feasible any more . For a comprehensive introduction to the various constraints , including conflict constraints , studied in fair division , we refer the interested reader to the recent survey of Suksompong [ 2021 ] . 1.1 Main Results . We study the fair interval scheduling problem ( FISP ) , where fairness is captured by MMS and EF . For each of them , we design approximation algorithms to compute MMS or EF1 schedules . Maximin Share . Informally , a machine ’ s MMS is defined to be her optimal worst-case utility in an imaginary experiment : she partitions the items into m bundles but was the last to select one , where m is the number of agents . It is noted that as the machines are heterogeneous , they may not have the same MMS value . Our task is to investigate the extent to which everyone agrees on the final allocation . A job assignment is called α-approximate MMS fair if every machine ’ s utility is no less than α fraction of her MMS value . Our main result in this part is an algorithmic framework which ensures a 1/3-approximate MMS schedule , and thus improves the best known approximation of 1/5 which is proved for a broader class of valuation functions – XOS Ghodsi et al . [ 2018 ] . Interestingly , in the independent and parallel work Hummel and Hetland [ 2021 ] , the authors also show the existence of 1/3-approximate MMS for graphically conflicting items . With XOS valuation oracles , Ghodsi et al . [ 2018 ] also designed a polynomial-time algorithm to compute a 0.125-approximate MMS allocation . As a comparison , by slightly modifying our algorithm , it returns a 0.24-approximate MMS allocation in polynomial time , without valuation oracles . When all jobs are rigid , i.e. , processing time = deadline - release time , our problem degenerates to finding a partition of an interval graph such that the minimum weight of the independent set for each subgraph is maximized . Recently , a pseudo-polynomial-time algorithm is given in Chiarelli et al . [ 2020 ] for constant number of agents . In this sense , we generalize this problem to flexible jobs and design approximation algorithms for arbitrary number of agents . Main Result 1 . For an arbitrary FISP instance , there exists a 1/3-approximate MMS schedule , and a ( 0.24− ) -approximate MMS schedule can be found in polynomial time , for any constant > 0 . EF1+PO . EF is actually a demanding fairness notion , in the sense that any approximation of EF is not compatible with PO . Instead , initiated by Lipton et al . [ 2004 ] , most research is focused on its relaxation , envy-freeness up to one item ( EF1 ) , which means the envy between two agents may exist but will disappear if some item is removed . Unfortunately , EF1 and PO are still not compatible even if all jobs are rigid and agents have unary valuations . However , the good news is , if all jobs have unit processing time , an EF1 and PO schedule is guaranteed to exist and can be found in polynomial time . This result continues to hold when agent valuations are weighted but identical . It is shown in Biswas and Barman [ 2018 ] that under laminar matroid constraint an EF1 and PO allocation exists when agents have identical utilities , but finding it may need exponential time . We improve this result in two perspectives . First , our feasibility constraints , even for unit jobs , are not necessarily laminar matroid . Second , our algorithm runs in polynomial time . Main Result 2 . No algorithm can return an EF1 and PO schedule for all FISP instances , even if all jobs are rigid and valuations are unary . When all jobs have unit processing time and valuations are ( weighted ) identical , an EF1 and PO schedule can be computed in polynomial time . Although exact EF1 and PO are not compatible , we prove that for an arbitrary FISP instance , there always exists a 1/4-approximate EF1 and PO schedule , which coincides with Wu et al . [ 2021 ] . If all jobs have unit processing time , a 1/2-approximate EF1 and PO schedule exists . To prove this result , we consider Nash social welfare – the geometric mean of all machines ’ utilities . We show that a Nash social welfare maximizing schedule satisfies the desired approximation ratio . This result is in contrast to the corresponding one in Caragiannis et al . [ 2016 ] , which shows that without any feasibility constraints , such an allocation is EF1 and PO . We also show that both approximations are tight . Main Result 3 . For any FISP instance , the schedule maximizing Nash social welfare is PO and 1/4-approximate EF1 . If all jobs have unit processing time , it is 1/2-approximate EF1 . EF1+IO By above results , we observe that PO is too demanding to measure efficiency in our model . One milder requirement is individual optimality ( IO ) . Intuitively , an allocation is called IO if every agent gets the best feasible subset of jobs from the union of her current jobs and unscheduled jobs . We show that EF1 is still not compatible with IO in the general case . But for unary valuations , we obtain positive results and design polynomial time algorithms for ( 1 ) computing an EF1 and IO schedule for rigid jobs , and ( 2 ) computing an EF1 and 1/2-approximate IO schedule for flexible jobs . To prove these results , we utilize two classic algorithms Earliest Deadline First and Round-Robin . We defer this part completely to the full version Li et al . [ 2021 ] . 1.2 Other Related Works . Since computing feasible job sets to maximize the total weight is NP-hard Garey and Johnson [ 1979 ] , various approximation algorithms have been proposed Bar-Noy et al . [ 2001 ] ; Berman and DasGupta [ 2000 ] ; Chuzhoy et al . [ 2006 ] , and the best known approximation ratio is 0.644 Im et al . [ 2020 ] . For rigid instances , the problem is polynomial-time solvable Schrijver [ 1999 ] . Recently , scheduling has been studied from the perspective of machine learning , including developing learning algorithms to empirically solve NP-hard scheduling problem Zhang et al . [ 2020 ] ; Paleja et al . [ 2020 ] , and predicting uncertain data in order to optimize the performance in the online setting Purohit et al . [ 2018 ] . Fairness has been concerned in the scheduling community in past decades Ajtai et al . [ 1998 ] ; Baruah and Lin [ 1998 ] ; Baruah [ 1995 ] . Most of these works aim at finding a fair schedule for the jobs , such as balancing the waiting and completion time Bilò et al . [ 2016 ] ; Im and Moseley [ 2020 ] . MMS allocation for indivisible resources has been widely studied since Budish [ 2010 ] . Unfortunately , it is shown in Kurokawa et al . [ 2018 ] ; Ghodsi et al . [ 2018 ] ; Feige et al . [ 2021 ] that an exact MMS fair allocation may not exist . Thereafter , a string of approximation algorithms for various valuation types are proposed , such as additive Garg and Taki [ 2020 ] , matroid-rank function Babaioff et al . [ 2021 ] ; Barman and Verma [ 2021 ] , submodular Barman and Krishnamurthy [ 2020 ] ; Ghodsi et al . [ 2018 ] , XOS and subadditive Ghodsi et al . [ 2018 ] . Regarding EF1 , in the unconstrained setting , an allocation that is both EF1 and PO is guaranteed to exist Caragiannis et al . [ 2016 ] ; Barman et al . [ 2018 ] . However , when there are constraints , such as cardinality and knapsack , the general compatibility is still open Biswas and Barman [ 2018 , 2019 ] ; Wu et al . [ 2021 ] ; Gan et al . [ 2021 ] ; Dror et al . [ 2021 ] .

Summary: Problem considered is the following: Given a m intelligent agents or machines and a set of Jobs (with release times and deadlines, processing time requirement and a utility) the goal is to allocate a maximum utility subset of jobs feasibly under some fairness criteria. They consider two fairness criteria namely: Minimax fairness and envy freeness.   The paper obtains novel approximation algorithms for various problems arising in this setting, improving upon the existing best results for some of them.