Online bipartite matching inoffline time

Bartłomiej Bosek1 Dariusz Leniowski2

Piotr Sankowski2 Anna Zych2

1Jagiellonian University

2University of Warsaw

FIT 2015, 31 January 2015

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Problem statement

• A bipartite graph G = 〈U ] V,E〉is revealed online.

• The vertices in U are fixed.

• The vertices from V arrive one per turn,each with all its incident edges.

• We want to maintain the maximumcardinality matching, and if possible,keep the number of reallocations low.

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Problem statement — example

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Problem statement — example

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Problem statement — example

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Problem statement — example

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Problem statement — example

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Our results

• Bound: it is possible to maintain maximummatching using 2

√n reallocations per vertex.

• Exact: in O(m√

n) total timemaintains the maximum cardinality matching.

• Approximation: in O(ε−1m) total timemaintains (1− ε)-approximation factor.

• Reductions: vertex-decremental (same bounds),weighted (pseudopolynomial bound).

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Motivation

• Servers and new jobs to process,edges denote eligibility.

• We want to admit as many requests aspossible — we reallocate them when necessary.

• Applications include streaming contentdelivery, web hosting, remote data storage,job scheduling, hashing, etc.

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History

• Grove, Kao, Krishnan and Vitter, ’95:at most 2 servers per client using O(log n)reallocations per vertex, O(n log n) total.

• Chaudhuri, Daskalakis, Kleinberg and Lin ’09:forests or particular random graphs,total bound of O(n log n).

• General case:only the trivial O(n2) bound was known.

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Relation to other models

• Karp, Vazirani and Vazirani, ’90:no reallocations, O(m) total time,approximation factor of (1− 1/e) ≈ 0.63.

• Gupta, Peng ’13: (1− ε)-approximationin O(m

√m) total time.

• Our result: (1− ε)-approximationin O(ε−1m) total time, exact in O(m

√n)

time, matches the running time of theoffline Hopcroft-Karp algorithm.

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Main idea

• For each u ∈ U we keep track of its rank,i.e., the number of times it was used byaugmenting paths up to this point.

• Suppose that vertex of V arrives.

• Pick an augmenting path that minimizesthe maximum rank for its every suffix.

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Ranks and tiers

RankGiven a vertex u ∈ U

rankt(u) = how many times augmenting paths used u,rankt(P) = max

u∈Prankt(u).

TierGiven a vertex w ∈ V ]U,

tiert(w) = the rank of the lowest-ranked alternatingpath to any unmatched vertex u ∈ U

= minunmatched u∈U

minalternating path P : w→u

rankt(P).

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Tiered paths

A path is called tiered if the tiers are non-increasing.

0

( )

tu

1

2

3

0

1

2

3 3

3

rank

because of this rank tier here is 3

because of this rise ( )

yellow path is not tiered

0 0

1

2 3

1

3

0

1

2

green path is tiered:

there is a rise in the ranks,

but the tier is non-increasing

1 2

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Tiers — a few notes

• Tiers do not decrease in time.

• tier(w) ≤ distALT(w,unmatched u).

• Tiers are hard to maintain.

• Instead, for any w we keep tierLB(w),a number such that tierLB(w) ≤ tiert(w);set tierLB(w) = 0 and update as necessary.

• Theorem: both tiers and ranks are O(√

n).

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Algorithm — a sketch

1. Add a vertex vt and assume its tierLB is 0.

2. Try to find a tiered augmenting pathof rank at most tierLB(vt).

3a. Failed search means that some tierLBwas too low, so raise it and repeat.

3b. For a successful search raise the ranks.

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Algorithm — example

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Algorithm — example

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Algorithm — example

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Algorithm — example

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Algorithm — approximation

• We set an artificial upper bound tierLB ≤ k.

• Vertices that went over are left unmatched.

• If the ranks of all augmenting paths are ≥ k,the matching is (1− 2/k)-approximate.

• With O(ε−1) changes per vertex, we get(1− ε)-approximation in O(ε−1m) time.

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Concluding remarks

• New type of augmenting path algorithms.

• The reallocation problem has bound 2√

n.

• Incremental or decremental,exact or approximate versions.

• It’s simple and combinatorial.

• Can the bound of O(n log n) be reached?

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Concluding remarks

• New type of augmenting path algorithms.

• The reallocation problem has bound 2√

n.

• Incremental or decremental,exact or approximate versions.

• It’s simple and combinatorial.

• Can the bound of O(n log n) be reached?

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Thank you!