24
Semi-Online Bipartite Matching Zoya Svitkina with Ravi Kumar, Manish Purohit, Aaron Schild, Erik Vee Google TTIC Summer Workshop on Learning-Based Algorithms August 13, 2019
Semi-Online Bipartite Matching
Zoya Svitkina
with Ravi Kumar, Manish Purohit, Aaron Schild, Erik Vee
TTIC Summer Workshop on Learning-Based Algorithms August 13, 2019
Semi-online algorithms
• Future is partly known, partly adversarial
• Pre-process the known part
• Then make irrevokable decisions at each step
• Interpolates between offline and online models
Offline bipartite matching• Polynomial-time solvable using max flow
Online bipartite matching• Nodes in known in advance
• Nodes in arrive one by one
• Match at each step
• Competitive ratio compares to offline OPT
U
V U V
Online bipartite matching• RANKING algorithm [1] is competitive:
• Fix a random permutation of offline nodes
• For each online node:
• Match to the first available neighbor in the permutation
1 − 1/e
[1] Richard Karp, Umesh Vazirani, Vijay Vazirani. An optimal algorithm for on-line bipartite matching. STOC 1990
Semi-online bipartite matching
• Know and part of in advance
• All of arrives one by one in arbitrary order
• Match at each step
• Competitive ratio compares to offline OPT
• Integral or fractional matching
U V
V
Notation• Bipartite graph
•
• : known (predicted) part of
• : unknown (adversarial) part of
• Known subgraph
G = (U, V, EG)
V = VP ∪ VA
VP V
VAV
H = (U, VP, EH)
H VP
VA
U V
Online/offline parameter δ• Simplifying assumption for this talk: perfect matching in
• , fraction of adversarial nodes
• : offline, : online
• Competitive ratio in terms of
• General case:
• Other definition doesn't work if many isolated nodes
G
δ =|VA ||V |
δ = 0 δ = 1
δ
δ = 1 −OPT(H)OPT(G)
Results• Integral matching:
• Algorithm with competitive ratio
• Hardness of
• Fractional matching:
• Algorithm and hardness of
1 − δ + δ2(1 − 1/e)
1 − δe−δ( ≈ 1 − δ + δ2 − δ3/2 + . . . )
1 − δe−δ
Related settings• Optimal online assignment with forecasts
Erik Vee, Sergei Vassilvitskii, and Jayavel Shanmugasundaram. EC 2010
• Uncertainty in demands, not in graph structure
• Online allocation with traffic spikes: Mixing adversarial and stochastic models Hossein Esfandiari, Nitish Korula, and Vahab Mirrokni. EC 2015
• Forecast is a distribution, not a fixed graph
• Large degree assumption
• Same hardness result
• Maximum matching in the online batch-arrival model Euiwoong Lee and Sahil Singla. IPCO 2017
• Online nodes arrive in batches
Observations• Worst case: predicted nodes before adversarial
• Algorithm for this case can be transformed into one for arbitrary order
• Should select a maximum matching on
• No benefit to leaving predicted nodes unmatched
• Do this as preprocessing
H
Selecting a matching for H
• Any deterministic algorithm would do badly
Algorithm outline
• Find a (randomized) maximum matching in
• Which nodes to "reserve" for ?
• Run RANKING for adversarial nodes
H
VA
Analysis outline• : not matched in .
• : matched to by OPT.
• Suppose
• Matching size
• Competitive ratio
• Aim for
Reserved ⊆ U H|Reserved | = n − |VP | = δn
Marked ⊆ U VA|Marked | = |VA | = δn
𝔼[ |Reserved ∩ Marked | ] = x ⋅ n
n − δn + (1 − 1/e)xn
1 − δ + (1 − 1/e)x
x = δ2
Reserving nodes• Goal: sample a matching in s.t.
• Special case: is complete
• Reserve each node with probability
• In general, a distribution over matchings s.t. may not exist
• Want a distribution making nodes' probabilities of being reserved as equal as possible
H𝔼[ |Reserved ∩ Marked | ] = δ2n
H
δ
∀u ∈ U, Pr[u is reserved] = δ
Matching skeleton decomposition1
• Decomposition of (poly-time)
• ,
•
•
• Fractional matching in each component
• ,
H
U = ∪i Ti VP = ∪i Si
Γ(∪i
Dependent rounding• Apply dependent rounding [1] to
each component of the matching skeleton
• Let
• Probability of being
reserved is
di = |Ti | − |Si |
u ∈ Tidi
|Ti |
[1] Rajiv Gandhi, Samir Khuller, Srinivasan Parthasarathy, Aravind Srinivasan. Dependent rounding and its applications to approximation algorithms. JACM 53(3):324–360, 2006.
Si Ti
23
23
13
13
Marked nodes• Adversary's goal:
• Mark nodes in whose complement has a matching in
• Minimize overlap with reserved nodes
• Best strategy:
• Select nodes per component
•
(by Cauchy-Schwarz)
• competitive ratio
δn U H
di = |Ti | − |Si | i
𝔼[ |Reserved ∩ Marked | ] = ∑i
di ⋅di
|Ti |≥
(δn)2
n
⇒ 1 − δ + δ2(1 − 1/e)
Hardness bound
• Predicted: complete graph; adversarial: block upper triangular
• Hardness of 1 − δe−δ
v1 v2 v3 v4 v5 v6 v7 v8 v9 v10 v11 v12
u1 u2 u3 u4 u5 u6 u7 u8 u9 u10 u11 u12U :
UA
VA
V :
VP
Fractional matching• Online model
• Nodes of arrive one at a time, have to be fractionally matched to
• Water-level algorithm [1] gives optimal ratio
• Match to the neighbor with lowest existing amount
• Semi-online fractional bipartite matching
• We get tight bounds of
• Primal-dual analysis extension of [2]
V U
1 − 1/e
1 − δe−δ
[1] Bala Kalyanasundaram and Kirk Pruhs. An optimal deterministic algorithm for online b-matching. Theor. Comput. Sci., 233(1-2):319–325, 2000 [2] Nikhil R. Devanur, Kamal Jain, Robert D. Kleinberg. Randomized primal-dual analysis of RANKING for online bipartite matching. SODA 2013
Algorithm for semi-online fractional matching
• For predicted nodes :
• Take fractional matching from skeleton decomposition of
• For adversarial nodes :
• Use water-level algorithm
VP
H
VA
12
12
13
13
13
16
+5
12
5
12
Primal-dual analysis
• For found by our algorithm, set and such that
• primal objective = dual objective
• for all edges
x αu βv
αu + βv ≥ 1 − δe−δ
Summary• Semi-online bipartite matching
• Algorithm:
• Hardness:
• Open problem: close the gap
• Fractional case
• Algorithm and hardness:
1 − δ + δ2(1 − 1/e)
1 − δe−δ
1 − δe−δ
Sets puzzle• Ground set with elements
• Collection of sets
• Each contains elements of
• Player 1: pick , maximize
• Player 2: pick , minimize
• Show: there is a randomized strategy for player 1 to guarantee
n
𝒮
S ∈ 𝒮 d [n]
A ∈ 𝒮 |A ∩ B |
B ∈ 𝒮 |A ∩ B |
𝔼[ |A ∩ B | ] ≥ d2/n
![} o o ] } v } ( v l } ] } v , Ì / v Æ ~ ,/ v E v l ^ ~E ^ µ ] v P …...' v ( µ v ] v P ~ o v t D v P u v d µ & µ v X D l o v µ P } µ v Ç ^ u Z } ] } v Z v l ] v P ^ Ç u](https://static.documents.pub/doc/80x56/5f7bc98a66f64a089a2d66eb/-o-o-v-v-l-v-oe-v-v-e-v-l-e-v-p-v-.jpg)
![Ã 1 p » 2 · 2019. 9. 26. · > v P l Z r o v P l Z u v P Z ] h i ] v E ] } v oH / o u µ < Á v v l. > v P l Z r o v P l Z u v P Z ] h i ] v E ] } v oH / o u µ < Á](https://static.documents.pub/doc/80x56/5fbd83a431870c110202f2e3/f-1-p-2-2019-9-26-v-p-l-z-r-o-v-p-l-z-u-v-p-z-h-i-v-e-v-oh.jpg)
![D v P u v ' µ ] ( } W µ o ] W ] } v ] v ^ ] v . Z Z · 2015-03-18 · À ] ] v P v µ v P v Æ ] v P } v X t Z v } u } o o Z } Z µ ] u v } ( µ v l ] v P ] Ì v ] v](https://static.documents.pub/doc/80x56/5ede49ebad6a402d66699c79/d-v-p-u-v-w-o-w-v-v-v-z-z-2015-03-18-v-p-v.jpg)
![P µ ] v P Z ] o v µ ] v P > } l } Á v€¦ · ^ ( P µ ] v P Z ] o v µ ] v P > } l } Á v d Z o } } ( ] + v Á Ç Ç } µ v P Z o ] ( Ç } µ Á } ] Title: C:UserscscfsgsmAppDataLocalTempmso9EAB.tmp](https://static.documents.pub/doc/80x56/5f483ee86fe8343e605bd708/p-v-p-z-o-v-v-p-l-v-p-v-p-z-o-v-v-p-.jpg)
![Z ] o Á Ç h v l ] v P [ , v } } l ( } / v v ] } v o } v ] v P v Ç D v P u v · 2019-12-19 · Z ] o Á Ç h v l ] v P [ , v } } l ( } / v v ] } v o } v ] v P v Ç D v P u v À](https://static.documents.pub/doc/80x56/5f2eab6074991d12ea1c794e/z-o-h-v-l-v-p-v-l-v-v-v-o-v-v-p-v-d-v-p-u-v-2019-12-19.jpg)
![o ] v P v P ] v & ] ] } v v ] P v · o ] v P v P ] v & ] ] } v v ] P v ... e\](https://static.documents.pub/doc/80x56/5e7a62a857e1734ede40a010/o-v-p-v-p-v-v-v-p-v-o-v-p-v-p-v-v-v-p-v-.jpg)
![11 Ashok Leyland Hosur - greenbusinesscentre.comgreenbusinesscentre.com/energyaward2017presentations/Auto Engg/11... · v P Ç D v P u v K P v ] Ì ] } v ^ µ µ 9D } v ] } ] v P](https://static.documents.pub/doc/80x56/5b3f933b7f8b9af6438c5004/11-ashok-leyland-hosur-engg11-v-p-c-d-v-p-u-v-k-p-v-i-v-.jpg)
![FINAL for Public Release - Oranga Tangata Oranga Whanau ... · K v P d v P U K v P t Z v µ í K v P d v P U K v P t Z v µ K v P d v P U K v P t Z v µ î Æ µ ] À ^ µ u u Ç](https://static.documents.pub/doc/80x56/5f8a7988b7732707fe0132ae/final-for-public-release-oranga-tangata-oranga-whanau-k-v-p-d-v-p-u-k-v-p.jpg)
![& o ^ o µ P v ^ µ o o P D v P u v ] v h v D Z Z](https://static.documents.pub/doc/80x56/615afd2c0eadf003b0110506/amp-o-o-p-v-o-o-p-d-v-p-u-v-v-h-v-d-z-z.jpg)
![Dh^d ] P v } v >> ð P · 2020-02-20 · Dh^d ] P v } v >> ð P / ( Ç } µ Á ] Z } ] v P v Ç v v } , } v P < } v P U r Ç } µ ] o o Z À } } } µ Z ] ] v Z ] (](https://static.documents.pub/doc/80x56/5ec03c42e4b78026ce4b524a/dhd-p-v-v-p-2020-02-20-dhd-p-v-v-p-.jpg)
![Policy paper MITRE MDS V1 · , u } v ] ] v P P } À v u v P µ ] v À ] µ } ( P o } o o Ç } P v ] Ç µ ] Ç](https://static.documents.pub/doc/80x56/5b9811bf09d3f2e3488ce6a0/policy-paper-mitre-mds-v1-u-v-v-p-p-a-v-u-v-p-v-a-.jpg)
![Asylum, relocation and return of migrants: Time to step up ... · ^ l ] o } Ç o U } u o ] v P } Z À v v v P o o u ] P v W P î / v v Z o o ( l v ] v P W µ v l ^ u u v ( v ] v P](https://static.documents.pub/doc/80x56/5f926edf874c123aca2c4281/asylum-relocation-and-return-of-migrants-time-to-step-up-l-o-o-u.jpg)
![v ] u ' µ ] v ( } ^ ( ^ o ] v P ( } h v Z o W } v Æ ] v ... · 3DJH RI À ] ( ( } À } ] Z v o ] v P o ] v o } v P ] v P ] (](https://static.documents.pub/doc/80x56/5fa4ae4b8e1f4f7b7057a033/v-u-v-o-v-p-h-v-z-o-w-v-v-3djh-ri-.jpg)
