Lower Bounds for Maximal Matchings and Maximal Independent Sets - PowerPoint PPT Presentation

Lower Bounds for Maximal Matchings and Maximal Independent Sets Alkida Balliu Aalto University, Finland

Joint work with Sebastian Brandt · ETH Zurich Juho Hirvonen · Aalto University Dennis Olivetti · Aalto University Mikaël Rabie · LIP6 - Sorbonne University Jukka Suomela · Aalto University 2

Overview Maximal matching Maximal independent set We will talk about lower bounds for solving these problems in the distributed setting 3

Distributed setting Graph = communication network; synchronous rounds; time = number of communication rounds 4

Maximal matching problem Input Output • Matching : edges in the matching do not share a node • Maximality : if we add any other edge in the matching, than it is not a matching anymore • We say that a node is matched : it is an endpoint of an edge in the matching 5

Maximal independent set problem Input Output • Independent set : nodes in the IS do not share an edge • Maximality : if we add any other node in the IS, than it is not independent anymore 6

Two classical graph problems Maximal matching Maximal independent set Easy linear-time centralized algorithm: add edges/nodes until stuck 7

Two classical graph problems Maximal matching Maximal independent set Can be verified locally : if it looks correct everywhere locally, it is also feasible globally Can these problems be solved locally ? 8

Locality = how far do I need to see to produce my own part of the solution? 9

Locality = how far do I need to see to produce my own part of the solution? I will output I will output I will output in out in 10

Locality = how far do I need to see to produce my own part of the solution? Local outputs form a globally consistent solution 11

Warmup: toy example Bipartite graphs & port-numbering model 12

1 1 2 2 computer output: 1 2 1 2 network with maximal 3 3 3 3 port numbering matching bipartite, 2 2 1 1 2-colored 3 3 23 3 graph 1 2 1 Δ-regular (here Δ = 3) 1 1 1 1 2 3 2 3 2 3 3 2 1 3 1 3 1 3 3 1 2 2 2 2 13

1 2 Very simple algorithm 1 2 3 3 unmatched white nodes: send proposal to port 1 2 1 3 23 1 1 1 2 3 2 3 1 1 3 3 2 2 14

1 2 Very simple algorithm 1 2 3 3 unmatched white nodes: send proposal to port 1 2 1 3 black nodes: 23 1 accept the first proposal you get, reject everything else (break ties with port numbers) 1 1 2 3 2 3 1 1 3 3 2 2 15

1 2 Very simple algorithm 1 2 3 3 unmatched white nodes: send proposal to port 1 2 1 3 black nodes: 23 1 accept the first proposal you get, reject everything else (break ties with port numbers) 1 1 2 3 2 3 1 1 3 3 2 2 16

1 2 Very simple algorithm 1 2 3 3 unmatched white nodes: send proposal to port 2 2 1 3 23 1 1 1 2 3 2 3 1 1 3 3 2 2 17

1 2 Very simple algorithm 1 2 3 3 unmatched white nodes: send proposal to port 2 2 1 3 black nodes: 23 1 accept the first proposal you get, reject everything else (break ties with port numbers) 1 1 2 3 2 3 1 1 3 3 2 2 18

1 2 Very simple algorithm 1 2 3 3 unmatched white nodes: send proposal to port 2 2 1 3 black nodes: 23 1 accept the first proposal you get, reject everything else (break ties with port numbers) 1 1 2 3 2 3 1 1 3 3 2 2 19

1 2 Very simple algorithm 1 2 3 3 unmatched white nodes: send proposal to port 3 2 1 3 23 1 1 1 2 3 2 3 1 1 3 3 2 2 20

1 2 Very simple algorithm 1 2 3 3 unmatched white nodes: send proposal to port 3 2 1 3 black nodes: 23 1 accept the first proposal you get, reject everything else (break ties with port numbers) 1 1 2 3 2 3 1 1 3 3 2 2 21

1 2 Very simple algorithm 1 2 3 3 unmatched white nodes: send proposal to port 3 2 1 3 black nodes: 23 1 accept the first proposal you get, reject everything else (break ties with port numbers) 1 1 2 3 2 3 1 1 3 3 2 2 22

1 2 Very simple algorithm 1 2 3 3 Finds a maximal matching in O (Δ) communication rounds 2 1 Note: running time does 3 23 not depend on n 1 1 1 2 3 2 3 1 1 3 3 2 2 23

Bipartite maximal matching • Maximal matching in very large 2-colored Δ-regular graphs • Simple algorithm: O (Δ) rounds , independently of n • Is this optimal? • o (Δ) rounds? • O (log Δ) rounds? • 4 rounds?? 24

Big picture Bounded-degree graphs & LOCAL model 25

LOCAL model • Each node has a unique identifier from 1 to poly( n ) 1 17 22 16 21 12 • No bounds on the computational 31 30 8 4 7 29 5 power 10 6 35 19 3 20 33 14 42 34 • No bounds on the bandwidth 24 36 13 18 23 25 27 • Synchronous model 15 40 28 32 26 38 • Everything can be solved in 2 9 44 41 11 Diameter time Strong model — lower bounds widely applicable 26

Maximal matching, LOCAL model, O(f( Δ ) + g(n)) Algorithms: deterministic randomized Lower bounds: deterministic g ( n ) randomized f (Δ) 27

Maximal matching, LOCAL model, O(f( Δ ) + g(n)) Algorithms: Israeli & Itai (1986) log n deterministic randomized O (log n ) randomized Lower bounds: deterministic randomized o (log* n ) impossible log ∗ n Linial (1987, 1992), Naor (1991) 28

log 7 n Hanckowiak et al. (1998) Maximal matching, LOCAL model, log 4 n Hanckowiak et al. (2001) O(f( Δ ) + g(n)) log 3 n Fischer (2017) Algorithms: Israeli & Itai (1986) log n deterministic randomized Lower bounds: deterministic polylog( n ) deterministic randomized log ∗ n Linial (1987, 1992), Naor (1991) 29

log 7 n Hanckowiak et al. (1998) Maximal matching, LOCAL model, log 4 n Hanckowiak et al. (2001) O(f( Δ ) + g(n)) log 3 n Fischer (2017) Algorithms: Israeli & Itai (1986) log n deterministic randomized Lower bounds: deterministic randomized O (Δ + log* n ) deterministic Panconesi & Rizzi (2001) log ∗ n Linial (1987, 1992), Naor (1991) ∆ 30

log 7 n Hanckowiak et al. (1998) Maximal matching, LOCAL model, log 4 n Hanckowiak et al. (2001) O(f( Δ ) + g(n)) log 3 n Fischer (2017) Algorithms: Israeli & Itai (1986) log n deterministic randomized s log n log log n Lower bounds: deterministic randomized Kuhn et al. (2004, 2016) Panconesi & Rizzi (2001) log ∗ n log ∆ Linial (1987, 1992), Naor (1991) ∆ log log ∆ 31

log 7 n Hanckowiak et al. (1998) Maximal matching, LOCAL model, log 4 n Hanckowiak et al. (2001) O(f( Δ ) + g(n)) log 3 n Fischer (2017) Algorithms: Israeli & Itai (1986) log n deterministic randomized s log n O (log Δ + polylog log n ) log log n Lower bounds: deterministic Barenboim et al. log 4 log n (2012, 2016) randomized log 3 log n Fischer (2017) Kuhn et al. (2004, 2016) Panconesi & Rizzi (2001) log ∗ n log ∆ log ∆ Linial (1987, 1992), Naor (1991) ∆ log log ∆ 32

log 7 n Hanckowiak et al. (1998) Maximal matching, LOCAL model, log 4 n Hanckowiak et al. (2001) O(f( Δ ) + g(n)) log 3 n Fischer (2017) Algorithms: Israeli & Itai (1986) log n deterministic randomized s log n log log n Lower bounds: deterministic Barenboim et al. log 4 log n (2012, 2016) randomized log 3 log n Fischer (2017) Kuhn et al. (2004, 2016) Panconesi & Rizzi (2001) log ∗ n log ∆ log ∆ Linial (1987, 1992), Naor (1991) ∆ log log ∆ 33

log 7 n Hanckowiak et al. (1998) Maximal matching, LOCAL model, log 4 n Hanckowiak et al. (2001) O(f( Δ ) + g(n)) log 3 n Fischer (2017) Algorithms: Israeli & Itai (1986) log n deterministic randomized s log n log log n Lower bounds: deterministic Barenboim et al. log 4 log n (2012, 2016) randomized log 3 log n Fischer (2017) O (log Δ + log* n ) ??? Kuhn et al. (2004, 2016) Panconesi & Rizzi (2001) log ∗ n log ∆ log ∆ Linial (1987, 1992), Naor (1991) ∆ log log ∆ 34

log 7 n Hanckowiak et al. (1998) Maximal matching, LOCAL model, log 4 n Hanckowiak et al. (2001) O(f( Δ ) + g(n)) log 3 n Fischer (2017) Algorithms: Israeli & Itai (1986) log n deterministic randomized s log n log log n Lower bounds: deterministic Barenboim et al. log 4 log n (2012, 2016) randomized log 3 log n Fischer (2017) Kuhn et al. ??? (2004, 2016) Panconesi & Rizzi (2001) log ∗ n log ∆ log ∆ Linial (1987, 1992), Naor (1991) ∆ log log ∆ 35

log 7 n Hanckowiak et al. (1998) Maximal matching, LOCAL model, log 4 n Hanckowiak et al. (2001) O(f( Δ ) + g(n)) log 3 n Fischer (2017) Algorithms: Israeli & Itai (1986) log n deterministic randomized s log n log log n Lower bounds: deterministic Barenboim et al. log 4 log n (2012, 2016) randomized log 3 log n Fischer (2017) log log n log log log n Kuhn et al. New (2004, 2016) Panconesi & Rizzi (2001) log ∗ n log ∆ log ∆ Linial (1987, 1992), Naor (1991) ∆ log log ∆ 36