Online Bipartite Matching with Augmentations Presentation by Henry Lin Joint work with Kamalika Chaudhuri, Costis Daskalakis, and Robert Kleinberg

Overview The online bipartite matching problem Background and previous work Our latest results Conclusion and open questions

Recall Basic Bipartite Matching Model: Bipartite graph between n clients and n servers Goal: Find a matching between clients and servers

Recall Basic Bipartite Matching Model: Bipartite graph between n clients and n servers Goal: Find a matching between clients and servers

The Online Matching Problem Model: Clients arrive online, and reveal edges to servers Goal: Maintain matching, while minimizing client- server switches

The Online Matching Problem Model: Clients arrive online, and reveal edges to servers Goal: Maintain matching, while minimizing client- server switches

The Online Matching Problem Model: Clients arrive online, and reveal edges to servers Goal: Maintain matching, while minimizing client- server switches

The Online Matching Problem Model: Clients arrive online, and reveal edges to servers Goal: Maintain matching, while minimizing client- server switches

The Online Matching Problem Model: Clients arrive online, and reveal edges to servers Goal: Maintain matching, while minimizing client- server switches

The Online Matching Problem Model: Clients arrive online, and reveal edges to servers Goal: Maintain matching, while minimizing client- server switches

Simplifying Assumptions For the purposes of this talk, we assume: Each server can serve at most one client There is a matching at each time step There are exactly n clients and n servers

A Few Sample Applications Web service provision Clients are website owners Servers are machines for hosting websites Job scheduling Clients are persistent job requests Servers are machines for servicing job requests Caching Clients are data objects Servers are locations in a hash table

Previous Work When each client has degree at most 2: [Grove, Kao, Krishnan, and Vitter 1995] The greedy algorithm has switching cost O( n log n ) For any algorithm, the switching cost is Ω(n log n) For general graphs No upper bounds on better than O(n 2 ) No lower bounds better than Ω(n log n)

Relaxing the Worst Case Model Good bounds in the worst case seems hard, but what about “on average”? Assume an arbitrary bipartite graph G, but what if the clients arrive randomly? What if G is a random graph where each client has O(log n) random edges?

Our Work When clients arrive uniformly at random, the greedy algorithm has cost O(n log n) w.h.p. When each client has O(log n) random edges, the total switching cost is O(n) w.h.p. When the bipartite graph is a tree, there is an algorithm with switching cost O(n log n)

An easier version of first theorem Let G be any bipartite graph between n clients and n servers (with a perfect matching) Theorem: If the clients of G arrive uniformly at random, the greedy algorithm has expected cost O(n log n). Result is tight as there is a graph, where any algorithm must have cost Ω(n log n)

Main lemma for easier theorem Lemma: If i clients have yet to arrive, the expected cost of the next arriving client is O(n/i) The lemma proves the theorem because the expected total cost is

Proving the Lemma Note: if remaining i clients all arrive, we can connect them with total switching cost ≤ n In this example: i=2 To match remaining clients, we only need to switch the 3 existing clients

Proving the Lemma Note: if remaining i clients all arrive, we can connect them with total switching cost ≤ n In this example: i=2 To match remaining clients, we only need to switch the 3 existing clients

Proving the Lemma There are also i augmenting paths of total length O(n), which can connect the i remaining clients Furthermore, the total length of the i shortest augmenting paths from the i remaining clients is O(n)

Proving the Lemma If total length of the i shortest augmenting paths from the i remaining clients is O(n) The expected length of these i augmenting paths is O(n/i) Thus, the expected cost of the greedy algorithm is O(n/i), when i clients remain

Recap: The lemma and theorem Lemma: If i clients have yet to arrive, the expected cost of the next arriving client is O(n/i) Theorem: When clients arrive uniformly at random, the greedy algorithm has expected cost O(n log n)

Related Work Online matching without re-assignment [Karp, Vazirani, Vazirani; Goel, Mehta; Saberi, et al.] Online load balancing [Azar, Broder, Karlin; Phillips, Westbrook; etc.] Cuckoo hashing [Pagh, Rodler; etc.]

Open Questions Can one derive an algorithm with O(n log n) switching cost? Can one prove a lower bound better than Ω(n log n)? Can our work be used to derive faster bipartite matching algorithms?

Thanks! Questions?