Perturbed Greedy on Oblivious Matching Problems

07/11/2019
by   Zhihao Gavin Tang, et al.
0

We study the maximum matching problem in the oblivious setting, i.e. the edge set of the graph is unknown to the algorithm. The existence of an edge is revealed upon the probe of this pair of vertices. Further, if an edge exists between two probed vertices, then the edge must be included in the matching irrevocably. For unweighted graphs, the Ranking algorithm by Karp et al. (STOC 1990) achieves approximation ratios 0.696 for bipartite graphs and 0.526 for general graphs. For vertex-weighted graphs, Chan et al. (TALG 2018) proposed a 0.501-approximate algorithm. In contrast, the edge-weighted version only admits the trivial 0.5-approximation by Greedy. In this paper, we propose the Perturbed Greedy algorithm for the edge-weighted oblivious matching problem and prove that it achieves a 0.501 approximation ratio. Besides, we show that the approximation ratio of our algorithm on unweighted graphs is 0.639 for bipartite graphs, and 0.531 for general graphs. The later improves the state-of-the-art result by Chan et al. (TALG 2018). Furthermore, our algorithm can be regarded as a robust version of the Modified Randomized Greedy (MRG) algorithm. By implication, our 0.531 approximation ratio serves as the first analysis of the MRG algorithm beyond the (1/2+ϵ) regime.

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1 Introduction

This paper studies the oblivious matching problem [ADFS95, ACC16, CCWZ18, CCW18] that is motivated by kidney exchange applications [RSÜ05]. Consider a graph in which the edge set is unknown to the algorithm, i.e., only the set of vertices is revealed. In each step, the algorithm picks a pair of unmatched vertices and probes whether edge exists. If it exists, the edge must be included into the solution irrevocably, and the two endpoints are removed from the graph. The algorithm determines a sequence of pairs to probe and aims at maximizing the size of the matching. The approximation ratio of an algorithm is the ratio between the (expected) number of edges matched by the algorithm, and the size of a maximum matching. It is straightforward to see that any algorithm that probes all pairs of vertices produces a maximal matching, and thus is -approximate. Indeed, this is the best possible ratio for deterministic algorithms. The major question is hence, to design a randomized algorithm with approximation ratio strictly greater than .

The first non-trivial theoretical guarantee for the problem is provided by Aronson et al. [ADFS95], who proved that the Modified Randomized Greedy (MRG) algorithm is -approximate. The MRG algorithm draws a random permutation over all vertices, and then for each vertex , independently draws another random permutation over all vertices. Then the MRG algorithm keeps probing pairs such that is the first unmatched vertex according to the order , and is ’s favorite unmatched neighbor according to its preference . This was later improved by Poloczek and Szegedy [PS12] to .111 It was pointed out by Chan et al. [CCWZ18] that their paper contains some gaps in their proof.

Another classic algorithm for the problem is the Ranking algorithm proposed by Karp et al. [KVV90]. Compared to MRG, the algorithm also uses a random order over the vertices. But instead of having an independent random preference for every vertex, Ranking uses as the common preference of all vertices. It is observed [GT12, PS12] that the approximation ratio of Ranking for the oblivious matching problem on bipartite graphs is the same as for online bipartite matching with random arrival order. Hence the result of Karande et al. [KMT11] and Mahdian and Yan [MY11] directly translate to and approximations for the problem on bipartite graphs. On general graphs, it is shown that Ranking is -approximate [CCWZ18, CCW18].222Goel and Tripathi (FOCS 2012) claimed that Ranking is 0.56-approximate, but later withdrew the paper when they discovered a bug in their proof. Very recently, the Ranking algorithm has been applied in the fully online matching model [HKT18] and achieves competitive ratios and for bipartite graphs [HPT19] and general graphs [HKT18] respectively. We remark that their results also apply to the oblivious matching problem, with the same approximation ratio333Their analysis is based on a different version of the Ranking algorithm that uses an arbitrarily order and a common random preference for all vertices..

Despite the successful progresses on unweighted graphs, the weighted version of the problem is less well understood. In the edge-weighted oblivious matching problem, the graph is associated with a weight function defined on all pairs of vertices that is also known by the algorithm. In particular, if edge exists between a probed pair of vertices , then the weight of the edge is given by . The objective of the algorithm is to maximize the total weight of edges matched. It is easy to show that the greedy algorithm that probes edges in descending order of the edge weights is -approximate. However, no algorithm with approximation ratio strictly larger than has been proposed yet. Indeed, prior to our work, the only positive result is a -approximation algorithm [ACC16, CCW18] for the vertex-weighted version. That is, each vertex is associated with a weight and the edge weight is given by .

Another closely related problem is the stochastic matching problem, which can be regarded as an easier version of the oblivious matching problem. In addition to the weight associated with every edge

, there is also an existence probability

that is known by the algorithm. That is, every edge exists independently with probability when it is probed. Obviously, any algorithm for the oblivious matching problem applies to the stochastic matching problem with the same approximation ratio by simply ignoring the information of existence probabilities of edges. Indeed, the approximation ratio is guaranteed for every realization of the random graph, where each edge exists with probability .

Very recently, Gamlath et al. [GKS19] proposed a -approximate algorithm for the stochastic matching problem on bipartite graphs, which is the first result bypassing the barrier. They proposed444The open questions are raised in their SODA 2019 conference presentation. two open questions for the stochastic matching problem: (1) does algorithm with approximation ratio strictly above exist on general graphs? (2) what approximation ratio can be obtained if the existence probabilities of edges are correlated?

It is easy to see that the edge-weighted oblivious matching problem is a generalization of the stochastic matching problem, even when the existence probabilities of edges are arbitrarily correlated. Thus an algorithm with approximation ratio strictly above for the former problem on general graphs solves both open questions.

1.1 Our Algorithm and Results

In this paper, we give the first algorithm with approximation ratio strictly larger than for the edge-weighted oblivious matching problem and thus, answer the open questions raised in [CCWZ18, GKS19] affirmatively. In particular, we propose the following greedy based algorithm, which is inspired by an idea used in the vertex-weighted version of the problem [AGKM11, CCW18]. In previous works, the algorithm perturbs vertex weights based on the random ranks of vertices.

Perturbed Greedy.

Each vertex draws a rank independently and uniformly at random. For each pair of vertices , let be its perturbed weight, where is a non-decreasing function we will fix later. Then the algorithm probes pairs in descending order of their perturbed weights.

Theorem 1.1

There exists a function so that Perturbed Greedy is -approximate for the edge-weighted oblivious matching problem.

As a by-product, we have the following improved result for the stochastic matching problem.

Corollary 1.1

There exists an algorithm for the edge-weighted stochastic matching problem on general graphs that is -approximate, even when the existence probabilities of the edges are arbitrarily correlated.

It is also interesting to look at the approximation ratio of our algorithm on unweighted graphs. Observe that for any increasing function , our algorithm probes pairs in ascending order of . Obviously, all edges adjacent to the same vertex will have the same perturbed weight. By breaking ties arbitrarily but consistently, we interpret our algorithm as follows.

Random Decision Time (Rdt).

Suppose each vertex has an arbitrarily fixed preference over all vertices. Each vertex draws a decision time independently and uniformly at random. At decision time (from the smallest to the largest), if is unmatched, the algorithm matches with the first unmatched neighbor according to ’s preference.

Theorem 1.2

RDT is -approximate for the oblivious matching problem.

It is quite interesting to make a comparison between our algorithm and existing algorithms in the literature. The MRG algorithm uses random decision times and independent random preferences; the Ranking algorithm uses the same random order as the decision times and preferences; the algorithm by [HKT18] uses arbitrary decision times and common random preferences. Our result shows that we can also achieve an approximation ratio larger than by using random decision times and arbitrary preferences. Furthermore, observe that our algorithm considers the preferences of vertices in the worst case while MRG uses random preferences. RDT is thus a robust version of MRG and our lower bound on the approximation ratio directly applies to MRG.

Corollary 1.2

MRG is -approximate for the oblivious matching problem.

This substantially improves the lower bounds for MRG by Aronson et al. [ADFS95] and Poloczek and Szegedy [PS12]. Besides, our result beats the state-of-the-art -approximation by Ranking [CCW18] for the unweighted oblivious matching problem.

Finally, we study the RDT algorithm when the underlying graph is bipartite. For bipartite graphs, researchers often break the symmetry of vertices, treating one side of the vertices as buyers and the other side as items. Consider the greedy algorithm that draws a random order over the buyers and let the buyers pick their favorite items sequentially based on the order. Karp et al. [KVV90] showed that this algorithm is -approximate555Ranking is equivalent to the greedy algorithm with random arrival by exchanging the two sides of vertices.. In contrast, RDT beats the barrier by treating both sides of vertices symmetrically and allows both sides of the vertices to make decisions.

Theorem 1.3

RDT is -approximate for the oblivious matching problem on bipartite graphs.

Last, in Appendix C, we show that RDT is at most -approximate on general graphs, which gives a separation on the approximation ratio of RDT on bipartite and general graphs.

1.2 Our Techniques

We will prove our results starting with RDT on bipartite graphs, then RDT on general graphs and finally Perturbed Greedy on edge-weighted general graphs, in progressive order of difficulty. Our analysis builds on the randomized primal dual framework introduced by Devanur et al. [DJK13] and recently developed in a sequence of results [HKT18, HTWZ18, HPT19]. Roughly speaking, we split the gain of each matched edge to its two endpoints. By proving that the expected combined gain of any pair of neighbors is at least , we have that the approximation ratio is at least . However, our approach differs from previous works in a way that we only look at the pairs that appear in some fixed maximum matching. We refer to such pairs as perfect partners. In order to prove the algorithm is -approximate, we observe that it suffices to show the expected combined gain of is at least for all perfect pairs . As far as we know, our result is the first successful application of the randomized primal dual technique to the oblivious matching problem, or to any edge-weighted maximum matching problems.

Unweighted Bipartite Graphs.

Consider a pair of neighbors . Suppose has an earlier decision time than . Existing analysis on Ranking relies on an important structural property that whenever is unmatched, then must be matched to some vertex with rank smaller than . However, this is not true in our case, since we do not have any control on the rank of the vertex chooses. Nevertheless, if but is already matched before ’s decision time, then it must be matched by some vertex with decision time earlier than .

For each edge added to the matching at ’s deadline, it is natural to define as the active vertex (who initiates the matching) and as the passive vertex. Furthermore, since we do not have any control of , we share the gain between by setting and . By fixing the decision times of all vertices other than arbitrarily and taking the randomness over , we give lower bounds on the expected combined gain of and .

For bipartite graphs, our analysis relies on a crucial structural property that if is matched when its neighbor is removed from the graph, then remains matched when is added back, no matter what the decision time has.

Unweighted General Graphs.

Going from bipartite graphs to general graphs takes away this nice property. The same problem also arises in [CCWZ18, HKT18]. The recent work by Huang et al. [HKT18] resolves this issue by introducing a notion of victim. Roughly speaking, they call an unmatched vertex the victim of its active neighbor if becomes matched when is removed. Then they define a compensation rule in which each active vertex sends an amount of gain, named compensation, to its victim (if any).

In this work, we propose a new definition of victim together with a compensation rule that is arguably more intuitive and has a clearer structure compared to that of [HKT18].

Suppose vertex actively matches in our algorithm. If is matched with its perfect partner when is removed, then we call the victim of . After sharing the gain as we have described for bipartite graphs, we let each active vertex send a portion of its gain to its victim if the victim is unmatched. This compensation rule is designed to retrieve some extra gain for when the aforementioned property for bipartite graphs fails to hold.

Weighted General Graphs.

For unweighted or vertex-weighted graphs, the contribution of each matched vertex to the algorithm is fixed. However, being matched is no longer a meaningful signature of a vertex on edge-weighted graphs. Indeed, we can add dummy vertices to the graph and add zero-weighted edges between them and original vertices. This shall not change the weight of maximum matching while now in any run of our algorithm, all vertices are guaranteed to be matched. This prevents the victim notion of Huang et al. [HKT18] from generalizing to edge-weighted graphs since being matched becomes meaningless.

In contrast, our definition of victim generalize immediately with a minimum change of the compensation rule. In particular, we fix a function that represents the amount of compensation one would like to send. The compensation rule works in the following way. Suppose is the victim of . We know that is matched with ’s perfect partner . Our compensation rule ensures that after the compensation, has gain at least . Note that the amount of compensation that sends depends on the current gain of . We believe this generalization further justifies the advantage of our new notion of victim and we believe the new notion will find more applications in other matching problems on general graphs.

Finally, we remark that our analysis directly guides us on designing the function in our algorithm. In contrast, all different choices of increasing function lead to the same algorithm when the graph is unweighted. For edge-weighted graphs, we explicitly construct a function (in Section 5) and show an analytical lower bound on the approximation ratio.

1.3 Other Related Works

For the hardness results of the problem, Goel and Tripathi [GT12] showed a upper bound on the approximation ratio of any algorithm for unweighted graphs. When restricted to the family of vertex iterative algorithms, they give a stronger bound of . This family of algorithms considers vertices one by one and probes all edges incident to a vertex until it is matched. Observe that all successful algorithms in the literature and our RDT algorithm fall into this family. For the MRG algorithm, Dyer and Frieze [DF91]’s bomb graphs give a upper bound on its approximation ratio. For the Ranking algorithm, the hard instance provided by Mahdian and Yan [MY11] implies a upper bound, which was later improved to by Chan et al. [CCWZ18].

The stochastic matching problem was first introduced by Chen et al. [CIK09], in which they have the constraints that each vertex can be probed no more than times. They gave an algorithm with approximation ratio , which is later improved to by a sequence of works [BGL12, AGM15, BCN18]. Recently, Gamlath et al. [GKS19] studied the stochastic matching problem on edge-weighted graphs but assuming unbounded . They give an -approximate algorithm and extend their result to the price-of-information model.

2 Preliminaries

The general algorithm for the edge-weighted oblivious matching problem is presented as follows.

Fix a non-decreasing function .
Each vertex independently draws a rank uniformly at random.
Probe pairs in descending order of their perturbed weights .
Algorithm 1 Perturbed Greedy

We use to denote the ranks of vertices, and to denote the matching produced by our algorithm with ranks . We use to denote the matching produced by our algorithm on , i.e. when vertex is removed from the graph. Fix any maximum weight matching , the approximation ratio is the ratio between the expected total weight of edges in , and the total weight of edges in .

The gain sharing framework we use in this paper is formalized as follows.

Lemma 2.1

If there exist non-negative random variables

depending on such that

  • for every , ;

  • for every , ,

then our algorithm is -approximate.

Proof.

The approximation ratio of Perturbed Greedy is given by

. ∎

Next, we define the notion of active and passive vertices fo edge-weighted general graphs and provide the monotonicity on the ranks (the proof can be found in Appendix A).

Definition 2.1 (Active, Passive)

For every edge added to the matching by Perturbed Greedy with , we say that is active and is passive.

Lemma 2.2 (Monotonicity)

Consider any matching and any vertex . If is passive or unmatched, then there exists some threshold such that if we reset to be any value in , the matching remains unchanged; if we reset to be any value in , then becomes active. Moreover, the weight of the edge actively matches is non-increasing w.r.t. .

2.1 Unweighted Graphs.

In this subsection, we focus on unweighted graphs and provide the standard alternating path property. An analog for edge-weighted graphs will be provided in Section 5.

As we have discussed in Section 1, we can equivalently interpret the algorithm as follows.

We call the rank of vertex as the decision time of , and let vertices act in ascending order of their decision times. If a vertex is not matched yet at its decision time, then it will match the unmatched neighbor according to its own preference order. We also call the decision time of edge , which is the only possible time the edge is included in the matching.

Each vertex independently draws a rank uniformly at random.
At decision time of , if is unmatched, chooses its favourite unmatched neighbor.
Algorithm 2 Random Decision Time

Note that for each vertex, the preference order of its neighbors is arbitrary but fixed for each vertex, i.e., it does not depend on the ranks of vertices.

It is easy to see that for the unweighted case, if a vertex is passively matched by , then the threshold of in Lemma 2.2 coincides with the rank of . Thus we have the following.

Corollary 2.1 (Unweighted Monotonicity)

Consider any matching and any vertex . If is passively matched by , then when has decision time in , the matching remains unchanged; when has decision time in , then becomes active. If is active, then the set of unmatched neighbors sees at its decision time grows when decreases.

The following important property characterizes the effect of the removal of a single vertex. For continuity of presentation we defer the proof of the lemma to Appendix A.

Lemma 2.3 (Alternating Path)

If is matched in , then the symmetric difference between and is an alternating path such that for all even ,

and for all odd

, . Moreover, the decision times of edges along the path are non-decreasing. Consequently, vertices are matched no later in than in .

Given the above lemma, we show the following useful property, which roughly says that any vertex can not affect the matching status of another vertex if is matched before .

Corollary 2.2

Suppose at time , vertex is matched while vertex is still unmatched. Then resetting to be any value in does not change the matching status of .

Proof.

Since is unmatched when gets matched, if is removed, then the matching status of is not affected. Now suppose that we insert with , then if is matched, then it must be matched at time later than (the time when gets matched). In other words, the insertion of triggers a (possibly empty) alternating path in which all edges (and therefore vertices) have decision time at most . Hence, is not included in the alternating path and its matching status is not affected. ∎

For bipartite graphs, Lemma 2.3 also implies the following nice property.

Corollary 2.3

For bipartite graphs, if is matched in and is a neighbor of , then is also matched in . Moreover, the time is matched in is no later than in .

Proof.

By Lemma 2.3, inserting (at any rank) creates a (possibly empty) alternating path . As the graph is bipartite, if appears in the path, then it must be one of , which is matched no later than in . Otherwise its matching status is not affected. ∎

3 Unweighted Bipartite Graphs

Recall that for unweighted case, vertices act in ascending order of their decision times, and each unmatched vertex chooses its favorite unmatched neighbor. We first give the gain sharing rule for every matched pair .

Gain Sharing.

Whenever actively chooses at time , let and , where is a non-decreasing function to be fixed later.

General Framework.

Recall that to prove an approximation ratio of , it suffices to show that for every . Fix such a pair , and fix the decision times of all vertices other than arbitrarily. For ease of notation, we use to denote the matching produced by our algorithm when ’s decision times are and , respectively. In the following, we will give a lower bound of for each , and show that there exists an appropriate function such that , finishing the proof of Theorem 1.3.

Consider matching , i.e., when have the latest decision times compared with other vertices. Depending on whether are matched together, we divide our analysis into two parts.

3.1 Symmetric Case:

In this case, and are not chosen by any other vertex. At time , and are matched together.

Fact 3.1

If , then when , is active in and is unmatched before time ; when , is active in , and is unmatched before time .

Proof.

Suppose . Consider the first time when one of (say, ) is matched. Obviously, . If , then must be chosen by a vertex at its decision time . By Lemma 2.2, we have , which violates the assumption of the fact. Thus , i.e, is active, and is unmatched before time . The case when is similar. ∎

We decrease gradually from to and study . By Corollary 2.1, the set of unmatched neighbors of at time grows when decreases. Hence there exists a transition time such that is no longer ’s favorite vertex when . In other words, when , actively matches in ; when , actively matches a vertex other than in . Moreover, by Corollary 2.2, the matching status of in is the same as in , as long as . Thus we have (refer to Figure 0(a))

  • when and ;

  • when and .

Similarly, we decrease gradually from to and study . Let be the transition time such that is ’s favourite neighbor if and only if . Then we have (refer to Figure 0(a))

  • when and ;

  • when and .

(a) Symmetric case: match each other
(b) Asymmetric case: is chosen at
Figure 3.1: Unweighted bipartite graphs: the horizontal and vertical axes correspond to respectively. For each region, the formula written serves as a lower bound of .

We refer to these gains as the basic gain of our analysis as they come immediately after we properly define . Next, we study the matching status of the vertex with later decision time and achieve some extra gains, where we crucially use the bipartiteness of the graph.

Lemma 3.1 (Extra Gain)

For all and , both and are matched in .

Proof.

By definition, when , actively matches a vertex other than in . Thus removing does not change the matching status of . In other words, is matched in . By Corollary 2.3, remains matched in . Similarly, we have is matched in for all , which finishes the proof. ∎

Let . It is easy to see that whenever is matched (actively or passively), . In summary, we have the following lower bound (refer to Figure 0(a)).

(3.1)

3.2 Asymmetric Case:

In this case, at least one of is matched before time . Without loss of generality, suppose is matched at time , and is strictly earlier than . Observe that must be passive since . Let be the vertex that actively matches . Then we have . Intuitively, is the “luckier” vertex compared with since it is favored by a vertex with early decision time. Indeed, would remain matched even when is removed from the graph.

First, observe that when both and have decision times larger than , is always matched by , and thus . When , must be active in , since at time , is unmatched and has unmatched neighbors and (with later decision times). Moreover, by Corollary 2.2, is active in as long as and . Thus . Similarly, for all and , is active and .

Again, we decrease gradually from to and study . Observe that at time , is an unmatched neighbor of . Then there exists a transition time such that is the favourite neighbor of when ; and matches a vertex other than when . In summary, we have the following basic gains (refer to Figure 0(b))

  • when ;

  • when and ;

  • when and ;

  • when and .

Next we retrieve some extra gains. Again, the following holds only for bipartite graphs.

Lemma 3.2 (Extra Gain)

When and , both and are matched in . When and , .

Proof.

Consider when and . According to the previous discussion, has two unmatched neighbors and at time . Thus would still be actively matched even if we remove from the graph. That is, is active in . Then by Corollary 2.3, after inserting at any rank, remains matched. In other words, is matched in for all .

By definition of , actively matches a vertex other than in for all . Thus is active in , and matched in for every by Corollary 2.3.

Now we consider the second statement, when and . Observe that in , matches at time

while at this moment

is unmatched. Removing does not affect and , i.e. actively matches in . Then by Corollary 2.3, inserting at any rank does not increase the time that gets matched. Hence in , is passively matched at time no later than , which implies by the monotonicity of . ∎

Adding these extra gains to the basic gains, we have (refer to Figure 0(b))

(3.2)
Analysis of Approximation Ratio.

To complete the analysis, it remains to find a non-decreasing function so that the minimum (over all possible values of ) of Equations (3.1) and (3.2) is at least . We prove Theorem 1.3 by fixing to be a step function and running a factor revealing LP. See Appendix B for a detailed discussion.

4 Unweighted General Graphs

Since Corollary 2.3 holds only for bipartite graphs, the extra gains we proved in the previous section cease to hold for general graphs. It is easy to check that applying the previous analysis while only having the basic gains, we are not able to beat the barrier on the approximation ratio.

The same difficulty arises in the fully online matching problem [HKT18]. The authors bypass it by introducing a novel concept of “victim”. They call a vertex the victim of in if (1) is a neighbor of ; (2) is active and is unmatched; (3) is matched in . Intuitively, is unmatched in because of the existence of . It is then shown that either are both matched for some recipe of , or is the victim of some vertex and receives compensation. In either case, the improved analysis beats the barrier.

In this paper, we introduce a new notion of victim and compensation, which is arguably clearer and more fundamental than the notion given in [HKT18]. Fix a maximum matching , we call and perfect partners of each other if .

Definition 4.1 (Victim)

Suppose in , actively matches and is the perfect partner of . Then we call the victim of if and match each other in .

Intuitively, the existence of prevents the algorithm from making the correct decision of matching together. Compared to the definition of Huang et al. [HKT18], we regard the victim of even when is matched in . The same definition will be applied to edge-weighted graphs in Section 5. Built upon this definition, we define the following gain sharing rule.

Gain Sharing.

Let be a non-decreasing function and be a function that is pointwise smaller than . Consider the following two-step gain sharing procedure in matching :

  • Whenever actively matches at time , let and .

  • For each active vertex that has an unmatched victim , decrease and increase by the same amount .

We refer to the second step of gain sharing as the compensation step, and the amount of gain as the compensation sent from to . Note that the compensation step does not change , which means that Lemma 2.1 can still be applied. It is easy to see that the passive gain of a vertex is at least and the active gain is at least .

Fact 4.1

If is matched in , then .

Moreover, if is the victim of vertex , either is matched, , or receives compensation from , . For analysis purpose, we choose so that , i.e., the gain of a matched vertex is at least the compensation of an unmatched vertex. To help understanding, one can imagine the compensation to be a very small amount of gain compared with . Consequently, we have the following.

Fact 4.2

If is the victim of vertex in , then .

Following the same framework as for bipartite graphs, we fix a pair of perfect partners , and fix the decision times of all vertices other than arbitrarily. Let denote the realized matching when have decision times and , respectively. Again, we consider whether and proceed differently.

4.1 Symmetric Case:

The analysis is similar to the bipartite case. Let be the transition time such that actively matches in when ; matches a vertex other than in when . The transition time of is defined analogously.

Following the same analysis for bipartite graphs, we have (refer to Figure 4.1)

  • when and ; is active when and ;

  • when and ; is active when and .

Observe that for general graphs, the gain of an active vertex is no longer , but is lower bounded by . However, if match each other, then the active vertex does not need to send compensations (recall that are perfect partners).

Figure 4.1: Unweighted general graphs: .

For bipartite graphs, we show that both and are matched in when and . Unfortunately, this is not guaranteed in general graphs. However, we manage to achieve a weaker version of the extra gains that if only one of is matched when and , then it need not send compensation.

Lemma 4.1 (Extra Gain)

For all and , we have in when and when .

Proof.

We first consider the case when . If is matched, then . Now suppose is unmatched. By definition, actively matches a vertex other than in when . Thus, is also active in . In other words, becomes unmatched after inserting at decision time . We show that in this case need not send compensation in , which implies .

Suppose matches in . By Lemma 2.3, removing triggers an alternating path that starts at and ends at . Thus the perfect partner of is either matched in both of and ; or unmatched in both. Consequently, does not have an unmatched victim.

Symmetrically, we have when . ∎

In summary, we have the following lower bound on . Note that are symmetric. We safely assume that (refer to Figure 4.1).

(4.1)

4.2 Asymmetric Case:

As before, at least one of is matched before time . We assume without loss of generality that is matched strictly earlier than in , and let be the active vertex that matches with decision time . First, when , is always matched by , and thus . When , we know that is active in , and thus active in as long as (by Corollary 2.2). Now consider when . Following the same analysis as for bipartite case, let be the transition time such that chooses when and chooses a vertex other than when . Moreover, since is active in when , following the same analysis as in Lemma 4.1, it can be shown that when and . Similarly, it is easy to show that when and 666The key observation is, is matched in , and thus in every . This implies that after inserting with , either is matched, or need not send compensation..

In summary, we have (refer to Figure 1(a))

  • when and ;

  • when and ;

  • when and ;

  • when and ;

  • when and .

(a) basic gains
(b) is matched earlier
(c) is matched earlier
Figure 4.2: Simple lower bounds and the case when .

Unsurprisingly, the above basic gains do not yield an approximation ratio strictly above .

Extra Gains.

For convenience of discussion, we define Zone-A to be the matchings when and (where lower bound (L1) is applied), and Zone-B to be the matchings when (where lower bound (L5) is applied). In the following, we show that better lower bounds can be obtained for either (L1) or (L5). Roughly speaking, if is unmatched in Zone-A, then either it is compensated by (in which case (L1) can be improved), or will be matched in Zone-B (in which case (L5) can be improved). Hence depending on the matching status of and in , i.e., when is removed, we divide our analysis into two cases.

4.2.1 Case 1:

In this case, at least one of is matched passively before time in . We first consider the case when is matched strictly earlier than . We show that in this case is matched in Zone-B, and thus (L5) can be improved (see Figure 1(b)).

Lemma 4.2

If is matched strictly earlier than in , then is matched in Zone-B.

Proof.

To show that is matched in Zone-B, by Lemma 2.2 it suffices to show that is matched in , for all . Suppose otherwise, i.e., is unmatched in for some .

Since matches in , the symmetric difference between and is an alternating path that starts from and ends with . Moreover, is the second last vertex in the alternating path. Now suppose we remove and simultaneously in . Then in the resulting matching, all vertices between and in the alternating path recover their matching status in , while all other vertices remain the same matching status. In particular, remains unmatched when both and are removed from . However, since is matched strictly earlier than in , should remain matched to the same vertex when we further remove , which is a contradiction. ∎

Given the lemma, we improve (L5) and obtain the following (refer to Figure 1(b)). As we will show later, this is not the bottleneck case since this lower bound is strictly larger than (4.5).

(4.2)

Next, we consider the case when is matched strictly earlier than in . We show that in this case is matched in Zone-A, which improves (L1).

Lemma 4.3

If is matched earlier than in , then is matched in Zone-A.

Proof.

Consider adding back to . Since chooses , remains matched in . Moreover, all matchings for are the same and hence, is matched in for all . Finally, by Lemma 2.2 is matched in Zone-A. ∎

Thus, we obtain the following lower bound (refer to Figure 1(c)). As we will show later, this is neither the bottleneck case since the lower bound is strictly larger than (4.4).

(4.3)

4.2.2 Case 2:

Now we study the second case when match each other in . This is where the notion of victim applies. Formally, we have the following lower bound of in Zone-A.

Lemma 4.4 (Compensation)

For all , if matches in , then we have in .

Proof.

When , actively matches in . Hence is the victim of and . When , either actively matches in and thus