Graphical Resource Allocation with Matching-Induced Utilities
Anonymous Author(s)
AffiliationAddress email
Abstract
Motivated by the real-world applications, we study the fair allocation of graphicalresources, where the resources are the vertices in a graph.
Upon receiving a setof resources, an agent’s utility equals the weight of the maximum matching inthe induced subgraph.
We care both maximin share (MMS) fairness and envyfreeness up to one item (EF1).
Regarding MMS, the problem does not admit 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 constant fraction 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.
However, in many real-world deployment of (social) network-based resource allocationmanagement (e g., Graph Neural Networks
[Ying et al. , 2019], Knowledge Graphs [Sadeghian etal. , 2019], Clustering [Pan and Kang, 2021] and more), the resources are graphical. To illustrateour model, consider the following real-world scenario.
When a department in some company getsassigned a number of engineers who need to work in pairs, the utility of the department may dependon the way to pair these engineers. Due to the engineers’ diverse expertise and personalities, anengineer can make different contribution by pairing with different people. We can use a graphto describe the above problem where the engineers are the vertices and the weight on the edgebetween two vertices represents the collective contribution by pairing these two engineers.
Thus thedepartment’s utility on a set of vertices is decided by the maximum matching in the induced subgraphof these vertices. To create a better working environment, we want to fairly allocate the engineers todifferent departments.
Similar pairwise joint work also appears as long-trip coach driver vs co-driverand accountant vs cashier. In this work, we formalize this scenario as a graphical resource allocationproblem, and explore the extent to which we can compute fair allocations with high social welfare.
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,the graph is used to characterize feasible allocations (such as the resources allocated to each agent should be connected), but the agents still have additive utilities over allocated items.
With graphicalresources, 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 ourproblem.
This problem 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]that partition a graph when the subgraphs are evaluated by maximum matchings, these algorithmsdo not have theoretical guarantees.
Our first fairness criterion is the maximin share (MMS) fairnessproposed by Budish [2011], which generalizes the max-min objective in Santa Claus problem [Bansaland Sviridenko, 2006].
Informally, the MMS value of an agent is her best guarantee in an imaginaryexperiment when she got the opportunity to partition the graph into several subgraphs but was thelast to select a subgraph.
We aim at designing efficient algorithms with provable approximationguarantees.
As will be clear later, to achieve (approximate) MMS fairness, significant amount ofsocial welfare has to be inevitably sacrificed.
Our second fairness notion is envy-freeness (EF)
[Foley, 1967].
In an EF allocation, no agent prefersthe allocation of another agent to her own.
Since the resources are indivisible, such an allocationbarely exists, and recent research in fair division focuses on achieving its relaxations instead.
Oneof the most widely accepted and studies relaxations is envy-freeness up to one item (EF1) [Budish,2011], which requires the envy to be eliminated after removing one item.
Lipton et al.
[2004] provedthat an EF1 allocation always exists even with combinatorial valuations.
It is noted that an arbitrary EF1 allocation may have low social welfare, and our goal is to compute an EF1 allocation whichpreserves a large fraction of the maximum social welfare without fairness constraints.
The socialwelfare loss by enforcing the allocations 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 two-agent case, the approximation ratio is the best possible.
1.2 Related Works
Two separate research lines are closely related to our work, namely graph partition and fair 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 weight of 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 ) or I =
( G, N, w ) if the weight function is highlighted.
Next we introduce the solution concepts that are studied in this work.
Our first fairness notion ismaximin share (MMS) [Budish, 2011].
Let Π n ( V ) be the set of all n -partitions of V . For each agenti ∈ N , consider the following imaginary experiment where i can partition all the items into n bundlesbut is the last one to select a bundle. Then her best strategy is to maximize the value of the smallestbundle. Formally, 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 agent i ∈ N ,
[Equation]
The allocation is called MMS fairness 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, people mostlystudy the relaxations of EF in the literature. Among the various ways to relax the requirement of EF,envy-free up to one item (EF1) is one of the most widely accepted and studied notions.
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.
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 ratioof 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. [Figure]
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 ∗ .
The greedy partition fails because w ( e ∗ ) istoo large so that very few edges or edges with very small weights can be put in M
i ’s for i ≥ 2 . Oneway to overcome this difficulty is to decrease the weight of e ∗ and re-compute a maximum matching,through which the advantage of e ∗ can diminish.
For simplicity, assume all edge weights are powersof 2.
This is without much loss of generality 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 , then we are safe to output the corresponding partitionof vertices so that the approximation ratio is at least 1 / 4 .
If | M 1 | = 1 , we consider two cases.
Ifw ( M n ) ≥ 1 / 2 · w ( M 1 ) , w ( M n ) is still not too small and we can stop the algorithm with a decentapproximation ratio.
However, if w ( M n ) < 1 / 2 · w ( M 1 ) , it means the smallest bundle is far smallerthan the largest bundle.
Then we update the edge weights: Let H be the edges with weights no smallerthan w ( e 1 ) where e 1 is the edge in M 1 , and decrease their weights to 1 / 2 · w ( e 1 ) .
By repeating theabove procedure, eventually we reach an allocation such that w ( M n ) ≥ 1 / 2 · w ( M 1 ) or | M 1 | ≥ 2 . 1 be the in M
∈ w
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 ) .
The first 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 happens.
• 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 execute Case 2 once following several timesof Case 1, and return an allocation.
If Case 2 is not executed, then the allocation is 1/2-approximate MMS fair and the analysis 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 any 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 Let X V ( M n ) ∪ V ′ . 7: end
9: Set 10: for i ∈ do 11: Let e i = ( v i 1 , v i 2 ) be the last edge added to M i in the greedy-partition procedure.
V = V 13: end for 14: for v ∈ V ′ do 15:
Let i = arg min i ∈ N u
( X i ) .
Set X i =
X i ∪ { v } .
17: end for 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 binary weight functions on the graph.
Theorem 4.2.
No algorithm has better than 1 /n approximation of social welfare for EF1 fairness.
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.
Move and Allocate 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 anedge (i.e., two items) to the agent i with 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 envies i for more than one item, we check whether i can maintain her utility by selecting a bundle from unallocateditems.
If so, we do exchange by asking j to (properly) select a bundle from X i and i to (properly)select a bundle from unallocated items so that the social welfare is increased by 1.
All the items in X iand the items in X j that are not selected by i are returned to the algorithm.
If not, we try to allocatean edge to the agent with second smallest value by executing the above procedures, and so on so forth.
We formally describe the algorithm in Algorithm 3 and prove the following theorem in the appendix.
Theorem 4.3.
For any instance I =
( G, N ) , Algorithm 3 returns an EF1 allocation with socialwelfare 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.
Future Directions
In this work, we study the fair (and efficient) allocation of graphical resources when the agent utilitiesare evaluated by the weights of the maximum matchings in the obtained subgraphs.
We providea string of algorithmic results regarding MMS and EF1, but also leave some problems open.
Forexample, for the cases of homogeneous agents and binary valuations, we believe EF1 allocations havebetter social welfare guarantee.
It is also interesting to identify hard instances and study the efficiencylimit of EF1 allocations.
We can also improve the 1/8 approximation ratio for the MMS allocationamong homogeneous agents.
Our work also uncovers some interesting future directions.
Firstly,regarding MMS, although we show that there is no bounded multiplicative approximation, it mayadmit good additive or bi-factor approximations.
Secondly, we only focused on the matching-inducedutilities in this work, and it is intriguing to consider other combinatorial structures such as independentset, network flow and more.
Thirdly, we can extend the framework to the fair allocation of graphicalchores when agents have costs to complete the assigned items.