Graphical Resource Allocation with Matching-Induced Utilities
Anonymous Author(s)
AffiliationAddress email
Abstract
Motivated by real-world applications, we study the fair allocation of graphicalresources, where the resources are the vertices in a graph.
Upon receiving a set ofresources, an agent’s utility equals the weight of the maximum matching in theinduced subgraph.
We care about maximin share (MMS) fairness and envy-freenessup to one item (EF1).
Regarding MMS fairness, the problem does not admit a finiteapproximation ratio for heterogeneous agents.
For homogeneous agents, we designconstant-approximation polynomial-time algorithms, and also note that significantamount of social welfare is sacrificed inevitably in order to ensure (approximate)
MMS fairness.
We then consider EF1 allocations whose existence is guaranteed.
We show that for homogeneous agents, there is an EF1 allocation that ensures atleast a constant fraction of the maximum possible social welfare.
However, thesocial welfare guarantee of EF1 allocations degrades to 1 /n for heterogeneousagents, where n is the number of agents.
Fortunately, for two special yet typicalcases, namely binary-weight and two-agent, we are able to design polynomial-timealgorithms ensuring a constant fractions of the maximum social welfare.
Introduction
Resource allocation has been actively studied due to its practical applications
[Moulin, 2003; Goldmanand Procaccia, 2014; Flanigan et al. , 2021].
Traditionally, the utilities are assumed to be additivewhich means an agent’s value for a bundle of resources equals the sum of each single item’s marginalutility.
But in many real-word problems, the resources have graph structures and thus the agents’utilities are not additive but depend on the structural properties of the received resources. For example,
Peer Instruction (PI) has been shown to be an effective learning approach based on a project conductedat Harvard University, and one of the simplest ways to implement PI is to pair the students [Crouchand Mazur, 2001].
Consider the situation when we partition students to advisors, where the advisorswill adopt PI for their assigned students. Note that the advisors may hold different perspectives onhow to pair the students based on their own experience and expertise, and they want to maximizethe efficiency of conducting PI in their own assigned students.
How should we assign the studentsfairly to the advisors? How can we maximize the social welfare among all (approximately) fairassignments?
In this work, we take an algorithm design perspective to solve these two questions. Similar pairwise joint work also appears as long-trip coach driver vs co-driver and accountant vscashier, which is widely investigated in matching theory [Lovász and Plummer, 2009].
The graphical nature of resources has been considered in the literature (see, e.g., [Bouveret et al. ,2017; Suksompong, 2019;
Bilò et al. , 2019; Igarashi and Peters, 2019]).
In this line of research, thegraph is used to characterize feasible allocations (such as the resources allocated to each agent shouldbe connected), but the agents still have additive utilities over allocated items.
With graphical resources,the value of a set of resources does not solely depend on the vertices or the edge weights, but decided by the combinatorial structure of the subgraph, namely, maximum matching in our problem.
Graphstructure is also considered in cooperative game theory (i.e., hedonic games) Bogomolnaia and
Jackson [2002]; Elkind and Wooldridge [2009]; Aziz et al.
but this is not a resource allocationproblem and its major concern is how stable coalition structure can be formed.
Our problem also aligns with the research of balanced graph partition [Miyazawa et al. , 2021].
Although there are heuristic algorithms in the literature [Kress et al. , 2015; Barketau et al. , 2015] thatpartition a graph when the subgraphs are evaluated by maximum matchings, these algorithms do nothave theoretical guarantees.
Our first fairness criterion is the maximin share (MMS) fairness proposedby Budish [2011], which generalizes the max-min objective in Santa Claus problem [Bansal and Sviridenko, 2006].
Informally, the MMS value of an agent is her best guarantee if she is to partitionthe graph into several subgraphs but receives the worst one.
We aim at designing efficient algorithmswith provable approximation guarantees.
As will be clear later, to achieve (approximate) MMSfairness, a significant amount of social welfare has to be inevitably sacrificed.
Our second fairnessnotion is envy-freeness (EF)
[Foley, 1967].
In an EF allocation, no agent prefers the allocation ofanother agent to her own.
Since the resources are indivisible, such an allocation barely exists, andrecent research in fair division focuses on achieving its relaxations instead.
One of the most widelyaccepted and studied relaxations is envy-freeness up to one item (EF1) [Budish, 2011], which requiresthe envy to be eliminated after removing one item.
Lipton et al.
[2004] proved that an EF1 allocationalways exists even with combinatorial valuations.
1 It is noted that an arbitrary EF1 allocation mayhave low social welfare, and our goal is to compute an EF1 allocation which preserves a large fractionof the maximum social welfare without fairness constraints.
The social welfare loss by enforcing theallocations to be EF1 is quantified by price of EF1 [Bei et al. , 2021].
1.1 Our Results
We study the fair allocation of graphical resources when the resources are indivisible and correspondto the vertices in the graph, and the agents’ valuations are measured by the weight of the maximummatchings in the induced subgraphs.
The fairness of an allocation is measured by maximin share(MMS) and envy-free up to one item (EF1).
We aim at designing efficient algorithms that computefair allocations with high social welfare.
Our main contributions are summarized as follows.
We first consider homogeneous agents when their valuations are identical.
For homogeneous agents,the MMS fairness degenerates to the max-min objective, i.e., partitioning the vertices so that theminimum weight of the maximum matchings in the subgraphs is maximized.
It is easy to seethat an MMS fair allocation always exists but finding it is NP-hard.
We design a polynomial-time1 / 8 -approximation algorithm for arbitrary number of agents, and show that when the problem onlyinvolves two agents, the approximation ratio can be improved to 2 / 3 .
It is noted that, to ensure anyfinite approximation of MMS fairness, significant amount of social welfare is inevitably sacrificed.
Regarding EF1 fairness, we design a polynomial-time algorithm that computes an EF1 allocationwhose social welfare is at least 2 / 3 + 2 / (9 n − 3) fraction of the maximum social welfare that can beachieved without fairness constraints, where n is the number of agents.
Note that when n = 2 , theapproximation ratio is 4/5, and we conjecture that there always exists an EF1 allocation that achievesthe maximum social welfare for any number of agents.
We then consider the case of heterogeneous agents.
Unfortunately, we show strong impossibilityresults for the general case.
Particularly, for MMS fairness, no algorithm has bounded approximationratio even if there are two agents with binary weights.
For EF1 fairness, no EF1 allocation can ensurebetter than 1 /n fraction of the maximum social welfare, but this result does not exclude the possibilityof constant approximations for two special cases.
In fact, for both two-agent case and binary-weightcase, we design polynomial-time algorithms that guarantee 1 / 3 fraction of the maximum socialwelfare.
Moreover, for the two-agent case, the approximation ratio is the best possible
1.2 Related Works
Two separate research lines are closely related to our work graph division
Graph Partition.
Partitioning graphs into balanced subgraphs has been extensively studied in operations research [Miyazawa et al. , 2021] and computer science [Buluç et al. , 2016].
There are severalpopular objectives for evaluating whether a partition is balanced.
Among the most prominent ones arethe max-min (or min-max) objectives, where the goal is to maximize (or minimize) the total weightof the minimum (or maximum) part.
Particularly, the vehicle routing problem (VRP)
[Koç et al. ,2016], which generalizes the travelling salesperson problem (TSP), is closely related to our work.
Itasks for an optimal set of routes for a number of vehicles, to visit a set of customers.
There are anumber of popular variants for the VRP, e.g., the so called heterogeneous vehicle routing problem[Yaman, 2006; Rathinam et al. , 2020].
There are many other combinatorial structures studied in graphpartitioning problems.
For example, in the min-max tree cover (a.k.a.
nurse station location) problem
,the task is to use trees to cover an edge-weighted graph such that the largest tree is minimized [Khaniand Salavatipour, 2014].
This problem also falls under the umbrella of a more general problem, thegraph covering problem, where a set of pairwise disjoint subgraphs (called templates) is used tocover a given graph, such as paths [Farbstein and Levin, 2015], cycles [Traub and Tröbst, 2020], andmatchings [Kress et al. , 2015].
Fair Division.
Allocating a set of indivisible items among multiple agents is a fundamental problemin the fields of multi-agent systems and computational social choice, and we refer the readers torecent surveys [Amanatidis et al. , 2022; Aziz et al. , 2022] for more detailed discussion.
Envyfreeness (EF) and maximin share fairness (MMS) are two well accepted and extensively studiedsolution concepts.
However, with indivisible items, these requirements are demanding and thusthe state-of-the-art research mostly studies their relaxations and approximations.
For example,
EF1 allocation is studied as a relaxation of EF which always exists [Lipton et al. , 2004].
Variousconstant approximation algorithms for MMS allocations are proposed in [Kurokawa et al. , 2018;
Garg and Taki, 2021] for additive valuations and in [Barman and Krishnamurthy, 2020; Ghodsi etal. , 2018] for subadditive valuations.
Our work focuses on indivisible graphical items where agentshave combinatorial valuations (neither subadditive nor superadditive) depending on the structuralproperties.
Moreover, all the existing algorithms for non-additive valuations run in polynomial timeonly if the computation of valuations is assumed to be effortless (i.e., oracles).
In contrast, in thiswork, we aim at designing truly polynomial-time approximation algorithms without valuation oracles.
Preliminaries
Denote by G =
( V, E ) an undirected graph without reflexive edges, where V contains all verticesand E contains all the edges.
The vertices are the items that are to be allocated to n heterogeneousagents, denoted by N .
Each agent i has an edge weight function w i : E → R + ∪ { 0 } , whichmay be different from others’.
If w i ( e ) ∈ { 0 , 1 } for all e ∈ E , then the weight function is calledbinary.
Let w
=
w 1 , · · · , w n ) .
A matching M ⊆
E is a set of vertex-disjoint edges, and letw i ( M )
= (cid:80) e
∈ M w i ( e ) .
For any subgraph G ′ , let V ( G ′ ) and E ( G ′ ) be the sets of vertices and edgesin G ′ , respectively.
An allocation X =
( X 1 , · · · , X n ) is a partition of V such that ∪ i ∈ N X i
=
Vand X i ∩
X j = ∅ for i ̸
= j .
If ∪ i ∈ N X i ⊊ V , the allocation is called partial .
Each agent i has autility function u
i : 2 V → R + ∪ { 0 } , where u i ( X i ) equals the weight of a maximum (weighted)matching in G [ X i ] .
When the agents have identical valuations (i.e., homogeneous agents), we omitthe subscript and use w ( · ) and u ( · ) to denote all agents’ weight and utility functions.
A probleminstance is denoted by I =
( G, N ) .
When we want to highlight the weight function, w is alsoincluded as a parameter, i.e., I = ( G, N, w ) .
Next we introduce the solution concepts.
Our first fairness notion is maximin share (MMS) [Budish,2011].
Letting Π n ( V ) be the set of all n -partitions of V , the maximin share of agent i is
[Equation]
We may write MMS i for short if I is clear from the context.
Therefore agent i is satisfied regarding
MMS fairness if her utility is no smaller than MMS i .
Definition 2.1 ( α -MMS) .
For any α ≥ 0 , an allocation X =
( X 1 , · · · , X n ) is called α -approximatemaximin share ( α -MMS) fair if for all agents i ∈ N ,
[Equation]
The allocation is called MMS fair if α = 1 .
The second fairness notion is about envy-freeness (EF).
An allocation X is called EF if no agentenvies any other agent’s bundle, i.e.,u i ( X i ) ≥ u i ( X j ) for all agents i, j ∈ N .
We can observe that it is very hard to satisfy EF for an arbitrary instance.
Consider a simple counterexample, where the graph is a triangle and two agents have weight 1 for all edges.
Then in everyallocation, there is one agent who gets at most one vertex (with utility 0) and the other agent getsat least two vertices (which contains an edge and thus has utility 1).
Accordingly, we focus on theenvy-free up to one item instead [Budish, 2011].
Definition 2.2 (EF1) .
An allocation X =
( X 1 , · · · , X n ) is called envy-free up to 1 item (EF1) if forany i and j , there exists g ∈ X
j such that u i ( X i ) ≥ u i ( X j \ { g } ) .
Besides fairness, we also want the allocation to be efficient.
Given an allocation X =
( X 1 , · · · , X n ) ,the social welfare of X is sw ( X ) =
(cid:80) i ∈
N u
i ( X i ) .
Note that given any instance I , the best possiblesocial welfare of any allocation is the weight of a maximum matching in the graph G by setting theweight of each edge to max i ∈ N w i ( e ) , which is denoted by sw ∗ ( I ) .
If the instance I is clear fromthe context, we also denote sw ∗ ( I ) as sw ∗ for short.
Homogeneous Agents
We start with the case of homogeneous agents when the agents have identical valuations.
3.1 MMS Fair Allocations for Homogeneous Agents
With identical valuations, the MMS fairness degenerates to the max-min objective, where the problemis to partition a graph into n subgraphs so that the smallest weight of the maximum matchings inthese subgraphs is maximized.
It is easy to see that finding such an allocation is NP-hard even whenthere are two agents and the graph contains a set of disjoint edges, which is essentially a Partitionproblem.
Therefore, we aim at designing polynomial-time approximation algorithms to achieve the
MMS fair objective.
Without loss of generality, in this section, we assume w ( e ) ≥ 1 for all e ∈ E .
Since the agents have identical valuations, we omit the subscript in MMS i and simply write MMS .
Our main result in this section is as follows.
Theorem 3.1.
We can compute a 1 / 8 -MMS allocation in polynomial time for homogeneous agents.
Before proving the theorem, we explain the intuition of Algorithm 1.
Given an instance I =
( G, N ) ,to ensure the maximum matching in every subset of vertices to be large, we first try to allocate amaximum matching in the original graph.
Specifically, we compute a maximum matching in Gdenoted by M ∗ ⊆
E , and then partition M ∗ into n bundles ( M 1 , · · · , M n ) where w ( M 1 ) ≥ · · · ≥w ( M n ) such that w ( M n ) is as large as possible.
This task is NP-hard and thus we instead use thefollowing simple greedy solution, which we call greedy partition of M ∗ .
Greedy Partition.
Given a matching M , partition M into Γ( M ) =
( M 1 , · · · , M n ) as follows.
• Sort and rename the edges in M such that w ( e 1 ) ≥ · · · ≥ w ( e k ) where k = | M | .
• Initially set M 1 = · · · = M n = ∅ .
• For i = 1 , · · · , k , select j such that w ( M j ) ≤ w ( M j ′ ) for all j ′ and set M j = M j ∪ { e i } .
• Sort and rename M 1 , · · · , M n so that w ( M 1 ) ≥ · · · ≥ w ( M n ) .
The greedy partition of M ∗ corresponds to an allocation of vertices where unmatched vertices
V ′ =
V \ ∪ i ∈ N V
( M i ) can be allocated arbitrarily.
The good news is that such an allocation achieves
MMS fairness when the graph is unweighted, i.e., w ( e ) =
w ( e ′ ) for all e, e ′ ∈ E .
Lemma 3.2.
If G is unweighted, the greedy partition ( M 1 , · · · , M n ) of M ∗ is an MMS allocation.
The bad news is that such an allocation does not have any bounded approximation guarantee whenthe edges have distinct weights.
Consider the following example with two agents and the graph isshown in Figure 1 where ∆ > 1 is arbitrarily large.
Any allocation with bounded approximation ratio [Figure] of MMS fairness ensures that every agent has value 1, but by partitioning the maximum matching(which contains a single edge with weight ∆ ) the smaller bundle has value 0.
However, if | M 1 | ≥ 2 ,such an allocation is 1 / 2 -MMS.
Lemma 3.3.
If | M 1 | ≥ 2 , Γ( M ∗ ) corresponds to an allocation that is 1 / 2 -MMS fair.
The tricky case is when M 1 contains a single edge e ∗ .
To use the approach in Lemma 3.3 to derive1 / 2 -MMS fair, we iteratively decrease the weight of e ∗ and re-compute a maximum matching until| M 1 | ≥ 2 .
For simplicity, assume all edge weights are powers of 2.
This is without much loss ofgenerality which decreases the approximation ratio by at most 1 / 2 .
Lemma 3.4.
Let I =
( G, N, w ) and I ′ =
( G, N, w ′ ) be two instances where I ′ is obtained from
I by rounding all edge weights down to the nearest power of 2.
If ( X 1 , · · · , X n ) is an α -MMSallocation of I ′ , then it is an α/ 2 -MMS allocation of I .
We prove Lemmas 3.2, 3.3, and 3.4 in the appendix.
Now we are ready to describe Algorithm 1.
We first compute a maximum matching M ∗ and its greedy partition Γ( M ∗ ) =
( M 1 , · · · , M n ) suchthat w ( M 1 ) ≥ · · · ≥ w ( M n ) .
If | M 1 | ≥ 2 , combining Lemmas 3.3 and 3.4, we are safe to outputthe corresponding partition of vertices so that the approximation ratio is at least 1 / 4 .
If | M 1 | = 1 ,we consider two cases.
If w ( M n ) ≥ 1 / 2 · w ( M 1 ) , w ( M n ) is still not too small and we can stop thealgorithm with a constant approximation ratio.
However, if w ( M n ) < 1 / 2 · w ( M 1 ) , it means theutility of the smallest bundle is much less than that of the largest bundle.
Then we update the edgeweights: Let H be the edges with weights no smaller than w ( e 1 ) where e 1 is the edge in M 1 , anddecrease their weights to 1 / 2 · w ( e 1 ) .
By repeating the above procedure, eventually we reach anallocation such that w ( M n ) ≥ 1 / 2 · w ( M 1 ) or | M 1 | ≥ 2 . Output w ( e ) =
⌊ log w ( e ) ⌋ .
We are now ready to prove Theorem 3.1.
Proof of Theorem 3.1.
First, we show Algorithm 1 is well-defined and runs in polynomial time.
Every time when the condition of the while loop holds, either the graph has different weights and anallocation is returned or the weights of the heaviest edges are decreased by 1 / 2 k with some k ≥ 1 .
Thus the while loop is executed O (max e ∈ E log w ( e )) rounds.
Next we prove the approximation ratio.
By Lemma 3.4, we only need to consider the instance wherethe edge weights are powers of 2 and show the allocation is 1/4-approximate MMS fair.
Denote by
O =
( O 1 , · · · , O n ) the optimal solution, where u ( O 1 ) ≥ · · · ≥ u ( O n ) and MMS ( I ) =
u ( O n ) .
Thefirst time when we reach the while loop, if w ( M 1 ) ≤ 2 · w ( M n ) ,
[Equation]
where the second inequality holds because M ∗ is a maximum matching in G .
Thus the allocation is1/2-MMS.
If all edges have the same weight, then by Lemma 3.2, the allocation is optimal.
We move into the while loop if w ( M 1 ) > 2 · w ( M n ) and the edge weights are not identical.
Notethat w ( M 1 ) > 2 · w ( M n ) implies M 1 contains a single edge denoted by e 1 .
Otherwise considerthe last edge added to M 1 in the greedy partition, denoted by e ′ .
Then w ( M 1 \ { e ′ } ) ≤ w ( M n )and w ( e ′ ) ≤ w ( M n ) , which implies w ( M 1 ) ≤ 2 · w ( M n ) .
After the while loop, denote by I ′ theinstance, by w ′ ( · ) the new weights with new utility function u ′ ( · ) , by O ′ =
( O ′ 1 , · · · , O ′ n ) the newoptimal solution and by M ′ the maximum matching with greedy partition ( M ′ 1 , · · · , M ′ n ) .
Then wehave the following claim, which is proved in the appendix.
Claim 3.5.
After each while loop, one of the following two cases holds true.
• Case 1.
w ( e 1 ) ≥ 2 · MMS ( I ) , then MMS ( I ′ )
=
MMS ( I ) ;• Case 2.
w ( e 1 ) < 2 · MMS ( I ) , then 2 · MMS ( I ′ ) >
MMS ( I ) and w ′ ( M ′ 1 ) ≤ 2 · w ′ ( M ′ n ) .
By Claim 3.5, the while loop will not execute Case 2 or it executes Case 1 for several times andthen Case 2 for exactly once.
If Case 2 is not executed, then the allocation is 1/2-MMS fair and theanalysis is the same with the case when the while loop is not executed.
If Case 2 is executed once, then by Claim 3.5,
[Equation]
Finally, by Lemma 3.4, the allocation is 1/8-MMS for any instance with arbitrary weights.
Remark.
When n = 2 , we can improve Algorithm 1 and obtain a better approximation ratio of 2/3.
Due to the space limit, we provide the refined algorithm in the appendix.
3.2 Efficient and EF1 Allocations for Homogeneous Agents
Recall the example shown in Figure 1.
The maximum social welfare is sw ∗ =
∆ , but any allocationwith bounded approximation ratio for MMS fairness has social welfare 2 ≪ ∆ , which means toensure MMS, we lose significant amount of efficiency.
Note that the existence of EF1 allocations isguaranteed by the envy-cycle elimination algorithm designed by Lipton et al.
But the socialwelfare of the returned allocation does not have any guarantee.
In this section, we aim at computingan EF1 allocation that also preserves high social welfare.
Theorem 3.6.
For any instance I =
( G, N ) , Algorithm 2 returns an EF1 allocation with socialwelfare at least (2 / 3 + 2 / (9 n − 3))
· sw ∗ ( I ) in polynomial time.
We prove Theorem 3.6 in the appendix, and in the following we briefly discuss the idea of Algorithm 2.
We first introduce the EF1-graph , inspired by the envy-graph introduced in [Lipton et al. , 2004].
Given a (partial) allocation ( X 1 , · · · , X n ) , we construct the corresponding EF1-graph G =
( N, E ) ,where the nodes are agents (and thus are used interchangeably)
and there is a directed edge from i toj if i envies j (or X j ) for more than one item,u i ( X i ) <
u i ( X j \ { v } ) for every v ∈ X j .
When the agents have identical utility functions, we have the following simple observation.
Observation 3.7.
The EF1-graph is acyclic; The in-degree of the agent with smallest utility is zero.
Similar with Algorithm 1, in Algorithm 2, we first compute a maximum weighted matching M ∗and let the corresponding unmatched vertices be V ′ .
If | M ∗ | ≤ n , by allocating each edge in M ∗to a different agent and V ′ to one agent who has the smallest utility is EF1, since by removing avertex from an edge, the remaining subgraph does not have edges any more.
If | M ∗ | > n , we find a i e i =
M i in the greedy-partition procedure
14: for v V ′ do 15:
Let i = arg min i ∈ N u
( X i ) .
Set X i =
i
17: end 18: Return greedy-partition Γ( M ∗ ) =
( M 1 , · · · , M n ) of M ∗ such that w ( M 1 ) ≥ · · · ≥ w ( M n ) .
However, bysimply assigning X i
= V ( M i ) for every i , it may not be EF1, which is illustrated in the appendix.
To overcome this difficulty, we utilize the EF1-graph G =
( N, E ) on the partial allocation( V ( M 1 ) , · · · , V ( M n )) .
Let Q ⊆
N be the set of agents who have positive in-degree, i.e., areenvied by some agent for more than one item.
By Observation 3.7, if G is nonempty, Q ̸ = ∅ andn / ∈ Q .
Moreover, since M n has the smallest weight in the greedy partition Γ( M ∗ ) , n has an edgeto every agent in Q .
We first consider the partial allocation after the for loop in Step 10, which isdenoted by Y =
( Y 1 , · · · , Y n ) .
We can prove that Y is EF1, and moreover, it ensures the desiredsocial welfare guarantee.
Finally, the remaining steps preserve the EF1ness and can only increase thesocial welfare of the allocation.
The formal analysis is deferred to the appendix.
Heterogeneous Agents
In this section, we discuss the general case of heterogeneous agents.
We first show the negativeresults for MMS and EF1 allocations, and then focus on the special cases when we are able to obtainpositive results.
Due to space limit, all the results in this section are proved in the appendix.
4.1 Negative Results for MMS and EF1 Allocations
We present the main theorems below whose proofs are in the appendix.
Theorem 4.1.
No algorithm has bounded approximation guarantee for MMS fairness, even for thecase of two agents with non-identical binary weight functions on the graph.
Theorem 4.2.
No algorithm has better than 1 /n approximation of social welfare for EF1 fairnessfor heterogeneous agents.
Theorem 4.1 is very strong in the sense that it excludes the possibility of designing algorithms withbounded approximation ratio for MMS even for the special cases of two-agent or binary weightfunctions.
However, Theorem 4.2 retains this possibility for EF1, and we design polynomial-timealgorithms to compute EF1 allocations that ensure constant fractions of the maximum social welfarefor these two cases.
In the appendix, we complement Theorem 4.2 with a positive result where wedesign an algorithm that has Ω(1 /n 2 ) approximation guarantee of social welfare for the general case.
4.2 Binary Weight Functions
Input: Instance I =
( G, N, w ) with G =
( V, E ) .
Output: Allocation X =
( X 1 , · · · , X n ) .
1: Initialize X i ← ∅ , i ∈ N .
Let M i be the maximum matching in G [ X i ] for agent i .
Denote by
G ′ =
( N, E ) the envy-graph on X .
2: Let P = V \
( X 1 ∪ · · · ∪ X n ) be the set of unallocated items (called pool ).
3: Partition agents i ∈ N into k groups
A ( X ) =
( A 1 , · · · , A k ) such that agents in the same group have the same value, i.e., u i ( X i ) =
u j ( X j ) for i, j ∈
A l and l ∈ [ k ] .
Assume A l ’s are ordered, i.e., u i ( X i ) <
u j ( X j ) for agents i ∈
A t 1 , j ∈
A t 2 and t 1 < t 2 .
4: Let t ← 1 and τ ← | A | .
5: while { t ≤ τ } do 6: // Case 1.
Directly Allocate 7:
if there exists an agent i ∈
A t such that (1) there is an edge e in G [ P ] with w i ( e ) = 1 and (2) allocating the two endpoints v 1 , v 2 of e to agent i does not break EF1 then 8:
X i ←
X
i ∪ { v 1 , v 2 } , P ← P \ { v 1 , v 2 } .
Update u
i
( X i ) for i ∈ N and the envy-graph G ′ .
Update the partition of agents in A .
Reset t ← 1 and τ ← | A | .
12: // Case 2.
Exchange and Allocate 13: else if there exists agent j ∈ N and i ∈
A t such that j envies i and there exists a subset with minimum size V ∗ ⊆
P in graph G such that u i ( V ∗ ) =
u i ( X i ) then 14:
Let V ∗ ⊆
P be a set with minimum size such that u i ( V ∗ )
=
u i ( X i ) .
Let V ∗ j ⊆
X i be a set with minimum size such that u j ( V ∗ j ) =
u j ( X j ) + 1 .
P ← ( P \ V ∗ ) ∪ X j ∪ ( X i \ V ∗ j ) .
X i ← V ∗ , X j ← V ∗ j .
Update u
i
( X i ) for i ∈ N and the envy-graph G ′ .
Update the partition of agents in A .
Reset t ← 1 and τ ← | A | .
21: else 22: // Case 3.
Skip the Current Agent 23: t ← t + 1 .
24: end if 25: end while 26: Execute the envy-cycle elimination procedure on the remaining items P .
27: Return the allocation ( X 1 , · · · , X n ) .
We first show that if the agents have binary weight functions, we can compute an EF1 allocation whosesocial welfare is at least 1/3 fraction of the optimal social welfare.
Before introducing our algorithm,we recall the envy-cycle elimination algorithm proposed by Lipton et al.
[2004], which always returnsan EF1 allocation.
Given a (partial) allocation ( X 1 , · · · , X n ) , we construct the corresponding envygraph G ′
=
( N, E ) , where the nodes are agents (and thus are used interchangeably)
and there is adirected edge from agent i to agent j if and only if u i ( X i ) <
u i ( X j ) .
The envy-cycle eliminationalgorithm runs as follows.
We first find an agent who is not envied by the others, and allocate a newitem to her.
If there is no such an agent, there must be a cycle in the corresponding envy graph.
Thenwe resolve this cycle by reallocating the bundles: every agent gets the bundle of the agent that sheenvies in the cycle.
We repeat resolving cycles until there is an unenvied agent.
The above procedurescontinue until all the items are allocated.
Note that in the execution of the algorithm, the agents’utilities can only increase, and the returned allocation is EF1.
It is not hard to verify that the envy-cycle elimination algorithm does not have any social welfareguarantee.
There are several reasons.
First, the algorithm does not control which item should beallocated to the unenvied agent so that the agent may receive a set of independent vertices.
Second,once an item is allocated it cannot be recalled so that we are not able to revise any bad decision wehave made.
To increase the social welfare, in each round of our algorithm, we try to allocate an edge(i.e. , two items) to the agent i with the smallest value so that the social welfare can increase by 1.
However, we need to be very careful by allocating two items which may break the EF1 requirementeven if i is not envied by the others.
If allocating an edge e to i makes some agent j envy i for more than one item, we check whether i can maintain her utility by selecting a bundle from unallocateditems.
If so, we execute exchange procedure by asking j to (properly) select a bundle from X i and ito (properly) select a bundle from unallocated items so that the social welfare is increased by 1.
Allthe items in X i and the items in X j that are not selected by i are returned to the algorithm.
If not, wetry to allocate an edge to the agent with the second smallest value by executing the above procedures,and so on.
The description is in Algorithm 3 and we prove the following theorem in the appendix.
Theorem 4.3.
For any instance I =
( G, N ) where agents have binary weights, Algorithm 3 returnsan EF1 allocation with social welfare at least 1 / 3 · sw ∗ ( I ) in polynomial time.
4.3 Two Heterogeneous Agents
We then discuss the case of two agents, and show that Algorithm 4 ensures at least 1/3 fraction ofthe optimal social welfare.
Intuitively, in Algorithm 4, we first check whether there is a single edgee for which some agent i has value at least 1 / 3 · sw ∗ ( I ) .
If so, allocating e to i already ensures1 / 3 · sw ∗ ( I ) .
Moreover, this partial allocation is EF1 since the removal of one item in e results in noedges, and thus we can use the envy-cycle elimination algorithm to allocate the remaining vertices,which returns an EF1 allocation and can only increase the social welfare.
Otherwise, we computea social welfare maximizing allocation ( M 1 , M 2 ) , i.e., u 1 ( M 1 ) + u 2 ( M 2 ) =
sw ∗ ( I ) .
Without lossof generality, assume u 1 ( M 1 ) ≤ u 2 ( M 2 ) .
We temporarily allocate M i to agent i for i = 1 , 2 .
If theallocation is not EF1, since u 1 ( M 1 ) ≤ u 2 ( M 2 ) , it can only be the case that agent 1 envies agent 2but agent 2 does not envy agent 1.
Then we move items in agent 2’s bundle one by one to agent 1.
It can be shown that there must be a time after which the allocation is EF1, and the first time whenthe allocation becomes EF1, the resulting social welfare must be at least 1 / 3 · sw ∗ ( I ) .
Formally, wehave the following theorem.
Interestingly, despite the simplicity of Algorithm 4, we can also showthat there is no algorithm that has better than 1/3 approximation.
Theorem 4.4.
For any instance I with two heterogeneous agents, Algorithm 4 returns an EF1allocation with social welfare at least 1 / 3 · sw ∗ ( I ) .
Moreover, the approximation of 1/3 is optimal.
Conclusion and Future Directions
In this work, we study the fair (and efficient) allocation of graphical resources when the agents’utilities are determined by the weights of the maximum matchings in the obtained subgraphs.
Weprovide a string of algorithmic results regarding MMS and EF1, but also leave some problems open.
For example, for the cases of homogeneous agents and binary valuations, we believe EF1 allocationshave better social welfare guarantee.
It is also interesting to identify hard instances and study theefficiency limit of EF1 allocations.
We can also improve the approximation ratio for the MMSallocation among homogeneous agents.
Our work also uncovers many interesting future directions.
Firstly, regarding MMS, although we show that there is no bounded multiplicative approximation,it may admit good additive or bi-factor approximations.
Secondly, we only focus on the matchinginduced utilities in this work, and it is intriguing to consider other combinatorial structures such asindependent set, network flow and more.
Thirdly, we can extend the framework to the fair allocationof graphical chores when agents have costs to complete the assigned items.