Tradeoffs Between Information and Ordinal Approximation for Bipartite Matching
TTradeoffs Between Information and Ordinal Approximation forBipartite Matching ∗ Elliot Anshelevich [email protected]
Wennan Zhu [email protected]
Rensselaer Polytechnic Institute110 8th Street, Troy NY 12180
Abstract
We study ordinal approximation algorithms for maximum-weight bipartite matchings. Suchalgorithms only know the ordinal preferences of the agents/nodes in the graph for their preferredmatches, but must compete with fully omniscient algorithms which know the true numericaledge weights (utilities). Ordinal approximation is all about being able to produce good resultswith only limited information. Because of this, one important question is how much better thealgorithms can be as the amount of information increases. To address this question for forminghigh-utility matchings between agents in X and Y , we consider three ordinal information types:when we know the preference order of only nodes in X for nodes in Y , when we know thepreferences of both X and Y , and when we know the total order of the edge weights in theentire graph, although not the weights themselves. We also consider settings where only the toppreferences of the agents are known to us, instead of their full preference orderings. We designnew ordinal approximation algorithms for each of these settings, and quantify how well suchalgorithms perform as the amount of information given to them increases. Many important settings involve agents with preferences for different outcomes. Such settingsinclude, for example, social choice and matching problems. Although the quality of an outcometo an agent may be measured by a numerical utility, it is often not possible to obtain these exactutilities when forming a solution. This can occur because eliciting numerical information fromthe agents may be too difficult, the agents may not want to reveal this information, or evenbecause the agents themselves do not know the exact numerical values. On the other hand, eliciting ordinal information (i.e., the preference ordering of each agent over the outcomes) is often muchmore reasonable. Because of this, there has been a lot of recent work on ordinal approximationalgorithms : these are algorithms which only use ordinal preference information as their input, andyet return a solution provably close to the optimum one (e.g., [3–5, 9–12, 17]). In other words, theseare algorithms which only use limited ordinal information, and yet can compete in the qualityof solution produced with omniscient algorithms which know the true (possibly latent) numericalutility information.Ordinal approximation is all about being able to produce good results with only limited in-formation. Because of this, it is important to quantify how well algorithms can perform as more ∗ This work was partially supported by NSF award CCF-1527497. a r X i v : . [ c s . G T ] J u l nformation is given. If the quality of solutions returned by ordinal algorithms greatly improveswhen they are provided more information, then it may be worthwhile to spend a lot of resourcesin order to acquire such more detailed information. If, on the other hand, the improvement issmall, then such an acquisition of more detailed information would not be worth it. Thus the mainquestion we consider in this paper is: How does the quality of ordinal algorithms improve as theamount of information provided increases?
In this paper, we specifically consider this question in the context of computing a maximum-utility matching in a metric space. Matching problems, in which agents have preferences for whichother agents they want to be matched with, are ubiquitous. The maximum-weight metric matchingproblem specifically provides solutions to important applications, such as forming diverse teamsand matching in friendship networks (see [4, 5] for much more discussion of this). Formally, thereexists a complete undirected bipartite graph for two sets of agents X and Y of size N , with an edgeweight w ( x, y ) representing how much utility x ∈ X and y ∈ Y derive from their match; these edgeweights satisfy the triangle inequality. The algorithms we consider, however, do not have access tosuch numerical edge weights: they are only given ordinal information about the agent preferences.The goal is to form a perfect matching between X and Y , in order to approximate the maximumweight matching as much as possible using only the given ordinal information. We compare theweight of the matching returned by our algorithms with the true maximum-weight perfect matchingin order to quantify the performance of our ordinal algorithms. Types of Ordinal Information
Ordinal approximation algorithms for maximum weight match-ing have been considered before in [4,5], although only for complete graphs; algorithms for bipartitegraphs require somewhat different techniques. Our main contribution, however, lies in consideringmany types of ordinal information, forming different algorithms for each, and quantifying how muchbetter types of ordinal information improve the quality of the matching formed. Specifically, weconsider the following types of ordinal information. • The most restrictive model we consider is one-sided preferences . That is, only preferences foragents in X over agents in Y are given to our algorithm. These preferences are assumed tobe consistent with the (hidden) agent utilities, i.e., if x prefers y to y , then it must be that w ( x, y ) ≥ w ( x, y ). Such one-sided preferences may occur, for example, when X representspeople and Y represents houses. People have preferences over different houses, but houses donot have preferences over people. These types of preferences also apply to settings in whichboth sides have preferences, but we only have access to the preferences of X , e.g., becausethe agents in Y are more secretive. • The next level of ordinal information we consider is two-sided preferences , that is, both pref-erences for agents in X over Y and agents in Y over X are given. This setting could applyto the situation that two sets of people are collaborating, and they have preferences overeach other, or of a matching between job applicants and possible employers. As we considerthe model in a metric space, the distance (weight) between two people could represent thediversity of their skills, and a person prefers someone with most diverse skills from him/herin order to achieve the best results of collaboration. • The most informative model which we consider in this paper is that of total-order . That is, theorder of all the edges in the bipartite graph is given to us, instead of only local preferences foreach agent. In this model, global ordinal information is available, compared to the preferencesof each agent in the previous two models. Studying this setting quantifies how much efficiencyis lost due to the fact that we only know ordinal information, as opposed to the fact that weonly know local information given to us by each agent.2omparing the results for the above three information types allows us to answer questions like:“Is it worth trying to obtain two-sided preference information or total order information whenonly given one-sided preferences?” However, above we always assumed that for an agent x , we aregiven their entire preferences for all the agents in Y . Often, however, an agent would not givetheir preference ordering for all the agents they could match with, and instead would only give anordered list of their top preferences. Because of this, in addition to the three models describedabove, we also consider the case of partial ordinal preferences, in which only the top α fraction ofa preference list is given by each agent of X . Thus for α = 0 no information at all is given to us,and for α = 1 the full preference ordering of an agent is given. Considering partial preferences tellsus when, if there is a cost to buying information, we might choose to buy only part of the ordinalpreferences. We establish tradeoffs between the percentage of available preferences and the possibleapproximation ratio for all three models of information above, and thus quantify when a specificamount of ordinal information is enough to form a high-quality matching. Our Contributions
We show that as we obtain more ordinal information about the agentpreferences, we are able to form better approximations to the maximum-utility matching, evenwithout knowing the true numerical edge weights. Our main results are shown in Figure 1.Figure 1: α vs. approximation ratio for par-tial information. As we obtain more infor-mation about the agent preferences ( α in-creases), we are able to form better approx-imation to the maximum-weight matching.The tradeoff for one-sided preferences is lin-ear, while it is more complex for two-sidedand total order.Using only one-sided preference information, with only the order of top αN preferences givenfor agents in X , we are able to form a (3 − (2 − √ α )-approximation. We do this by combiningrandom serial dictatorship with purely random matchings. When α = 1, the algorithm yieldsa ( √ RSD onmaximum bipartite matching in a metric space, and this analysis is one of our main contributions.Given two-sided information, with the order of top αN preferences for agents in both X and Y ,we can do significantly better. When α ≥ , adopting an existing framework in [4], by mixing greedyand random algorithms, and adjusting it for bipartite graphs, we get a (3 − α )(3 − α )2 α − α +3 -approximation.When α ≤ , the framework would still work, but would not produce a good approximation. Weinstead design a different algorithm to get better results. Inspired by RSD , we take advantageof the information of preferences from both sets of agents, adjust
RSD to obtain “undominated”edges in each step, and finally combine it with random matchings to get a (3 − α )-approximation.When α ≥ , the algorithm yields a 1 . αN heaviest edges in the bipartite graph is given.We use the framework in [4] again to obtain a √ − α −√ − α -approximation. Here we must re-design the3ramework to deal with the cases that α ≤ N , which is not a straight-forward adjustment. When α ≥ N the algorithm yields a -approximation.Finally, in Section 6 we analyze the case when edge weights cannot be too different: the highestweight edge is at most β times the lowest weight edge in one-sided model. When the edge weightshave this relationship, we can extend our analysis to give a ( (cid:113) β − + )-approximation, evenwithout assuming that edge weights form a metric. Discussion and Related Work
Previous work on forming good matchings can largely beclassified into the following classes. First, there is a large body of work assuming that numericalweights or utilities don’t exist, only ordinal preferences. Such work studies many possible objectives,such as forming stable matchings (see e.g., [15,16]), or maximizing objectives determined only by theordinal preferences (e.g., [2, 8]). Second, there is work assuming that numerical utilities or weightsexist, and are known to the matching designer. Unlike the above two settings, we consider thecase when numerical weights exist , but are latent or unknown , and yet the goal is to approximatethe true social welfare, i.e., maximum weight of a perfect matching. Note that although someprevious work assumes that all numerical utilities are known, they often still use algorithms whichonly require ordinal information, and thus fit into our framework; we discuss some of these resultsbelow.Similar to our one-sided model, house allocation [1] is a popular model of assigning n agents to n items. [6] studied the ordinal welfare factor and the linear welfare factor of RSD and other ordinalalgorithms. [14] studied both maximum matching and maximum vertex weight matching using anextended RSD algorithm. These either used objectives depending only on ordinal preferences,such as the size of the matching formed, or used node weights (as opposed to edge weights).[11] and [9] assumed the presence of numerical agent utilities and studied the properties of RSD.Crucially, this work assumed normalized agent utilities, such as unit-sum or unit-range. Thisallowed [9, 11] to prove approximation ratios of Θ( √ n ) for RSD. Instead of assuming that agentutilities are normalized, we consider agents in a metric space; this different correlation betweenagent utilities allows us to prove much stronger results, including a constant approximation ratiofor RSD. Kalyanasundaram et al. studied serial dictatorship for maximum weight matching in ametric space [13], and gave a 3-approximation for RSD in this, while we are able to get a tighterbound of 2.41-approximation. Besides maximizing social welfare, minimizing the social cost of a matching is also popular. [7]studied the approximation ratio of RSD and augmentation of serial dictatorship (SD) for minimumweight matching in a metric space. Their setting is very similar to ours, except that we considerthe maximization problem, which has different applications [4, 5], and allows for a much betterapproximation factor (constant instead of linear in n ) using different techniques.Another area studying ordinal approximation algorithms is social choice, where the goal is todecide a single winner in order to maximize the total social welfare. This is especially related toour work when the hidden utilities of voters are in a metric space (see e.g., [3, 10, 12, 17]),The work most related to ours is [4, 5]. As mentioned above, we use an existing framework [4]for the two-sided and the total-order model. While the goal is the same: to approximate themaximum weight matching using ordinal information, this paper is different from [4] in severalaspects. [4] only considered approximating the true maximum weight matching for non-bipartitecomplete graphs. We instead focus on bipartite graphs, and especially on considering differentlevels of ordinal information by analyzing three models with increasing amount of information, and Note that many of the papers mentioned here specifically attempt to form truthful algorithms. While RSD iscertainly truthful, in this paper we attempt to quantify what can be done using ordinal information in the presenceof latent numerical utilities, and leave questions of truthfulness to future work.
For all the problems studied in this paper, we are given as input two sets of agents X and Y with |X | = |Y| = N . G = ( X , Y , E ) is an undirected complete bipartite graph with weights on theedges. We assume that the agent preferences are derived from a set of underlying hidden edgeweights w ( x, y ) for each edge ( x, y ), x ∈ X , y ∈ Y . w ( x, y ) represents the utility of the matchbetween x and y , so if x prefers y to y , then it must be that w ( x, y ) ≥ w ( x, y ). Let OP T ( G )denote the complete bipartite matching that gives the maximum total edge weights. w ( G ) of anybipartite graph G is the total edge weight of the graph, and w ( M ) of any matching M is the totalweight of edges in the matching. The agents lie in a metric space, by which we will only mean that, ∀ x , x ∈ X , ∀ y , y ∈ Y , w ( x , y ) ≤ w ( x , y ) + w ( x , y ) + w ( x , y ). We assume this property inall sections except for Section 6.For the setting of one-sided preferences, ∀ x ∈ X , we are given a strict preference ordering P x over the agents in Y . When dealing with partial preferences, only top αN agents in P x are given tous in order. We assume αN is an integer, α ∈ [0 , α = 0, nothing can be doneexcept to form a completely random matching. For two-sided partial preferences, we are given boththe top α fraction of preferences P x of agents x in X over those in Y , and vice versa. For the totalorder setting, we are given the order of the highest-weight αN edges in the complete bipartitegraph G = ( X , Y , E ). For one-sided preferences, our problem becomes essentially a house allocation problem to maximizesocial welfare, see e.g., [9, 11, 14]. Before we proceed, it is useful to establish a baseline for whatapproximation factor is reasonable. Simply picking a matching uniformly at random immediatelyresults in a 3-approximation (see Theorem 2), and there are examples showing that this bound istight. Other well-known algorithms, such as Top Trading Cycle, also cannot produce better than a3-approximation to the maximum weight matching for our setting. Serial Dictatorship, which usesonly one-sided ordinal information, is also known to give a 3-approximation to the maximum weightmatching for our problem [13]. Serial Dictatorship simply takes an arbitrary agent from x ∈ X ,assigns it x ’s favorite unallocated agent from Y , and repeats. Unfortunately, it is not difficult toshow that this bound of 3 is tight. Our first major result in this paper is to prove that Random
Serial Dictatorship always gives a ( √ lgorithm 1: Random Serial Dictatorship for Perfect Matching of one-sided ordering.Initialize M = ∅ , G = ( X , Y , E ) ; while E (cid:54) = ∅ do Pick an agent x uniformly at random from X ;Let y denote x ’s most preferred agent in Y ;Take e = ( x, y ) from E and add it to M ;Remove x , y , and all edges containing x or y from the graph G ; endFinal Output: Return M . Theorem 1.
Suppose G = ( X , Y , E ) is a complete bipartite graph on the set of nodes X , Y with |X | = |Y| = N . Then, the expected weight of the perfect matching M returned by Algorithm 1 is E [ w ( M )] ≥ √ w ( OP T ( G )) .Proof. Notation : Consider a bipartite subgraph S ⊆ G , that satisfies S = ( X (cid:48) , Y (cid:48) , E (cid:48) ) , X (cid:48) ⊆X , Y (cid:48) ⊆ Y , and |X (cid:48) | = |Y (cid:48) | . Let M in ( S ) denote a minimum weight perfect matching on S , and RSD ( S ) denote the expected weight returned by Algorithm 1 on graph S .For any x ∈ X (cid:48) , we use λ ( S, x ) to denote the edge between x and its most preferred agent in Y (cid:48) . Define R ( S, x ) as the remaining graph after removing x , x ’s most preferred agent, and all theedges containing x or x ’s most preferred agent from S .We begin by simply expressing RSD ( S ) in terms of these quantities. Lemma 1.
For any subgraph S as decribed above, RSD ( S ) = |X (cid:48) | (cid:80) x ∈X (cid:48) w ( λ ( S, x )) + |X (cid:48) | (cid:80) x ∈X (cid:48) RSD ( R ( S, x )) .Proof. This simply follows from definition of expectation. In the first round of Algorithm 1, anagent x is selected uniformly at random from X (cid:48) . Given that x is selected, the edge added to thematching is exactly λ ( S, x ), and the expected weight of the matching for the remaining graph isexactly
RSD ( R ( S, x )). Each of these occurs with probability 1 / |X (cid:48) | .We now state the main technical lemma which allows us to prove the result. This lemma givesa bound on the maximum weight matching in terms of the quantities defined above. Lemma 2.
For any given graph G = ( X , Y , E ) , one of the following two cases must be true: Case 1 : w ( OP T ( G )) ≤ |X | (cid:80) x ∈X w ( OP T ( R ( x ))) + √ |X | (cid:80) x ∈X w ( λ ( x )) Case 2 : w ( OP T ( G )) ≤ ( √ w ( M in ( G ))We will prove this lemma below, but first we discuss how the rest of the proof will proceed.When Case 1 above holds, we know that at any step of the algorithm, the change in the weight of theoptimum solution in the remaining graph is not that different from the weight of the edge selectedby our algorithm. This allows us to compare the weight of OP T with the weight of the matchingreturned by our algorithm. In fact, this is the technique used in a previous paper [5] to analyzeRSD for complete graphs (i.e., non-bipartite graphs), and show that RSD gives a 2-approximationfor perfect matching on complete graphs. Similar to Case 1 in Lemma 2, this was done by provingthat in each step, the expected loss of optimal matching is at most twice the expected weight ofthe chosen edge, and thus the emtire algorithm gives a 2-approximation.It is important to note here that this does not work for bipartite graphs. In bipartite matching,using only this method will not give an approximation ratio better than 3. To see this, consider thebipartite graph in Figure 2. Suppose G = ( X , Y , E ) is a complete bipartite graph, |X | = |Y| = N .6he edges shown in the Figure are the maximum weight matching of G ; all the other edges haveweight of 1. It is easy to see that these edge weights form a metric. ∀ x ∈ X , x ’s most preferredagent in Y is y , second preferred agent is y , ..., least preferred agent is y n (we can always perturbthe edge weights by an infinitesimal amount to remove ties for this example). Then the weight ofthe optimum solution is w ( OP T ( G )) = ( N −
1) + 3. In this example, the expected decrease in theweight of the optimal matching in the first step of RSD is 3: choosing x loses 3, and choosing anyother agent x i in X loses 3 since ( x , y ) and ( x i , y i ) can no longer ne used (decrease of 4), butthe edge ( x , y i ) can be used (increase of 1). On the other hand, the expected weight of the edgechosen by RSD is N − N . In this case, almost 3 times the expected weight of the chosen edge isneeded to compensate for the loss of optimal matching, so the inequality in “Case 1” above onlyholds if we replace √ √ √ Proposition 1.
As long as Lemma 2 is obeyed for every S , Algorithm 1 provides a ( √ -approximation to the Maximum weight perfect matching: RSD ( G ) ≥ √ w ( OP T ( G )) .Proof. We proceed by induction. Clearly when G only has two agents, RSD produces the op-timum matching. Now consider a bipartite graph G = ( X , Y , E ) with |X | = |Y| = N , andsuppose that the claim is true for all smaller graphs, i.e., ∀ x ∈ X , we know that RSD ( R ( G, x )) ≥ √ w ( OP T ( R ( G, x ))).If Case 2 in Lemma 2 holds for G , then because M in ( G ) is the minimum weight perfect match-ing, we know that w ( M in ( G )) ≤ RSD ( G ). So RSD ( G ) ≥ √ w ( OP T ( G )). Otherwise Case 1 inLemma 2 must be true. w ( OP T ( G )) ≤ N (cid:88) x ∈X w ( OP T ( R ( G, x ))) + √ N (cid:88) x ∈X w ( λ ( G, x ))7y our assumption, w ( OP T ( G )) ≤ √ N (cid:88) x ∈X RSD ( R ( G, x )) + √ N (cid:88) x ∈X w ( λ ( G, x ))This completes the proof by Lemma 1.We now proceed with the main technical part of the proof, i.e., the proof of Lemma 2.
Proof of Lemma 2
For compactness of notation, since S is fixed, we will omit S and simplywrite λ ( x ) and R ( x ) instead of λ ( S, x ) and R ( S, x ). For any fixed x ∈ X (cid:48) , denote x ’s most preferredagent in Y (cid:48) as y (so λ ( x ) = ( x, y )). In OP T ( S ), suppose x is matched to b ∈ Y (cid:48) , and y is matchedto a ∈ X (cid:48) . In M in ( S ), suppose b is matched to m ∈ X (cid:48) . ∀ x ∈ X (cid:48) , there exist y , a , b , m as describedabove. As shown in Figure 3, denote edge ( x, y ) by λ ( x ), ( x, b ) by P ( x ), ( a, y ) by ¯ P ( x ), and ( a, b )by D ( x ). Figure 3: Notation of λ ( x ), P ( x ), ¯ P ( x ), D ( x ).We’ll prove Lemma 2 by showing that if Case 2 is not true, then
Case 1 must be true. Suppose
Case 2 is not true, i.e., w ( OP T ( S )) > ( √ w ( M in ( S )).Suppose that random serial dictatorship picks x ∈ X (cid:48) . Then OP T ( R ( S, x )) is at least as goodas the matching obtained by removing P ( x ) and ¯ P ( x ), and adding D ( x ) to OP T ( S ) (the rest staythe same): w ( OP T ( R ( x ))) ≥ w ( OP T ( S )) − w ( P ( x )) − w ( ¯ P ( x )) + w ( D ( x ))Note that when λ ( x ) ∈ OP T ( S ), ¯ P ( x ) = P ( x ) = D ( x ), and the inequality still holds. Summingthis up over all nodes x , we obtain:1 |X (cid:48) | (cid:88) x ∈X (cid:48) w ( OP T ( R ( x ))) ≥ |X (cid:48) | (cid:88) x ∈X (cid:48) ( w ( OP T ( S )) − w ( P ( x )) − w ( ¯ P ( x )) + w ( D ( x )))= w ( OP T ( S )) − |X (cid:48) | (cid:88) x ∈X (cid:48) ( w ( P ( x )) + w ( ¯ P ( x )) − w ( D ( x )))= (1 − |X (cid:48) | ) w ( OP T ( S )) − |X (cid:48) | (cid:88) x ∈X (cid:48) ( w ( ¯ P ( x )) − w ( D ( x ))) (1)In Figure 3, by the triangle inequality, we know that w ( a, y ) ≤ w ( a, b ) + w ( m, b ) + w ( m, y )8ote that when y = b the inequality still holds, because w ( a, y ) = w ( a, b ). It also holds when a = m for the same reason.Because λ ( m ) is the edge to m ’s most preferred agent, w ( m, y ) ≤ w ( λ ( m )), and thus w ( ¯ P ( x )) ≤ w ( D ( x )) + w ( m, b ) + w ( λ ( m )))Summing this up for all x ∈ X (cid:48) , note that each x is matched to a unique b in OP T ( S ), andeach b is matched to a unique m in M in ( S ), so each agent in Y (cid:48) appears as b exactly once and eachagent in X (cid:48) appears as m exactly once. (cid:88) x ∈X (cid:48) w ( ¯ P ( x )) ≤ (cid:88) x ∈X (cid:48) w ( D ( x )) + w ( M in ( S )) + (cid:88) x ∈X (cid:48) w ( λ ( x ))) (cid:88) x ∈X (cid:48) ( w ( ¯ P ( x )) − w ( D ( x ))) ≤ w ( M in ( S )) + (cid:88) x ∈X (cid:48) w ( λ ( x ))) (2)Combining Inequality 1 and Inequality 2,1 |X (cid:48) | (cid:88) x ∈X (cid:48) w ( OP T ( R ( x ))) ≥ (1 − |X (cid:48) | ) w ( OP T ( S )) − |X (cid:48) | [ w ( M in ( S )) + (cid:88) x ∈X (cid:48) w ( λ ( x ))] (3) ∀ x ∈ X (cid:48) , w ( P ( x )) ≤ w ( λ ( x )) since λ ( x ) is the most preferred edge of x , so it is obvious that w ( OP T ( S )) ≤ (cid:80) x ∈X (cid:48) w ( λ ( x )).By our assumption, w ( M in ( S )) < √ w ( OP T ( S )) ≤ √ (cid:88) x ∈X (cid:48) w ( λ ( x ))Thus, putting this together with Inequality 3, we obtain that,1 |X (cid:48) | (cid:88) x ∈X (cid:48) w ( OP T ( R ( x ))) ≥ w ( OP T ( S )) − |X (cid:48) | (2 + 1 √ (cid:88) x ∈X (cid:48) w ( λ ( x )))= w ( OP T ( S )) − √ |X (cid:48) | (cid:88) x ∈X (cid:48) w ( λ ( x )) Partial One-sided Ordinal Preferences
In this section, we consider the case when we are given even less information than in the previ-ous one, i.e., only partial preferences. We begin by establishing the following easy result for thecompletely random algorithm.
Algorithm 8:
Random Algorithm for Perfect Bipartite Matching.Initialize M = ∅ , G = ( X , Y , E ) ; while E (cid:54) = ∅ do Pick an edge e = ( x, y ) from E uniformly at random and add it to M ;Remove x , y , and all edges containing x or y from G ; endFinal Output: Return M . 9 emma 3. Suppose G = ( X , Y , E ) is a complete bipartite graph on the set of nodes X , Y with |X | = |Y| = N . Then, the expected weight of the random perfect matching returned by Algorithm 8for the input G is E [ w ( M )] = N (cid:80) ( x,y ) ∈ E w ( x, y ) . This lemma was proved in [4].
Theorem 2.
The uniformly random perfect matching is a 3-approximation to the maximum-weightmatching.Proof.
Suppose G = ( X , Y , E ) is a complete bipartite graph on the set of nodes X , Y ⊆ N with |X | = |Y| = N . Let OP T be the optimal perfect matching. Suppose ( x, y ) is an edge in
OP T .Then for any edge ( a, b ) ∈ E , by triangle inequality, w ( x, y ) ≤ w ( x, b ) + w ( a, y ) + w ( a, b )Summing up for all ( a, b ) ∈ E , N w ( x, y ) ≤ N (cid:88) b ∈Y w ( x, b ) + N (cid:88) a ∈X w ( a, y ) + (cid:88) ( a,b ) ∈ E w ( a, b )Summing up for all ( x, y ) ∈ OP T , N w ( OP T ) ≤ N (cid:88) ( a,b ) ∈ E w ( a, b ) + N (cid:88) ( a,b ) ∈ E w ( a, b ) + N (cid:88) ( a,b ) ∈ E w (( a, b ) = 3 N (cid:88) ( a,b ) ∈ E w ( a, b )Let M be the matching returned by Algorithm 8. Then, by Lemma 3, E [ w ( M )] = 1 N (cid:88) ( a,b ) ∈ E w ( a, b ) ≥ w ( OP T ) Algorithm 2:
Algorithm for Perfect Matching given partial one-sided ordering.Run Algorithm 1, stop when | M | = αN , then form random matches until all agents arematched. Return M . Theorem 3.
Suppose G = ( X , Y , E ) is a complete bipartite graph on the set of nodes X , Y with |X | = |Y| = N . There is a strict preference ordering P x over the agents in Y for each agent x ∈ X .We are only given top αN agents in P x in order. Then, the expected weight of the perfect matching M returned by Algorithm 2 is E [ w ( M )] ≥ − (2 −√ α w ( OP T ( G )) , as shown in Figure 1.Proof. We use the same notation as in the proof of Theorem 1. We would like to apply our maintechnical result (Lemma 2) to analyze this algorithm. Define
Alg i ( S ) as the expected weight ofchosen edge in round i of RSD on any subgraph S . For any bipartite graph S , let Rand ( S ) denotethe expected weight of the perfect matching returned by Algorithm 8, and Avg ( S ) denote theaverage weight of edges in S .We begin by bounding w ( OP T ( G )) by the sum of expected weights of chosen edges in RSD,and the weight of the remaining subgraph. 10 emma 4. Let L ( G, (cid:96) ) be the subgraph of G after (cid:96) rounds of RSD, which has N − (cid:96) nodes both in X and Y . Note that L ( G, (cid:96) ) is a random variable. Then we have that: w ( OP T ( G )) ≤ ( √ (cid:96) (cid:88) i =1 Alg i ( G ) + 3 E [ Rand ( L ( G, (cid:96) ))]
Proof.
We prove this by induction on (cid:96) . For the Base Case, when (cid:96) = 0, then this simply reducesto Theorem 3. Now assume by the inductive hypothesis that, ∀ x ∈ X , w ( OP T ( R ( G, x ))) ≤ ( √ (cid:96) − (cid:88) i =1 Alg i ( R ( G, x )) + 3 E [ Rand ( L ( R ( G, x ) , (cid:96) − G , then w ( OP T ( G )) ≤ √ |X | (cid:88) x ∈X w ( λ ( G, x )) + 1 |X | (cid:88) x ∈X w ( OP T ( R ( G, x ))) ≤ ( √ (cid:96) (cid:88) i =1 Alg i ( G ) + 3 E [ Rand ( L ( G, (cid:96) ))]The last inequality is simply because of the inductive hypothesis, and the fact that E [ Rand ( L ( G, (cid:96) ))] = |X | (cid:80) x ∈X E [ Rand ( L ( R ( G, x ) , (cid:96) − G , then w ( OP T ( G )) ≤ ( √ w ( M in ( G ))Let’s consider a perfect matching on G generated by running RSD for (cid:96) rounds, and then obtainingthe minimum weight matching for the remaining subgraph. By the definition of M in ( G ), the weightof the matching described above is no less than M in ( G ): w ( OP T ( G )) ≤ ( √ w ( M in ( G )) ≤ ( √ (cid:96) (cid:88) i =1 Alg i ( G ) + ( √ E [ w ( M in ( L ( G, (cid:96) )))] ≤ ( √ (cid:96) (cid:88) i =1 Alg i ( G ) + ( √ E [ Rand ( L ( G, (cid:96) ))] ≤ ( √ (cid:96) (cid:88) i =1 Alg i ( G ) + 3 E [ Rand ( L ( G, (cid:96) ))]To finish the proof of the theorem, we need to be able to compare
Rand ( L ( G, (cid:96) )) and
Alg i .After all, if the random part of our matching is much larger in weight than the RSD part, then therandom part will dominate, resulting in only a 3 approximation. Fortunately, it is not hard to seethe following lemma. Let G (cid:48) = L ( G, αN ) be a random variable representing the graph obtained byrunning RSD on G for αN rounds, which we can always do if we are given the top αN preferencesof every agent. Lemma 5. ∀ i ≤ αN , Alg i ( G ) is heavier than the expected average edge weight in G (cid:48) , i.e., Alg i ( G ) ≥ E [ Avg ( G (cid:48) ] . roof. First notice that
Alg ( G ) ≥ Alg ( G ). This is true because: Alg ( G ) = 1 |X | (cid:88) x ∈X |X | − (cid:88) y ∈X − x w ( λ ( R ( G, x ) , y )) ≤ |X | (cid:88) x ∈X |X | − (cid:88) y ∈X − x w ( λ ( G, y ))= |X | − |X | ( |X | − (cid:88) y ∈X w ( λ ( G, y ))=
Alg ( G )The inequality above is simply because the best edge leaving y in a smaller graph R ( G, x ) isat most the best edge leaving it in a larger graph G . By the same argument, we know that Alg i ( G ) ≥ Alg i +1 ( G ) for all i .Now consider an arbitrary complete graph S = ( X (cid:48) , Y (cid:48) , E (cid:48) ) with |X (cid:48) | = |Y (cid:48) | . One way to thinkof Avg ( S ) is as an expected value of the following randomized algorithm: take a node x in X (cid:48) uniformly at random, and then take a random edge leaving that node, and return its weight. Theexpected value returned by this algorithm is exactly the expected weight of an edge in S takenuniformly at random, i.e., exactly Avg ( S ). Compare this algorithm with the performance of RSD;RSD does exactly the same thing in the first round, but chooses the best edge coming out of x instead of a random edge. Therefore, the first round of RSD on any graph always performs betterthan the average edge weight. In particular, this is true for every instantiation of the graph G (cid:48) ,and thus Alg αN +1 ( G ) ≥ E [ Avg ( G (cid:48) )]. This concludes the proof.Finally, let’s finish the proof of Theorem 3. By Lemma 4, w ( OP T ( G )) ≤ ( √ αN (cid:88) i =1 Alg i ( G ) + 3 E [ Rand ( G (cid:48) )]= ( √ αN (cid:88) i =1 Alg i ( G ) + 3(1 − α ) N × E [ Avg ( G (cid:48) )] , By Lemma 5, αN (cid:88) i =1 Alg i ( G ) ≥ αN × E [ Avg ( G (cid:48) )] , and thus, w ( OP T ( G )) ≤ (3 − (2 − √ α )( αN (cid:88) i =1 Alg i ( G ) + (1 − α ) N × E [ Avg ( G (cid:48) )])= (3 − (2 − √ α )( αN (cid:88) i =1 Alg i ( G ) + E [ Rand ( G (cid:48) )])Note that (cid:80) αNi =1 Alg i ( G ) + E [ Rand ( G (cid:48) )] is the expected weight of M , which completes the proof: w ( OP T ( G )) ≤ (3 − (2 − √ α ) E [ w ( M )] . Two-sided Ordinal Preferences
For two-sided preferences, we give separate algorithms for the cases when α ≥ and when α ≤ ,as these require somewhat different techniques. α ≥ While for the case when α < new techniques are necessary to obtain a good approx-imation, the approach for the case when α ≥ is essentially the same as the one used in [4].We adopt this approach to deal with bipartite graphs and with partial preferences, giving us a1.8-approximation for α = 1. To do this, we re-state the definition of Undominated Edges from [4],and a standard greedy algorithm for forming a matching of size k . Definition 1. (Undominated Edges) Given a set E of edges, ( x, y ) ∈ E is said to be an undominatededge if for all ( x, a ) and ( y, b ) in E , w ( x, y ) ≥ w ( x, a ) and w ( x, y ) ≥ w ( y, b ) . Note that an undominated edge must always exist: either there are two nodes x and y such thatthey are each other’s top preferences (and so ( x, y ) is undominated), or there is a cycle x , x , . . . inwhich x i +1 is the top preference of x i , in which case all edges in the cycle must be the same weight,and thus all edges in the cycle are undominated. This also gives us an algorithm for determining ifan edge ( x, y ) is undominated: either x and y prefer each other over all other agents, or it is partof such a cycle of top preferences. Lemma 6.
Given an edge set E of a complete bipartite graph G = ( X , Y , E ) , the weight of anyundominated edge is at least one third as much as the weight of any other edge in E , i.e., if e = ( x, y ) is an undominated edge in E , that x ∈ X , y ∈ Y , then for any ( a, b ) ∈ E , a ∈ X , b ∈ Y , w ( x, y ) ≥ w ( a, b ) .Proof. Since e = ( x, y ) is an undominated edge, w ( x, y ) ≥ w ( x, b ) and w ( x, y ) ≥ w ( a, y ). Bytriangle inequality, we know that w ( a, b ) ≤ w ( x, y ) + w ( x, b ) + w ( a, y ) ≤ w ( x, y ). Algorithm 3:
Greedy Algorithm for Max k -Matching of two-sided ordering.Initialize M = ∅ , E is the valid set of edges initialized to the complete bipartite graph G ; while E (cid:54) = ∅ do Pick an undominated edge e = ( x, y ) from E and add it to M ;Remove x , y , and all edges containing x or y from E ; if | M | = k then break ; endendFinal Output: Return M .Before stating the full algorithm for the case when α ≥ , we mention two lemmas which willbe useful to establish its approximation ratio. These lemmas are essentially the same as the similarones from [4], except that we must adjust all the factors to deal with bipartite graphs, while [4]considered only non-bipartite graphs. The basic analysis techniques remain the same, however, andwe only provide proofs of these lemmas for completeness.13 emma 7. Suppose G = ( X , Y , E ) is a complete bipartite graph on the set of nodes X , Y with |X | = |Y| = N . Given k = γN , the performance of the greedy k -matching returned by Algorithm 3with respect to the optimal perfect matching OPT is given by − γγ .Proof. The analysis here is essentially identical to that of a similar lemma in [4], except thatLemma 6 gives a ratio of 3 instead of 2 between any edge and an undominated edge for bipartitegraphs. We include the whole analysis of the framework for completeness.Let M be the greedy k -matching, and M ∗ be the optimal perfect matching. We show the claimby charging every edge in M ∗ to one or more edges in the greedy matching M . Consider any edge e ∗ = ( a, b ) in M ∗ , the edge must belong to one of the following two types.1. (Type I) Some edges consisting of a or b (both a and b ) are present in M .2. (Type II) No edge in M has a or b as an endpoint.Suppose that M ∗ contains m Type I edges, and m Type II edges. We know that m + m = N .Let T ⊂ M denote the heaviest m edges in M . Initialize U as all the edges in M . We describeour charging algorithm in three phases.(First Phase) We can charge all Type I edges in M ∗ to the edges in T , so that (cid:80) e ∈ T s e w e ≥ (cid:80) e ∈ T ypeI ( M ∗ ) w e , s e ≤
2. We charge the edges as follows: Repeat until U contains no Type I edge:pick a type I edge e ∗ = ( a, b ) from U . Suppose that e = ( a, c ) is the first edge containing either aor b that was added to M , Since w e ≥ w ∗ e , charge e ∗ to e , increase s e by one and remove e ∗ fromU. In the end, all the edges that are charged in M have s e ≤
2, and (cid:80) e s e = m . We can transferthe slots to the heaviest m edges in M , each has s e ≤
2. Keep transfering the slots to the heaviest m µ edges in M , so that each edge has s e ≤ µ .(Second Phase) Repeat until s e = µ for all e ∈ M \ T or until U is empty: pick any arbitraryedge e ∗ from U and the smallest edge e ∈ M \ T such that s e < µ By Lemma 6 w e ∗ ≤ w e , charge e ∗ using three slots of e , transfer slots to the heaviest edges e ∈ M \ T such that s e < µ . So e ∗ ischarged by three slot from edges in M \ T .At the end of the second phase, | U | = max(0 , m − ( k − m µ ) × µ ).(Third Phase) Repeat until U is empty: pick any arbitrary edge e ∗ from U . Since w ∗ e ≤ w e forall e ∈ M , charge e ∗ uniformly to all edges in M , i.e., increase s e by k for every e ∈ M and remove e ∗ from U .At the end of the third phase, for every e ∈ M , s e ≤ µ + 3 k max(0 , m − ( k − m µ ) × µ m + m = N , s e ≤ max( µ, m + Nk )Type II edges don’t share nodes with any of the k edges in M , so m + k ≤ N , s e ≤ max( µ, N − kk ) s e ≤ max ( µ, − γγ )14et µ = − γγ , s e ≤ − γγ Lemma 8.
Let G T = ( X T , Y T , E T ) be a complete bipartite subgraph on the set of nodes X T ⊆ X , Y T ⊆ Y , with |X T | = |Y T | = n , and let M be any perfect matching on G = ( X , Y , E ) . Then, thefollowing is an upper bound on the weight of M , nw ( M ) ≤ (2 + Nn ) (cid:88) x ∈X T y ∈Y T w ( x, y ) + (cid:88) x ∈X T y ∈Y\Y T w ( x, y ) + (cid:88) x ∈X \X T y ∈Y T w ( x, y ) Proof.
For e = ( x, y ) ∈ M, e (cid:48) = ( a, b ) ∈ E T , by triangle inequality, w ( a, y ) + w ( a, b ) + w ( x, b ) ≥ w ( x, y )Sum up for all ( a, b ) ∈ E T , n (cid:88) a ∈X T w ( a, y ) + (cid:88) a ∈X T b ∈Y T w ( a, b ) + n (cid:88) b ∈Y T w ( x, b ) ≥ n w ( x, y )Sum up for all ( x, y ) ∈ M , n (cid:88) a ∈X T y ∈Y w ( a, y ) + N (cid:88) a ∈X T b ∈Y T w ( a, b ) + n (cid:88) b ∈Y T x ∈X w ( x, b ) ≥ n w ( M )Because Y = Y T ∪ {Y\Y T } , and X = X T ∪ {X \X T } , n ( (cid:88) a ∈X T y ∈Y T w ( a, y ) + (cid:88) a ∈X T y ∈Y\Y T w ( a, y )) + N (cid:88) a ∈X T b ∈Y T w ( a, b ) + n ( (cid:88) b ∈Y T x ∈X T w ( x, b ) + (cid:88) b ∈Y T x ∈X \X T w ( x, b )) ≥ n w ( M )Replace a with x , and b with y ,2 n (cid:88) x ∈X T y ∈Y T w ( x, y ) + n (cid:88) x ∈X T y ∈Y\Y T w ( x, y ) + n (cid:88) x ∈X \X T y ∈Y T w ( x, y ) + N (cid:88) x ∈X T y ∈Y T w ( x, y ) ≥ n w ( M ) nw ( M ) ≤ (2 + Nn ) (cid:88) x ∈X T y ∈Y T w ( x, y ) + (cid:88) x ∈X T y ∈Y\Y T w ( x, y ) + (cid:88) x ∈X \X T y ∈Y T w ( x, y )We can now state the algorithm for α ≥ . The algorithm is a mix of greedy and randomalgorithms: for graph G = ( X , Y , E ), given top αN of P ( X ) and top αN of P ( Y ), run Algorithm 3on k = αN , to obtain the matching M . This is possible using the preference we are given.One method we could do at this point is to form a random matching on the rest of the agents.However, this will not form a good approximation, as there are examples when all the high-weightedges are between nodes matched in M and nodes which are unmatched. Another method isto randomly choose some matched nodes from M , make then unmatched, and form a random15ipartite matching between all the agents which were not matched in M , and the nodes whichwe chose from M to become unmatched. This second method is likely to add high-weight edgesbetween nodes in M and nodes outside of it to our matching. Mixing over these two methodsactually returns a high-weight matching in expectation.Note that for α > this algorithm does not seem to provide better guarantees than for α = .Because of this, for α > , we simply run the same algorithm for α = Algorithm 4:
Algorithm for two-sided matching with partial ordinal information ( ≤ α ≤ ). Input : X , Y , top αN of P ( X ), top αN of P ( Y ) Output:
Perfect Bipartite Matching MInitialize E to be complete bipartite graph on X , Y , and M = M = ∅ ;Let M be the output returned by Algorithm 3 for E , k = αN ;Let X T be the set of nodes in X matched in M , Y T be the set of nodes in Y matched in M ,and T be the complete bipartite graph on X T , Y T ;Let X B = X \X T , Y B = Y\Y T , and B be the complete bipartite graph on X B , Y B ; First Algorithm ; M = M ∪ (The perfect matching output by Algorithm 8 on B ); Second Algorithm ;Choose (2 α − N edges from M uniformly at random and add them to M ;Let X A be the set of nodes in X T and not in M , Y A be the set of nodes in Y T and not in M ;Let E AB be the edges of the complete bipartite graph ( X A , Y B ) and E (cid:48) AB be the edges of thecomplete bipartite graph ( X B , Y A ) ;Run random bipartite matching on the set of edges in E AB and E (cid:48) AB separately to obtainperfect bipartite matchings and add the edges returned by the algorithm to M ; Final Output:
Return M with probability − α − α and M with probability α − α .Note that for α > this algorithm does not seem to provide better guarantees than for α = .Because of this, for α > , we simply run the same algorithm for α = . Theorem 4.
Algorithm 4 returns a (3 − α )(3 − α )2 α − α +3 -approximation to the maximum-weight perfectmatching given two-sided ordering when ≤ α ≤ .Proof. |X T | = |Y T | = αN , |X B | = |Y B | = (1 − α ) N .By Lemma 7, w ( M ) ≥ α − α OP T . By Lemma 3, the perfect matching output by Algorithm 8 on B has expected weight at least − α ) N w ( B ). Therefore, E [ w ( M )] ≥ α − α OP T + 1(1 − α ) N w ( B )Because | X A | = | Y A | = (1 − α ) N , and they are leftover nodes after (2 α − N nodes are chosenuniformly at random from M , E [ w ( E AB ) + w ( E (cid:48) AB )] = 1 − αα w ( T, B ) . Recall that w ( T, B ) is the total weight of all edges between T and B . Let M AB be a random16ipartite matching formed on edges E AB and E (cid:48) AB . By Lemma 3, E [ w ( M AB )] = 1(1 − α ) N E [ w ( E AB )] + 1(1 − α ) N E [ w ( E (cid:48) AB )]= 1(1 − α ) N E [ w ( E AB ) + w ( E (cid:48) AB )]= 1 αN w ( T, B )By Lemma 8, with M = OP T, T = B, n = (1 − α ) N :(1 − α ) N w ( OP T ) ≤ (2 + 11 − α ) w ( B ) + w ( T, B ) E [ w ( M AB )] = 1 αN w ( T, B ) ≥ αN ((1 − α ) N w ( OP T ) − − α − α w ( B )) M contains α − α fraction of edges randomly chosen from M , together with M AB : E [ w ( M )] = 2 α − α × α − α w ( OP T ) + E [ w ( M AB )] ≥ α − − α w ( OP T ) + 1 αN ((1 − α ) N w ( OP T ) − − α − α w ( B ))= 4 α − α + 3 α (3 − α ) w ( OP T ) − − αα (1 − α ) N w ( B )Return M with probability − α − α and M with probability α − α . Then, the expected weight ofour final matching is3 − α − α E [ w ( M )] + α − α E [ w ( M )] ≥ α − α + 3(3 − α )(3 − α ) w ( OP T ) . α ≤ Unlike the case for α ≥ , this case requires different techniques than in [4]. While thetechniques above would still work, they will not give us a bound as good as the one we form below.The idea in this section is to do something similar to our one-sided algorithm for partial preferences:run the greedy algorithm for a while, and then switch to random. Unfortunately, if we simply runthe greedy Algorithm 3 and then switch to random, this will not form a good approximation.The reason why this is true is that an undominated edge which is picked by the greedy algorithmmay be much worse than the average weight of an edge, and so the approximation factor of therandom algorithm will dominate, giving only a 3-approximation. Even taking an undominated edgeuniformly at random has this problem. We can fix this, however, by picking each undominated edgewith an appropriate probability, as described below. Such an algorithm results in matchings whichare guaranteed to be better than either RSD or Random, thus allowing us to prove the result.This algorithm guarantees that an undominated edge is chosen for any x in any bipartite graph G . Now, before we reach an undominated edge, the weights of edges are non-decreasing in the17 lgorithm 5: Algorithm for two-sided matching with partial ordinal information (0 ≤ α ≤ ). Input : X , Y , top αN of P ( X ) and P ( Y )Initialize M = ∅ , G = ( X , Y , E ) ; while E (cid:54) = ∅ do Pick an agent x uniformly at random from X ;Let y denote x ’s most preferred agent in Y ; x ← x , y ← y , c ← y ; while ( x , y ) is not an undominated edge doif c = y then x ← y ’s most preferred agent in X ; c ← x ; else y ← x ’s most preferred agent in Y ; c ← y ; endend Take ( x , y ) from E and add it to M ;Remove x , y , and all edges containing x or y from the graph G ; if | M | = αN then break; endend Run Algorithm 8 for the remaining graph G , add the edges returned by the algorithm to M . Final Output:
Return M .order they are checked. Thus whenever a node x is picked, the algorithm adds an undominatededge ( x , y ) to the matching which is guaranteed to have higher weight than all edges leaving x .Note that it is not possible to apply this algorithm to one-sided matching because the preferencesof agents in Y are not given, and thus we cannot detect which edges are undominated. Theorem 5.
Algorithm 5 returns a (3 − α ) -approximation to the maximum-weight perfect matchinggiven two-sided ordering when ≤ α ≤ .Proof. We use a similar method and the same notation as in Section 3 to proof this theorem.Essentially, because we are always picking undominated edges, we can form a linear interpolationbetween a factor of 2 and a factor of 3 for random matching, instead of between factors √ x ∈ X (cid:48) , and end up with an undominated edge ( x , y ). Let λ D ( S, x ) denotethe undominated edge picked by the algorithm for x in graph S , λ D ( S, x ) = ( x , y ) = λ ( S, x )in this case. And let R D ( S, x ) denote the remaining graph after removing λ D ( S, x ) and the edgesconnected to both vertexes of λ D ( S, x ). 18e start with a lemma to bound the maximum weight matching,
Lemma 9.
For any given subgraph S = ( X (cid:48) , Y (cid:48) , E (cid:48) ) , w ( OP T ( S )) ≤ |X (cid:48) | (cid:80) x ∈X (cid:48) w ( OP T ( R D ( S, x )))+ |X (cid:48) | (cid:80) x ∈X (cid:48) w ( λ D ( S, x )) .Proof. Using the same notation as in the proofs of Theorems 1 and 3, suppose that Algorithm 5 picks x ∈ X (cid:48) , and end up with an undominated edge ( x , y ). Then OP T ( R D ( S, x )) =
OP T ( R ( S, x ))is at least as good as the matching obtained by removing P ( x ) and ¯ P ( x ), and adding D ( x ) to OP T ( S ) (the rest stay the same): w ( OP T ( R D ( S, x ))) ≥ w ( OP T ( S )) − w ( P ( x )) − w ( ¯ P ( x )) + w ( D ( x )) ≥ w ( OP T ( S )) − w ( P ( x )) − w ( ¯ P ( x ))Because λ D ( S, x ) is an undominated edge, w ( λ D ( S, x )) ≥ P ( x ), w ( λ D ( S, x )) ≥ ¯ P ( x ), w ( OP T ( R D ( S, x ))) ≥ w ( OP T ( S )) − w ( λ D ( S, x ))Summing up for all x in X (cid:48) , (cid:88) x ∈X (cid:48) w ( OP T ( R D ( S, x ))) ≥ |X (cid:48) | w ( OP T ( S )) − (cid:88) x ∈X (cid:48) w ( λ D ( S, x )) w ( OP T ( S )) ≤ |X (cid:48) | (cid:88) x ∈X (cid:48) w ( OP T ( R D ( S, x ))) + 2 |X (cid:48) | (cid:88) x ∈X (cid:48) w ( λ D ( S, x )) . Then we bound w ( OP T ( G )) by the sum of expected weights of chosen edges in Algorithm 5,and the weight of the remaining subgraph. We still use Alg i ( S ) as the expected weight of chosenedge in round i , but note that for any x , the chosen edge is λ D ( G, x ) instead of λ ( G, x ) as inTheorem 3. By an identical argument as in our Lemma 4, we have that the following holds: w ( OP T ( G )) ≤ (cid:96) (cid:88) i =1 Alg i ( G ) + 3 E [ Rand ( L ( G, (cid:96) ))] . We need to prove that a version of Lemma 5 still holds for Algorithm 5, as the edges are chosendifferently from RSD in each step. In other words, we need to be able to compare
Rand ( L ( G, (cid:96) ))and
Alg i . This is where we need to use the fact that each undominated edge is carefully chosen witha specific probability. Let G (cid:48) = L ( G, αN ) be a random variable representing the graph obtainedby running our greedy algorithm on G for αN rounds, which we can always do if we are given thetop αN preferences of every agent. Lemma 10. ∀ i ≤ αN , Alg i ( G ) is heavier than the expected average edge weight in G (cid:48) , i.e., Alg i ( G ) ≥ E [ Avg ( G (cid:48) ] .Proof. We must show that
Alg ( G ) ≥ Alg ( G ). To see this, Alg ( G ) = 1 |X | (cid:88) x ∈X |X | − (cid:88) y ∈X − λ D ( G,x ) w ( λ D ( R D ( G, x ) , y )) ≤ |X | (cid:88) x ∈X |X | − (cid:88) y ∈X − λ D ( G,x ) w ( λ D ( G, y ))19he inequality above is because the undominated edge found after selecting y and then followingthe agents’ top preferences in a smaller graph R D ( G, x ) is at most that in a larger graph G .Fix some x ∈ X , and let ( x , y ) be the edge λ D ( G, x ) be the edge added to the matching if x is picked by our algorithm, and thus x is the node removed from X . Note that for the case when x (cid:54) = x , we still have that w ( λ D ( G, x )) = w ( λ D ( G, x )), since if x is picked by our algorithm, thenthe undominated edge next to it ( x , y ) is immediately returned. Therefore, in the sum above,we can replace w ( λ D ( G, x )) (since x still remains in X − λ D ( G, x )) with w ( λ D ( G, x )), and thusequivalently make the sum be over X − x instead of over X − λ D ( G, x ). Alg ( G ) = ≤ |X | (cid:88) x ∈X |X | − (cid:88) y ∈X − x w ( λ D ( G, y ))= |X | − |X | ( |X | − (cid:88) y ∈X w ( λ D ( G, y ))=
Alg ( G )By the same argument, we know that Alg i ( G ) ≥ Alg i +1 ( G ) for all i .All that is left is to compare Alg αN +1 ( G ) with E [ Avg ( G (cid:48) )]. We know that the first round of RSDon any graph always performs better than the average edge weight. And for every x that is chosenuniformly at random in the first step of Algorithm 5, the weight of final chosen edge λ D ( x ) is nosmaller than λ ( x ). Therefore, the expected weight of chosen edge in the first round of Algorithm 5 isno smaller than that of RSD, thus better than the average edge weight, Alg αN +1 ( G ) ≥ E [ Avg ( G (cid:48) )].This concludes the proof.Finally, to finish the proof of Theorem 5. Similarly to the proof of Theorem 3, it is easyto show that there is a linear tradeoff from 3 to 2-approximation for α = 0 to α = 1, whichgives w ( OP T ( G )) ≤ (3 − α ) E [ w ( M )], in which M is a random variable representing the matchingreturned by Algorithm 5. For the setting in which we are given the top αN edges of G in order, we prove that for α = ,we can obtain an approximation of in expectation. For larger α , however, more information doesnot seem to help, and so we simply use the algorithm for α = for any α > . Algorithm 6:
Greedy Algorithm for Max k -Matching given the total ordering of edge weights.Initialize M = ∅ , E is the valid set of edges initialized to the complete bipartite graph G ; while E (cid:54) = ∅ do Pick the heaviest edge e = ( x, y ) from E and add it to M ;Remove x , y , and all edges containing x or y from E ; if | M | = k then break ; endendFinal Output: Return M . 20 emma 11. Suppose G = ( X , Y , E ) is a complete bipartite graph on the set of nodes X , Y with |X | = |Y| = N . Given k = γN , the performance of the greedy k-matching returned by Algorithm 6with respect to the optimal perfect matching OPT is at least γ , for γ ≤ .Proof. Let M be the matching returned by Algorithm 3 for k = N . From Lemma 7, w ( M ) ≥ w ( OP T ). In the proof of Lemma 7, each edge in M is charged at most twice by edges of OPT,and there are N charges in total. Transfer all the charges to the highest weight N edges in M ;this tells us that the highest weight N edges of M are at least w ( OP T ). Further transfer all thecharges to the highest weight γN edges in M ; this results in each such edge being charged to 1 /γ times by edges of OPT. Therefore, the highest weight γN edges of M are at least γ w ( OP T ) intotal.Same as Algorithm 3, Algorithm 6 also picks an undominated edge each round; the difference isthe edges in the matching are picked in non-decreasing order. So Algorithm 6 returns a k -matchingwith the same weight as the highest γN edges in the perfect matching returned by Algorithm 3 onthe same graph, which gives at least γ w ( OP T ). Lemma 12.
Suppose G = ( X , Y , E ) is a complete bipartite graph on the set of nodes X , Y with |X | = |Y| = N . Given the order of the top αN edges in the graph, we are able to run greedyk-matching by Algorithm 6 for k = (1 − √ − α ) N .Proof. In the first step of Algorithm 6, the heaviest edge is taken, and 2 N − N − αN edges. After the first k steps of Algorithm 6,the total number of removed edges is:2 N − N − − ... + 2( N − ( k − −
1= 2( N + N − ... + N − ( k − − k = 2 N k − k Given the order of top αN edges, we are able to run Algorithm 6 for at least k steps until2 N k − k = αN . Solve the equation for k , k = (1 − √ − α ) N .The algorithm for bipartite matching with partial ordinal information is similar to that withpartial two-sided ordinal information, except that we only need to consider the case that k ≤ N ,i.e., 1 − √ − α ≤ , α ≤ . In two-sided model, we are given the top αN preferences for both setsof agents, and able to run greedy algorithm for k = αN . While in total ordering model, we couldonly run greedy algorithm for k = (1 − √ − α ) N given the order of the top αN edges. Differentfrom two-sided model, α does not equal to the number of agent pairs we are able to match bygreedy algorithm in total ordering model. 21 lgorithm 7: Algorithm for matching given partial total ordering.
Input : X , Y , order of the top αN edges in the graph. Output:
Perfect Bipartite Matching MInitialize E to be complete bipartite graph on X , Y , and M = M = ∅ ;Let M be the output returned by Algorithm 6 for E , k = (1 − √ − α ) N . Let α = 1 − √ − α , then k = α N ;Let X T be the set of nodes in X matched in M , Y T be the set of nodes in Y matched in M ,and T be the complete bipartite graph on X T , Y T ;Let X B be the set of nodes in X not matched in M , Y B be the set of nodes in Y notmatched in M , and B is the complete bipartite graph on X B , Y B ; First Algorithm ; M = M ∪ (The perfect matching output by Algorithm 8 on B ); Second Algorithm ;Choose (1 − α ) N nodes both from X B and Y B uniformly at random, get the perfectmatching output by Algorithm 8 on these nodes and add the results to M ;Let X A be the set of nodes in X B and not in M , Y A be the set of nodes in Y B and not in M ;Let E AT be the edges of the complete bipartite graph ( X A , Y T ) and E (cid:48) AT be the edges of thecomplete bipartite graph ( X T , Y A ) ;Run random bipartite matching on the set of edges in E AT and E (cid:48) AT separately to obtainperfect bipartite matchings and add the edges returned by the algorithm to M ; Final Output:
Return M with probability √ − α and M with probability √ − α √ − α . Theorem 6.
Algorithm 7 returns a √ − α −√ − α -approximation to the maximum-weight matching inexpectation for α ≤ , as shown in Figure 1.Proof. By Lemma 12, we are able to run Algorithm 6 for k = (1 − √ − α ) N . We analyze thealgorithm when α ≤ , α = 1 − √ − α ≤ . |X T | = |Y T | = α N , |X B | = |Y B | = (1 − α ) N .By Lemma 11, w ( M ) ≥ α w ( OP T ). By Lemma 3, the perfect matching output by Algorithm 8on B has expected weight − α ) N w ( B ). Thus, E [ w ( M )] ≥ α w ( OP T ) + 1(1 − α ) N w ( B )Analysis of E [ w ( M )] is very similar to the case when α ≥ for Algorithm 4, except that now B is larger than T , and so we form a random bipartite matching using all of the nodes in T insteadof just some of them. Formally, because | X A | = | Y A | = α N , and they are leftover nodes after(1 − α ) N nodes are chosen uniformly at random from B , we know that E [ w ( E AT ) + w ( E (cid:48) AT )] = α − α w ( T, B ) . Let M AT be the random bipartite matching formed between sets A and T . By Lemma 3, E [ w ( M AT )] = 1 α N E [ w ( E AT )] + 1 α N E [ w ( E (cid:48) AT )]= 1(1 − α ) N w ( T, B )22y Lemma 8, setting M = OP T, T = B, n = (1 − α ) N ,(1 − α ) N w ( OP T ) ≤ (2 + 11 − α ) w ( B ) + w ( T, B ) . Thus, E [ w ( M AT )] = 1(1 − α ) N w ( T, B ) ≥ − α ) N ((1 − α ) N w ( OP T ) − − α − α w ( B )) E [ w ( M )] = 1 − α − α × − α ) N w ( B ) + E [ w ( M AT )] ≥ − α (1 − α ) N w ( B ) + 1(1 − α ) N ((1 − α ) N w ( OP T ) − − α − α w ( B ))= w ( OP T ) − − α ) N w ( B )Return M with probability − α = √ − α , and M with probability − α − α = √ − α √ − α ,23 − α E [ w ( M )] + 1 − α − α E [ w ( M )] ≥ α − α w ( OP T )= 2 − √ − α √ − α w ( OP T ) In previous sections, we made the assumption that the agents lie in a metric space, and thus theedge weights, although unknown to us, must follow the triangle inequality. In this section we onceagain consider the most restrictive type of agent preferences — that of one-sided preferences — butnow instead of assuming that agents lie in a metric space, we instead consider settings where edgesweights cannot be infinitely different from each other. This applies to settings where the agentsare at least somewhat indifferent and the items are somewhat similar; the least-preferred agent andthe most-preferred items differ only by a constant factor to any agent. Indeed, when for examplepurchasing a house in a reasonable market (i.e., once houses that almost no one would buy havebeen removed from consideration), it is unlikely that any agent would like house x so much morethan house y that they would be willing to pay hundreds of times more for x than for y .More formally, for each agent i ∈ X , we are given a strict preference ordering P i over the agentsin Y . In this section we assume that the highest weight edge e max is at most β times of the lowestweight edge e min . We normalize the lowest weight edge e min in the graph to w ( e min ) = 1; then forany edge e ∈ E , w ( e ) ≤ β . We use similar analysis as in Section 3, except that instead of gettingbounds by using the triangle inequality, the relationships among edge weights are bounded by ourassumption of the highest and lowest weight edge ratio. As stated above, we no longer assume theagents lie in a metric space in this section. 23 heorem 7. Suppose G = ( X , Y , E ) is a complete bipartite graph on the set of nodes X , Y with |X | = |Y| = N . w ( e min ) = 1 , ∀ e ∈ E , w ( e ) ≤ β . The expected weight of the perfect matchingreturned by Algorithm 1 is w ( M ) ≥ (cid:113) β − + w ( OP T ) . Figure 4: β vs. approximation ratio of RSD on restricted weight bipartite graph. For edges with asmall difference in weight, we still obtain a reasonable approximation to the optimum matching. Proof.
We use the same notation as in Section 3. Once again, our proof relies on the followingclaim, similar to Lemma 2. Once the statement below is proven, the rest of the proof proceedsexactly as in Theorem 1, simply replacing √ (cid:113) β − + . Lemma 13.
For any given subgraph S = ( X (cid:48) , Y (cid:48) , E (cid:48) ) , one of the following two cases must be true: Case 1 , w ( OP T ( S )) ≤ |X (cid:48) | (cid:80) x ∈X (cid:48) w ( OP T ( R ( S, x ))) + (cid:113) β − + |X (cid:48) | (cid:80) x ∈X (cid:48) w ( λ ( x )) Case 2 , w ( OP T ( S )) ≤ ( (cid:113) β − + ) w ( M in ( S )) Proof.
Again, we use the same notation as in Section 3.We’ll prove Lemma 13 by showing that if
Case 2 is not true, then
Case 1 must be true.Suppose
Case 2 is not true, w ( OP T ( S )) > ( (cid:113) β − + ) w ( M in ( S )).Suppose that random serial dictatorship picks x ∈ X (cid:48) . Just as in the proof of Lemma 2, weobtain that1 |X (cid:48) | (cid:88) x ∈X (cid:48) w ( OP T ( R ( x ))) ≥ (1 − |X (cid:48) | ) w ( OP T ( S )) − |X (cid:48) | (cid:88) x ∈X (cid:48) ( w ( ¯ P ( x )) − w ( D ( x ))) (4)24e know that ∀ e ∈ E (cid:48) , 1 ≤ w ( e ) ≤ β . So w ( D ( x )) ≥ w ( ¯ P ( x )) ≤ β , and thus1 |X (cid:48) | (cid:88) x ∈X (cid:48) ( w ( ¯ P ( x )) − w ( D ( x ))) ≤ β − ∀ x ∈ X (cid:48) , w ( P ( x )) ≤ w ( λ ( x )), so it is obvious that w ( OP T ( S )) ≤ (cid:80) x ∈X (cid:48) w ( λ ( x )). M in ( S ) is a perfect matching, so w ( M in ( S )) ≥ |X (cid:48) | . By our assumption, |X (cid:48) | ≤ w ( M in ( S )) < (cid:113) β − + w ( OP T ( S )) (6)Combining Inequalities 4, 5, and 6,1 |X (cid:48) | (cid:88) x ∈X (cid:48) w ( OP T ( R ( x ))) ≥ w ( OP T ( S )) − |X (cid:48) | w ( OP T ( S )) − |X (cid:48) | ( β − |X (cid:48) |≥ w ( OP T ( S )) − |X (cid:48) | w ( OP T ( S )) − |X (cid:48) | ( β −
1) 1 (cid:113) β − + w ( OP T ( S ))= w ( OP T ( S )) − |X (cid:48) | (1 + β − (cid:113) β − + ) w ( OP T ( S ))= w ( OP T ( S )) − (cid:113) β − + |X (cid:48) | w ( OP T ( S )) ≥ w ( OP T ( S )) − (cid:113) β − + |X (cid:48) | (cid:88) x ∈X (cid:48) w ( λ ( x ))This completes the proof of the theorem. In this section, we provide some example to study the lower bound of algorithms on maximumweight bipartite graph perfect matching, given two-sided or one-sided ordinal information.
Example
Consider a bipartite graph G = ( X , Y , E ), X = { a, b } , Y = { c, d } . Let (cid:15) be a verysmall positive number. Consider two sets of weight assignment that have the same two-sidedordinal preferences in metric space: W w ( a, c ) = w ( b, d ) = 1 + (cid:15) , w ( b, c ) = 3, w ( a, d ) = 1. W w ( a, c ) = w ( b, d ) = 1 − (cid:15) , w ( b, c ) = 1, w ( a, d ) = (cid:15) . The maximum weight perfect matchingfor W M { ( a, d ) , ( b, c ) } , while for W M { ( a, c ) , ( b, d ) } . Applying any randomizedalgorithm choosing M p and M − p to these two weightsettings, the optimal algorithm is when p = , gives a 1 . .2 Lower Bound of One-sided Ordinal Information Example
For one-sided ordinal information, consider a graph G = ( X , Y , E ), |X | = |Y| = N , X = { x , x , ...x N } , Y = { y , y , ..., y N } . Each agent in X have the same preferences over agentsin Y as y > y > ... > y N , because of this setting, no random algorithm could distinguish agentsand get a better performance than random algorithm. Assign the weights of the graph as: for acertain number ν ∈ [0 , i < = ν , w ( x i , y j ) = 3 for j < = i , all other edges have weight 1.The maximum matching is { ( a , b ) , ( a , b ) , ..., ( a N , b N ) } , with a total weight (2 ν + 1) N . Randommatching of this graph gets an expected weight of ( ν ( N + ν ) + 1) N , when N is large, the weightapproaches ( ν + 1) N . When ν = √ − , random algorithm gets a 1 . In this paper we quantified the tradeoffs between the amount of ordinal information available, andthe quality of solutions produced by our ordinal approximation algorithms, for metric maximum-weight bipartite matchings. For example, if we are able to collect preference data through surveys,but for each extra preference we must perform a certain extra amount of market research (i.e.,increasing α comes at a cost), then our findings would quantify how big we should make α in orderto form a good approximation to the best possible matching. All of this is without knowing thetrue numerical weights, only ordinal information.One thing to note here is that asking people to list their preference orderings, even partialpreference orderings for relatively small α , may be prohibitive. Agents are usually willing to nametheir top 3-10 choices, but not more than that. Notice, however, that all our algorithms can bethought of differently. For example, RSD does not actually require the preference ordering as aninput. It simply needs to ask each agent a single question: what is you favorite agent who hasnot been matched yet? Similarly, our other algorithms can be considered to ask agents a series ofquestions about their preferences, all of the same form. Such questions (determining their favoritefrom a set) are usually much easier for agents to answer than the question of specifying a preferenceordering.One clear research direction is to relax the assumption that we can only obtain ordinal informa-tion. What if we could also obtain some numerical information, but at further cost? What is thetradeoff between quality of solution formed and the amount of numerical information we obtain?What if we could ask the agents more complex questions than “Who is your favorite unmatchedagent?”, but were limited in the number of times we could ask such questions? We leave theseimportant directions for future work. References [1] Atila Abdulkadiro˘glu and Tayfun S¨onmez. Random serial dictatorship and the core fromrandom endowments in house allocation problems.
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