Online Bipartite Matching with Augmentations

1
Online Bipartite Matching with Augmentations

• Presentation by Henry Lin
• Joint work with Kamalika Chaudhuri, Costis
  Daskalakis, and Robert Kleinberg

2
Overview

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

3
Recall Basic Bipartite Matching

• Model Bipartite graph between n clients and n
  servers
• Goal Find a matching between clients and servers

4
Recall Basic Bipartite Matching

• Model Bipartite graph between n clients and n
  servers
• Goal Find a matching between clients and servers

5
The Online Matching Problem

• Model Clients arrive online, and reveal edges to
  servers
• Goal Maintain matching, while minimizing
  client-server switches

6
The Online Matching Problem

• Model Clients arrive online, and reveal edges to
  servers
• Goal Maintain matching, while minimizing
  client-server switches

7
The Online Matching Problem

• Model Clients arrive online, and reveal edges to
  servers
• Goal Maintain matching, while minimizing
  client-server switches

8
The Online Matching Problem

• Model Clients arrive online, and reveal edges to
  servers
• Goal Maintain matching, while minimizing
  client-server switches

9
The Online Matching Problem

• Model Clients arrive online, and reveal edges to
  servers
• Goal Maintain matching, while minimizing
  client-server switches

10
The Online Matching Problem

• Model Clients arrive online, and reveal edges to
  servers
• Goal Maintain matching, while minimizing
  client-server switches

11
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

12
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

13
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 O(n log
  n)
• For general graphs
• No upper bounds on better than O(n2)
• No lower bounds better than O(n log n)

14
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?

15
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)

16
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 ?(n log n)

17
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

18
Proving the Lemma

• Note if remaining i clients all arrive, we can
  connect them with total switching cost n
• In this example i2
• To match remaining clients, we only need to
  switch the 3 existing clients

19
Proving the Lemma

• Note if remaining i clients all arrive, we can
  connect them with total switching cost n
• In this example i2
• To match remaining clients, we only need to
  switch the 3 existing clients

20
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)

21
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

22
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)

23
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.

24
Open Questions

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

25
Thanks!

• Questions?