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 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 U • Nodes in known in advance U V V • Nodes in arrive one by one • Match at each step • Competitive ratio compares to offline OPT
Online bipartite matching 1 − 1/ e • 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] Richard Karp, Umesh Vazirani, Vijay Vazirani. An optimal algorithm for on-line bipartite matching. STOC 1990
Semi-online bipartite matching U V • Know and part of in advance V • All of arrives one by one in arbitrary order • Match at each step • Competitive ratio compares to offline OPT • Integral or fractional matching
Notation H G = ( U , V , E G ) • Bipartite graph V P V = V P ∪ V A • V P V • : known (predicted) part of V A V • : unknown (adversarial) part V A of H = ( U , V P , E H ) • Known subgraph U V
δ Online/offline parameter G • Simplifying assumption for this talk: perfect matching in δ = | V A | • , fraction of adversarial nodes | V | δ = 0 δ = 1 • : offline, : online δ • Competitive ratio in terms of δ = 1 − OPT ( H ) • General case: OPT ( G ) • Other definition doesn't work if many isolated nodes
Results • Integral matching: • Algorithm with competitive ratio 1 − δ + δ 2 (1 − 1/ e ) 1 − δ e − δ • Hardness of ( ≈ 1 − δ + δ 2 − δ 3 /2 + . . . ) • Fractional matching: 1 − δ e − δ • Algorithm and hardness of
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 H • Should select a maximum matching on • No benefit to leaving predicted nodes unmatched • Do this as preprocessing
H Selecting a matching for • Any deterministic algorithm would do badly
Algorithm outline H • Find a (randomized) maximum matching in V A • Which nodes to "reserve" for ? • Run RANKING for adversarial nodes
Analysis outline Reserved ⊆ U H • : not matched in . | Reserved | = n − | V P | = δ n Marked ⊆ U V A • : matched to by OPT. | Marked | = | V A | = δ n 𝔽 [ | Reserved ∩ Marked | ] = x ⋅ n • Suppose n − δ n + (1 − 1/ e ) xn • Matching size 1 − δ + (1 − 1/ e ) x • Competitive ratio x = δ 2 • Aim for
Reserving nodes H • Goal: sample a matching in s.t. 𝔽 [ | Reserved ∩ Marked | ] = δ 2 n H • Special case: is complete δ • Reserve each node with probability • In general, a distribution over matchings s.t. ∀ u ∈ U , Pr[ u is reserved ] = δ may not exist • Want a distribution making nodes' probabilities of being reserved as equal as possible
Matching skeleton decomposition 1 H • Decomposition of (poly-time) U = ∪ i T i V P = ∪ i S i • , Γ ( ∪ i < j S i ) = ∪ i < j T i • T i S i S j i < j ⇒ S i > • T i T j • Fractional matching in each component deg( u ) = 1 deg( v ) = | S i | / | T i | • , [1] Ashish Goel, Michael Kapralov, Sanjeev Khanna. On the communication and streaming complexity of maximum bipartite matching. SODA 2012.
Dependent rounding • Apply dependent rounding [1] to each component of the matching 2 skeleton 3 1 3 d i = | T i | − | S i | T i S i • Let 1 3 u ∈ T i • Probability of being 2 3 d i reserved is | T i | [1] Rajiv Gandhi, Samir Khuller, Srinivasan Parthasarathy, Aravind Srinivasan. Dependent rounding and its applications to approximation algorithms. JACM 53(3):324–360, 2006.
Marked nodes • Adversary's goal: δ n U H • Mark nodes in whose complement has a matching in • Minimize overlap with reserved nodes • Best strategy: d i = | T i | − | S i | i • Select nodes per component | T i | ≥ ( δ n ) 2 𝔽 [ | Reserved ∩ Marked | ] = ∑ d i d i ⋅ • n i (by Cauchy-Schwarz) ⇒ 1 − δ + δ 2 (1 − 1/ e ) • competitive ratio
Hardness bound U A U : u 1 u 2 u 3 u 4 u 5 u 6 u 7 u 8 u 9 u 10 u 11 u 12 V : v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 v 9 v 10 v 11 v 12 V P V A • Predicted: complete graph; adversarial: block upper triangular 1 − δ e − δ • Hardness of
Fractional matching • Online model V U • Nodes of arrive one at a time, have to be fractionally matched to 1 − 1/ e • Water-level algorithm [1] gives optimal ratio • Match to the neighbor with lowest existing amount • Semi-online fractional bipartite matching 1 − δ e − δ • We get tight bounds of • Primal-dual analysis extension of [2] [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 V P • For predicted nodes : 1 2 • Take fractional matching 1 2 from skeleton decomposition H of 1 3 V A • For adversarial nodes : 1 5 3 12 1 • Use water-level algorithm 3 1 6 + 5 12
Primal-dual analysis x α u β v • For found by our algorithm, set and such that • primal objective = dual objective α u + β v ≥ 1 − δ e − δ • for all edges
Summary • Semi-online bipartite matching 1 − δ + δ 2 (1 − 1/ e ) • Algorithm: 1 − δ e − δ • Hardness: • Open problem: close the gap • Fractional case 1 − δ e − δ • Algorithm and hardness:
Sets puzzle n • Ground set with elements 𝒯 • Collection of sets [ n ] S ∈ 𝒯 d • Each contains elements of | A ∩ B | A ∈ 𝒯 • Player 1: pick , maximize B ∈ 𝒯 | A ∩ B | • Player 2: pick , minimize • Show: there is a randomized strategy for player 1 to 𝔽 [ | A ∩ B | ] ≥ d 2 / n guarantee
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