1. Optimization Problems Online with Random Demands
Anupam Gupta Carnegie Mellon University

2. SICOMP, 30(1): , 2000

3. model ❶: random order arrivals
the adversary chooses the set of requests
the request sequence is just this set, in random order
mitigates some of the hardness arising from the uncertainty
[Kenyon (Mathieu) ’96] for bin-packing

4. maximum matching: random order
Adversary chooses bipartite matching instance (hidden from algo) Algo starts off with n blue nodes right/request/red nodes arrive online They arrive in random order Edges added to the matching irrevocably a p b q c r d s e t a b c d e p q r s t

5. maximum matching: random order
The greedy algorithm: match each request to “first” open blue node greedy is 2-competitive in general greedy is (1-1/e)-comp for random order Can we do better? a p b q c r d s e t
Karp Vazirani Vazirani ’90
Goel Mehta ‘08
Birnbaum Mathieu ‘08 a b c d e p
Devanur Jain Kleinberg ‘11 q r s t

6. maximum matching: random order
The greedy algorithm: match each request to “first” open blue node greedy is 2-competitive in general greedy is (1-1/e)-comp for random order The Ranking algorithm: pick a random ranking on the blue nodes run greedy with this ranking Ranking is comp for random order arrivals (but no better than 0.73) a p b q c r d s e t
[KVV 90] a b c d e p q r s t
Mahdian Yan ‘11
Karande Mehta Tripathi ‘11

7. e.g.: k-edge-connected networks online
Algorithm:
When terminal t arrives, find cheapest k edge-disjoint paths connecting t to the set of previously arrived terminals.

8. e.g.: k-edge-connected networks online
Theorem [uses Mader’s splitting-off theorem]
Given graph G that k-edge-connects a set of terminals S, can find k edge-disjoint paths P1s, P2s, …, Pks from each terminal t to the rest S – {t} such that
every edge in G is used at most twice.
 a random terminal in S can connect to others at cost 2 OPT/|S|
Condition on first j arrivals, call this S
S in random order, so cost of jth request is 2 OPT(S)/j
≤ 2 OPT / j
 total expected cost ≤ O( OPT log n )

9. some other problems
For Facility Location Access Network Design Buy-at-Bulk Network Design online algorithms are known for the random-order model that either beat the lower bounds for adversarial order online (Fac.Loc.) or beat the currently known algorithms for adversarial order. E.g., [Bahmani Chowdhury Goel 2011] show how to compute pagerank well using single pass over data, but viewing edges in random order. Work on streaming algorithms uses randomness in the order of the data stream And of course there’s the secretary algorithm (and its many variants)
O(1)-comp.
Meyerson ‘01
O(# of cables)-comp
Meyerson Munagala Plotkin ‘01
sub-polynomial-comp.
Charikar Karagiozova ‘05

10. model ❷: i.i.d. arrivals
we are given a probability distribution over single requests
each request is an i.i.d. draw from this distribution
(clearly, this is a more restricted model than random-order)
studied for paging/k-server/scheduling, here focus on different problems

11. maximum matching: i.i.d. arrivals
Algo is given a “template” graph on blue/red nodes In real graph, only blue nodes known right/request/red nodes arrive online Each request (red) node is an independent uniformly random copy of a red template vertex (with replacement) a p b q c r d s e t a b c d e
p_3
q_1
r_2
q_4

12. maximum matching: i.i.d. arrivals
Results in this model beat those in the random order model. a p b q c r d s e t
random order
i.i.d
easyness
0.673
0.702
“hardness”
0.823 a b c d e
p_3
q_1
r_2
q_4
Feldman Mehta Mirrokni Muthukrishnan ‘09
Bahmani Kapralov ‘10
Manshadi Oveis-Gharan Saberi ‘11

13. objective function: i.i.d. arrivals
Measure of Goodness:
Usual measure is competitive ratio
We can either consider
or: a p b q
EA [ cost of algorithm A on ¾ ]
OPT(set ¾)
max¾ c r d s e t a b c d e
p_3
E¾,A [ cost of algorithm A on ¾ ]
E¾ [ OPT(set ¾) ]
q_1
r_2
q_4
cost of algorithm A on ¾
OPT(set ¾)
E¾,A

14. Steiner tree: background
Input: a metric space
a root vertex r
a subset R of terminals
Output: a tree T connecting R to r of minimum length/cost.
Facts: NP-hard and APX-hard
MST is a 2-approximation cost(MST(R [ r)) ≤ 2 OPT(R)
[Byrka et al. STOC ’10] give a approximation

15. the online greedy algorithm
[Imase Waxman ’91] in the standard online setting, the greedy algorithm is O(log k) competitive for sequences of length k. and this is tight.

16. Steiner tree online: i.i.d. arrivals
Suppose demands are nodes in V drawn uniformly at random, independently of previous demands. uniformity: not important could have (given) probabilities p1, p2, …, pn independence: important, lower bounds otherwise Measure of goodness:
E¾,A [ cost of algorithm A on ¾ ]
E¾ [ OPT(set ¾) ]
≤ 4
Assume for this talk: know the length k of the sequence

17. Augmented greedy Sample k vertices S = {s1, s2, …, sk} independently.
[Garg G. Leonardi Sankowski]
Sample k vertices S = {s1, s2, …, sk} independently.
Build an MST T0 on these vertices S [ root r.
When actual demand points xt (for 1 · t · k) arrives, greedily connect xt to the tree Tt-1

18. Proof for augmented greedy
Let X = {x1, x2, …, xk} be the actual demands Claim 1: E[ cost(T0) ] ≤ 2 £ E[ OPT(X) ] Claim 2: E[ cost of k augmentations in Step 3 ] ≤ E[ cost(T0) ]
Sample k vertices S = {s1, s2, …, sk} independently.
Build an MST T0 on these vertices S [ root r.
When actual demand points xt (for 1 · t · k) arrives, greedily connect xt to the tree Tt-1
Proof: E[ OPT(S) ] = E[ OPT(X) ]
 Ratio of expectations ≤ 4

19. Proof for augmented greedy
Let X = {x1, x2, …, xk} be the sample Claim 2: ES,X[ augmentation cost ] ≤ ES[ MST(S [ r) ] Claim 2a: ES,X[  x2X d(x, S [ r) ] ≤ ES[ MST(S [ r) ] Claim 2b: ES,x[ d(x, S [ r) ] ≤ (1/k) ES[ MST(S [ r) ]
Sample k vertices S = {s1, s2, …, sk} independently.
Build an MST T0 on these vertices S [ root r.
When actual demand points xt (for 1 · t · k) arrives, greedily connect xt to the tree Tt-1

20. Proof for augmented greedy
Claim 2b: ES,x[ d(x, S [ r) ] ≤ (1/k) ES[ MST(S [ r) ]
= E[ distance from one random point to (k random points [ r) ]
≥ (1/k) * k * Ey, S-y[ distance(y, (S-y) [ r) ]
≥ E[ distance from one random point to (k-1 random points [ r) ]
≥ E[ distance from one random point to (k random points [ r) ]

21. Proof for augmented greedy
Let X = {x1, x2, …, xk} be the actual demands Claim 1: E[ cost(T0) ] ≤ 2 £ E[ OPT(X) ] Claim 2: E[ cost of k augmentations in Step 3 ] ≤ E[ cost(T0) ]
Sample k vertices S = {s1, s2, …, sk} independently.
Build an MST T0 on these vertices S [ root r.
When actual demand points xt (for 1 · t · k) arrives, greedily connect xt to the tree Tt-1
 Ratio of expectations ≤ 4

22. directions/questions
[Grandoni G. Leonardi Mettienen Sankowski Singh] set cover in i.i.d. model O(log m + log n) instead of O(log m log n)
(small) Markov chain generates the requests
Koutsoupias & Papadimitriou showed positive results for k-server where requests drawn from small Markov chains.
Bad news: For Steiner tree, this is as hard as general adversaries
Can something interesting be said for matchings? other problems?
how much randomness is “enough” randomness?
even the two models discussed above still not completely understood
don’t yet know the extent to which random order/i.i.d. models help us for various problems
thank you!