Towards a Complexity Theory for the Congested Clique Janne H. Korhonen Jukka Suomela Aalto University
The Congested Clique • a fully connected distributed model • specialisation of the standard CONGEST
The Congested Clique • n nodes • communication graph = clique • input graph = arbitrary graph • synchronous, error-free • O(log n ) bandwidth/edge/round • unlimited local computation • a fully connected distributed model • time measure: number of rounds • specialisation of the standard CONGEST
The Congested Clique local input: incident edges • n nodes • communication graph = clique • input graph = arbitrary graph • synchronous, error-free • O(log n ) bandwidth/edge/round • unlimited local computation • a fully connected distributed model • time measure: number of rounds • specialisation of the standard CONGEST
The Congested Clique • everything O( n /log n ) • very good upper bounds for many problems • a fully connected distributed model • specialisation of the standard CONGEST
The Congested Clique • everything O( n /log n ) • very good upper bounds for many problems • No lower bounds • no bottlenecks or distances • CONGEST/LOCAL techniques fail • connections to circuit complexity • a fully connected distributed model • specialisation of the standard CONGEST
The Congested Clique This work: more “traditional” complexity theory view • a fully connected distributed model • specialisation of the standard CONGEST
What does the complexity landscape of the congested clique look like? Turing LOCAL clique machines
Highlight 1: Time Hierarchy there are decision problems of any possible complexity in the congested clique Theorem. For increasing computable functions f , g such that f = o ( g ) , we have CLIQUE( f ( n )) ⊊ CLIQUE( g ( n ))
Nonuniform protocols [Applebaum, Kowalski, Patt-Shamir & Rosén] x 7 x 8 x 6 • fix n , bandwidth B x 9 x 5 • each node i gets k input bits x i • want to compute some binary function h ( x 1 ,…, x n ) x 4 x 10 x 11 x 3 x 12 x 2 x 1
Nonuniform protocols [Applebaum, Kowalski, Patt-Shamir & Rosén] x 7 x 8 x 6 • fix n , bandwidth B x 9 x 5 • each node i gets k input bits x i • want to compute some binary function h ( x 1 ,…, x n ) x 4 x 10 • most functions require k / B rounds x 11 x 3 (counting argument) x 12 x 2 x 1
Nonuniform protocols [Applebaum, Kowalski, Patt-Shamir & Rosén] x 7 x 8 x 6 • fix n , bandwidth B x 9 x 5 • each node i gets k input bits x i • want to compute some binary function h ( x 1 ,…, x n ) x 4 x 10 • most functions require k / B rounds x 11 x 3 (counting argument) x 12 x 2 x 1 “ Lifting” into time hierarchy theorem • for each n , pick a function h n with complexity g ( n ) in deterministically computable manner • similar to time hierarchy for circuit complexity
What is the correct notion of “interesting problem?” ( easy to verify, difficult to solve ) LOCAL: clique: centralised: LCL problems ??? NP-complete
Highlight 2: NCLIQUE(1) Problems a natural congested clique analogue for NP problems and LCL problems (LOCAL model)
Highlight 2: NCLIQUE(1) Problems a natural congested clique analogue for NP problems and LCL problems (LOCAL model) x 7 ,y 7 NCLIQUE(1) x 8 ,y 8 x 6 ,y 6 • Constant-round verifier that takes x 9 ,y 9 x 5 ,y 5 an input and a certificate • Yes-instance if and only if there is x 4 ,y 4 x 10 ,y 10 an accepting certificate x 11 ,y 11 x 3 ,y 3 • Corresponding search problem : find an accepting certificate x 12 ,y 12 x 2 ,y 2 x 1 ,y 1
Highlight 2: NCLIQUE(1) Problems a natural congested clique analogue for NP problems and LCL problems (LOCAL model) x 7 ,y 7 NCLIQUE(1) Certificate length? x 8 ,y 8 x 6 ,y 6 x 9 ,y 9 x 5 ,y 5 x 4 ,y 4 x 10 ,y 10 x 11 ,y 11 x 3 ,y 3 x 12 ,y 12 x 2 ,y 2 x 1 ,y 1
Highlight 2: NCLIQUE(1) Problems a natural congested clique analogue for NP problems and LCL problems (LOCAL model) x 7 ,y 7 Certificate length? x 8 ,y 8 x 6 ,y 6 x 9 ,y 9 x 5 ,y 5 • NCLIQUE(1) problems always have x 4 ,y 4 x 10 ,y 10 a verifiers/certificates with O ( n log n ) bits per node x 11 ,y 11 x 3 ,y 3 x 12 ,y 12 x 2 ,y 2 x 1 ,y 1
Highlight 2: NCLIQUE(1) Problems a natural congested clique analogue for NP problems and LCL problems (LOCAL model) x 7 ,y 7 Examples: x 8 ,y 8 x 6 ,y 6 x 9 ,y 9 x 5 ,y 5 Maximal independent set • Hamiltonian cycle • x 4 ,y 4 3-colouring x 10 ,y 10 • Canonical problem family: • x 11 ,y 11 x 3 ,y 3 edge labelling problems x 12 ,y 12 x 2 ,y 2 x 1 ,y 1
Highlight 2: NCLIQUE(1) Problems a natural congested clique analogue for NP problems and LCL problems (LOCAL model) x 7 ,y 7 Examples: x 8 ,y 8 x 6 ,y 6 x 9 ,y 9 x 5 ,y 5 Maximal independent set • ? Hamiltonian cycle • CLIQUE(1) = NCLIQUE(1) x 4 ,y 4 3-colouring x 10 ,y 10 • Canonical problem family: • x 11 ,y 11 x 3 ,y 3 edge labelling problems x 12 ,y 12 x 2 ,y 2 x 1 ,y 1
More on nondeterminism NCLIQUE( T ( n )) • Certificate size is bounded by the running time of the verifier • Allows extension of time hierarchy theorem to NCLIQUE : NCLIQUE( f ( n )) ⊊ NCLIQUE( g ( n )) Constant-round decision hierarchy • Constant rounds, alternating quantifiers: Σ 1 , Π 1 , Σ 2 , Π 2 , … • Analogue(s) of polynomial hierarchy • Certificate size matters much more
Reduction-based perspective to relationships between natural problems? ( subpolynomial vs. polynomial ) • Triangle: O ( n 0.157 ) • MST: O (1) • APSP: O ( n 1/3 ) • MIS: O (log log Δ ) • k -subgraph: O ( n 1-2/k )
Highlight 3: Fine-grained Complexity fine-grained complexity is a useful tool for understanding polynomial complexities
Highlight 3: Fine-grained Complexity fine-grained complexity is a useful tool for understanding polynomial complexities δ ( P ) = inf { δ : P ∈ CLIQUE( n δ ) }
Highlight 3: Fine-grained Complexity fine-grained complexity is a useful tool for understanding polynomial complexities δ ( P ) = inf { δ : P ∈ CLIQUE( n δ ) } δ (Ring-MM) ≤ 0.157… δ (APSP) ≤ 1/3 δ ( k -IS) ≤ 1 − 2/k δ ( k -COL) ≤ 1
Highlight 3: Fine-grained Complexity fine-grained complexity is a useful tool for understanding polynomial complexities δ ( P ) = inf { δ : P ∈ CLIQUE( n δ ) } δ (Ring-MM) ≤ 0.157… • Proving relationships of form δ (APSP) ≤ 1/3 δ ( P ) ≤ δ ( Q ) via δ ( k -IS) ≤ 1 − 2/k subpolynomial reductions δ ( k -COL) ≤ 1
0 1-2/ ω 1/3 1-2/k 1-1/k 1 0.2096 Semiring Ring MM MM (min,+) APSP APSP APSP MM w/ud/(1+ ε ) uw/d w/d APSP w/ud SSSP SSSP SSSP APSP w/ud/(1+ ε ) w/ud w/d w/ud/(2- ε ) Boolean APSP MinVC MM uw/ud Transitive SSSP SSSP size 3 size k k-cycle k-DS MaxIS closure uw/ud uw/d subgraph subgraph Triangle/ BFS tree k-IS k-COL 3-IS
0 1-2/ ω 1/3 1-2/k 1-1/k 1 0.2096 Semiring Ring MM MM (min,+) APSP APSP APSP MM w/ud/(1+ ε ) uw/d w/d APSP w/ud SSSP SSSP SSSP APSP w/ud/(1+ ε ) w/ud w/d w/ud/(2- ε ) Boolean APSP MinVC MM uw/ud Transitive SSSP SSSP size 3 size k k-cycle k-DS MaxIS closure uw/ud uw/d subgraph subgraph Triangle/ BFS tree k-IS k-COL 3-IS
0 1-2/ ω 1/3 1-2/k 1-1/k 1 0.2096 Semiring Ring MM MM (min,+) APSP APSP APSP MM w/ud/(1+ ε ) uw/d w/d APSP w/ud SSSP SSSP SSSP APSP w/ud/(1+ ε ) w/ud w/d w/ud/(2- ε ) Boolean APSP MinVC MM uw/ud Transitive SSSP SSSP size 3 size k k-cycle k-DS MaxIS closure uw/ud uw/d subgraph subgraph Triangle/ BFS tree k-IS k-COL 3-IS
0 1-2/ ω 1/3 1-2/k 1-1/k 1 0.2096 Semiring Ring MM MM (min,+) APSP APSP APSP MM w/ud/(1+ ε ) uw/d w/d APSP w/ud SSSP SSSP SSSP APSP w/ud/(1+ ε ) w/ud w/d w/ud/(2- ε ) Boolean APSP MinVC MM uw/ud Transitive SSSP SSSP size 3 size k k-cycle k-DS MaxIS closure uw/ud uw/d subgraph subgraph Triangle/ BFS tree k-IS k-COL 3-IS
0 1-2/ ω 1/3 1-2/k 1-1/k 1 0.2096 Semiring Ring MM MM (min,+) APSP APSP APSP MM w/ud/(1+ ε ) uw/d w/d APSP w/ud SSSP SSSP SSSP APSP w/ud/(1+ ε ) w/ud w/d w/ud/(2- ε ) Boolean APSP MinVC MM uw/ud Transitive SSSP SSSP size 3 size k k-cycle k-DS MaxIS closure uw/ud uw/d subgraph subgraph Triangle/ BFS tree k-IS k-COL 3-IS
0 1-2/ ω 1/3 1-2/k 1-1/k 1 0.2096 Semiring Ring MM MM (min,+) APSP APSP APSP MM w/ud/(1+ ε ) uw/d w/d APSP w/ud k -vertex cover: O ( k ) rounds SSSP SSSP SSSP APSP w/ud/(1+ ε ) w/ud w/d w/ud/(2- ε ) Boolean APSP MinVC MM uw/ud Transitive SSSP SSSP size 3 size k k-cycle k-DS MaxIS closure uw/ud uw/d subgraph subgraph Triangle/ BFS tree k-IS k-COL 3-IS
Closing remarks: Beyond the congested clique? message passing clique models ( k-machine model, BSP, MapReduce, MPC,… )
Closing remarks: Beyond the congested clique? message passing clique models n vertices in input n vertices in input n processors p < n processors
Closing remarks: Beyond the congested clique? message passing clique models n vertices in input n vertices in input n processors p < n processors
Thanks! Questions? (arXiv:1705.03284)
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