On Conceptually Simple Algorithms for Variants of Online Bipartite Matching

Abstract

We present a series of results regarding conceptually simple algorithms for bipartite matching in various online and related models. We first consider a deterministic adversarial model. The best approximation ratio in this model is 1/2, which is achieved by any greedy algorithm. Dürr et al. (2016) presented a 2-pass algorithm Category-Advice with approximation ratio 3/5. We extend their algorithm to multiple passes. We prove the exact approximation ratio for the k-pass Category-Advice algorithm for all k ≥ 1, and show that the approximation ratio converges quickly to the inverse of the golden ratio $2/(1+\sqrt {5}) \approx 0.618$ as k goes to infinity. We then consider a natural adaptation of a well-known offline MinGreedy algorithm to the online stochastic IID model, which we call MinDegree. In spite of excellent empirical performance of MinGreedy, it was recently shown to have approximation ratio 1/2 in the adversarial offline setting — the approximation ratio achieved by any greedy algorithm. Our result in the online known IID model is, in spirit, similar to the offline result, but the proof is different. We show that MinDegree cannot achieve an approximation ratio better than 1 − 1/e, which is guaranteed by any consistent greedy algorithm in the known IID model. Finally, following the work in Besser and Poloczek (Algorithmica 2017(1), 201–234, 2017), we depart from an adversarial or stochastic ordering and investigate a natural randomized algorithm (MinRanking) in the priority model. Although the priority model allows the algorithm to choose the input ordering in a general but well defined way, this natural algorithm cannot obtain the approximation of the Ranking algorithm in the ROM model.

Appendices

Appendix A: Relevant Algorithmic Models

We elaborate on the relavant models given in the Introduction.

Adversarial Online Model

In the online bipartite matching introduced by Karp, Vazirani, and Vazirani [37], nodes V are the offline nodes known to an algorithm beforehand, and nodes U arrive online in some adversarial order. When a node in U arrives, all its neighbors in V are revealed simultaneously. An online algorithm is required to make an irrevocable decision with regard to which neighbor (if any) the arrived node is matched. Any greedy algorithm (yielding a maximal matching) achieves a 1/2 approximation and Karp, Vazirani, and Vazirani showed that no deterministic algorithm can achieve an (asymptotic) approximation ratio better than 1/2 in the adversarial online model. They gave a randomized online algorithm called Ranking and showed that it achieves a 1 − 1/e ≈ 0.632 expected approximation ratio. Moreover, they proved that no randomized algorithm can beat 1 − 1/e in the adversarial online model. Seventeen years after the publication of the Ranking algorithm a mistake was found in the analysis of the Ranking algorithm. The mistake was discovered independently by Krohn and Varadarajan and by Goel and Mehta in [29], and a correct proof was provided by Goel and Mehta. Since then many different and simplified proofs of the approximation ratio of Ranking have been given (see [8, 20, 29]). Thus, the one-pass adversarial online setting for MBM is now reasonably well understood.

Online Stochastic Models

The Feldman et al. [27] known IID distributional model reflex the observation that in applications of MBM to online advertising, one often knows some statistics about the upcoming queries (online nodes). Feldman et al. model this by the notion of a type graph G = (U, V, E) and a probability distribution p : U → [0, 1]. The online nodes are sampled from p independently one at a time. An algorithm knows G and p beforehand. As a special case, they define the IID model with integral arrival rates where the probability distribution p is assumed to have the following property: for each type, the expected number of online nodes of that particular type appearing in a random instance is an integer. By repeating nodes, it is easy to see that the performance of an algorithm with respect to a uniform distribution p carries over to arbitrary integral rates. As before, an algorithm is required to make an irrevocable decision about which neighbor to match the newly arriving online node to. In this setting, the adversary can only choose the type graph and the distribution p but doesn’t have further control over the online sequence of nodes, as those are sampled in the IID fashion. Thus, the adversary is more restricted than in the adversarial online model. Feldman et al. [27] describe an algorithm beating the 1 − 1/e barrier and achieving approximation ratio ≈ 0.67 for the known IID model with integral arrival rates. This work was followed by a long line of work including [5, 15, 30, 34, 45]. So far, the best approximation ratio for arbitrary arrival rates is ≈ 0.706 due to Jaillet and Lu [34] and the best approximation for integral types .7299 is due to Brubach et al. [15]. Other online stochastic input models have been studied; e.g., the Random Order Model (ROM), and the Unknown IID model. Karande et al. [36] show that an online algorithm with respect to the unknown i.i.d. model (i.e., where the type graph is not known to the algorithm) can be veiwed as an algorithm in the ROM model (where the adversary creates the set of input items and the input order is a random permutation of those items).

In the known IID model with integral types, the probability distribution p is assumed to have the following property: for each type, the expected number of online nodes of that particular type appearing in a random instance is an integer. It is easy to see that the performance of an algorithm with respect to a uniform distribution p carries over to arbitrary integral types. From now on, whenever we mention “known IID” we refer to known IID with a uniform distribution. example of a problem that is at the intersection of practice and theory.

Semi-streaming Model

The semi-streaming model (with edge inputs) was introduced by Feigenbaum et al. [26] and subsequently studied in [23, 24, 38]. In particular, Eggert et al. [23] provide a FPTAS multi-pass semi-streaming algorithm for MBM using space ${\tilde O}(n)$ in the edge input model. In the vertex input semi-streaming model, Goel et al. [28] give a deterministic1 − 1/e approximation and Kapralov [35] proves that no semi-streaming algorithm can improve upon this ratio. (See also the recent survey by McGregor [46].) The difference between semi-streaming algorithms and online algorithms in the sense of competitive analysis is that streaming algorithms do not have to make online decisions but must maintain small space throughout the computation while online algorithms must make irrevocable decisions for each input item but have no space requirement. The Goel et al. result shows the power of deterministic semi-streaming over deterministic online algorithms. In some cases, streaming algorithms are designed so as to make results available at any time (after each input item) during the computation and hence some streaming algorithms can also be viewed both as a streaming algorithm and an online algorithm. Conversely, any online algorithm that restricts itself to ${\tilde O}(n)$ space can also be considered as a semi-streaming algorithm.

Priority Model

The priority model captures the idea that many offline greedy algorithms initially sort the input items, and then do a single pass over the items in the sorted order. More generally, priority algorithms can adaptively reorder so as to select the next input item to process. Many problems have been studied in the priority model [2, 7, 10, 11, 18, 51, 58]. The original deterministic priority model was extended to the randomized priority model in [3]. We shall use the term fully randomized priority algorithm to indicate that the ordering of the input items and the decisions for each item are both randomized. When only the decisions are randomized (and the ordering is deterministic) we will simply say randomized priority algorithm. With regards to maximum matching, Aronson et al. [4] proved that an algorithm that picks a random vertex and matches it to a random available neighbor (if it exists) achieves approximation ratio at least $\frac {1}{2} + \frac {1}{400000}$ in general graphs. This is improved by Poloczek and Szegedy [53] to $\frac {1}{2} + \frac {1}{256}$. Besser and Poloczek [7] show that the algorithm that picks a random vertex of minimum degree and matches it to a randomly selected neighbor cannot improve upon the 1/2 approximation ratio (with high probability) even for bipartite graphs. Pena and Borodin [50] show that no deterministic (respectively, fully randomized) priority algorithm can achieve approximation ratio better than 1/2 (respectively 53/54) for the MBM problem. (See also [6] with respect to the difficulty of proving inapproximation results for all randomized priority algorithms.)

Advice Model

Dürr et al. show that for 𝜖 ∈ (0, 1/e), Θ_𝜖(n) advice bits are necessary and sufficient for obtaining a (1 − 𝜖) approximation.⁸ where the high 𝜖 regime is due to Mikkelsen [48]. Dürr et al. also show that O(log n) advice bits are sufficient for a deterministic advice algorithm to guarantee a 1 − 1/e approximation ratio. This result is based on the randomization to advice transformation due to Böckenhauer et al. [9]. Thus there is an interesting sharp threshold phenomenon happening at approximation ratio 1 − 1/e, where the required advice length jumps from O(log n) to Ω(n). Construction of the O(log n) advice bits is based on examining the behavior of the Ranking algorithm on all n! possible random strings for a given input of length n, which requires exponential time. It is not known if there is an efficient way to construct O(log n) advice bits. More specifically, one may put computational or information-theoretic restrictions on the advice string, and ask what approximation ratios are achievable by online algorithms with restricted advice.

Appendix B: Consistent Greedy Algorithms

Informally, the consistency condition says that if an online node u is matched with u^∗ in some run of the algorithm and u^∗ is available in another “similar” run of the algorithm, u should still be matched with u^∗ (“similar” means that neighbors of u in the second run form a subset of the neighbors of u in the first run). More formally:

Definition 4

Let ALG be a greedy algorithm. Fix an instance graph $\widehat {G}$, so that it is no longer a random variable. Let $\widehat {G}(\pi )$ denote an instance graph with online nodes presented in the order given by π. Let N_π(u) denote the set of neighbors of u that are available to be matched when u is processed by ALG running on $\widehat {G}(\pi )$. We say that a greedy algorithm is consistent or thatit satisfies consistency conditions if the following holds: ∀π, π^′, u if $N_{\pi ^{\prime }}(u) \subseteq N_{\pi }(u)$ and u is matched with $u^{*} \in N_{\pi ^{\prime }}(u)$ by ALG$(\widehat {G}(\pi ))$ then u is also matched with u^∗ by ALG$(\widehat {G}(\pi ^{\prime }))$.