Towards a Complexity Theory for the Congested Clique Janne H. - - PowerPoint PPT Presentation

▶
towards a complexity theory for the congested clique
SMART_READER_LITE
LIVE PREVIEW

Towards a Complexity Theory for the Congested Clique Janne H. - - PowerPoint PPT Presentation

Oct 07, 2022 215 likes •573 views

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

slide-1
SLIDE 1

Towards a Complexity Theory for the Congested Clique

Janne H. Korhonen Jukka Suomela

Aalto University

slide-2
SLIDE 2

The Congested Clique

  • a fully connected distributed model
  • specialisation of the standard CONGEST
slide-3
SLIDE 3

The Congested Clique

  • a fully connected distributed model
  • specialisation of the standard CONGEST
  • n nodes
  • communication graph = clique
  • input graph = arbitrary graph
  • synchronous, error-free
  • O(log n) bandwidth/edge/round
  • unlimited local computation
  • time measure: number of rounds
slide-4
SLIDE 4

The Congested Clique

  • a fully connected distributed model
  • specialisation of the standard CONGEST

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
  • time measure: number of rounds
slide-5
SLIDE 5
  • everything O(n/log n)
  • very good upper bounds

for many problems

The Congested Clique

  • a fully connected distributed model
  • specialisation of the standard CONGEST
slide-6
SLIDE 6
  • 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

The Congested Clique

  • a fully connected distributed model
  • specialisation of the standard CONGEST
slide-7
SLIDE 7

This work: more “traditional” complexity theory view

The Congested Clique

  • a fully connected distributed model
  • specialisation of the standard CONGEST
slide-8
SLIDE 8

What does the complexity landscape of the congested clique look like?

LOCAL clique Turing machines

slide-9
SLIDE 9

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))

slide-10
SLIDE 10

Nonuniform protocols

[Applebaum, Kowalski, Patt-Shamir & Rosén]

x12 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11

  • fix n, bandwidth B
  • each node i gets k input bits xi
  • want to compute some binary 


function h(x1,…, xn)

slide-11
SLIDE 11

Nonuniform protocols

[Applebaum, Kowalski, Patt-Shamir & Rosén]

x12 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11

  • fix n, bandwidth B
  • each node i gets k input bits xi
  • want to compute some binary 


function h(x1,…, xn)

  • most functions require k/B rounds


(counting argument)

slide-12
SLIDE 12

Nonuniform protocols

[Applebaum, Kowalski, Patt-Shamir & Rosén]

x12 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11

  • fix n, bandwidth B
  • each node i gets k input bits xi
  • want to compute some binary 


function h(x1,…, xn)

  • most functions require k/B rounds


(counting argument)

“Lifting” into time hierarchy theorem

  • for each n, pick a function hn with complexity g(n) in

deterministically computable manner

  • similar to time hierarchy for circuit complexity
slide-13
SLIDE 13

What is the correct notion of “interesting problem?”

LOCAL: LCL problems clique: ??? centralised: NP-complete (easy to verify, difficult to solve)

slide-14
SLIDE 14

Highlight 2:

NCLIQUE(1) Problems

a natural congested clique analogue for NP problems and LCL problems (LOCAL model)

slide-15
SLIDE 15

a natural congested clique analogue for NP problems and LCL problems (LOCAL model)

x12,y12 x1,y1 x2,y2 x3,y3 x4,y4 x5,y5 x6,y6 x7,y7 x8,y8 x9,y9 x10,y10 x11,y11

Highlight 2:

NCLIQUE(1) Problems

  • Constant-round verifier that takes

an input and a certificate

  • Yes-instance if and only if there is

an accepting certificate

  • Corresponding search problem:

find an accepting certificate

NCLIQUE(1)

slide-16
SLIDE 16

a natural congested clique analogue for NP problems and LCL problems (LOCAL model)

x12,y12 x1,y1 x2,y2 x3,y3 x4,y4 x5,y5 x6,y6 x7,y7 x8,y8 x9,y9 x10,y10 x11,y11

Highlight 2:

NCLIQUE(1) Problems

NCLIQUE(1)

Certificate length?

slide-17
SLIDE 17

a natural congested clique analogue for NP problems and LCL problems (LOCAL model)

x12,y12 x1,y1 x2,y2 x3,y3 x4,y4 x5,y5 x6,y6 x7,y7 x8,y8 x9,y9 x10,y10 x11,y11

Highlight 2:

NCLIQUE(1) Problems

  • NCLIQUE(1) problems always have

a verifiers/certificates with O(n log n) bits per node Certificate length?

slide-18
SLIDE 18

a natural congested clique analogue for NP problems and LCL problems (LOCAL model)

x12,y12 x1,y1 x2,y2 x3,y3 x4,y4 x5,y5 x6,y6 x7,y7 x8,y8 x9,y9 x10,y10 x11,y11

Highlight 2:

NCLIQUE(1) Problems

  • Maximal independent set
  • Hamiltonian cycle
  • 3-colouring
  • Canonical problem family:


edge labelling problems

Examples:

slide-19
SLIDE 19

a natural congested clique analogue for NP problems and LCL problems (LOCAL model)

x12,y12 x1,y1 x2,y2 x3,y3 x4,y4 x5,y5 x6,y6 x7,y7 x8,y8 x9,y9 x10,y10 x11,y11

Highlight 2:

NCLIQUE(1) Problems

Examples:

  • Maximal independent set
  • Hamiltonian cycle
  • 3-colouring
  • Canonical problem family:


edge labelling problems

CLIQUE(1) = NCLIQUE(1)

?

slide-20
SLIDE 20
  • Certificate size is bounded by the running time of the verifier
  • Allows extension of time hierarchy theorem to NCLIQUE:

More on nondeterminism

NCLIQUE(T(n))

NCLIQUE(f(n)) ⊊ NCLIQUE(g(n))

Constant-round decision hierarchy

  • Constant rounds, alternating quantifiers: Σ1, Π1, Σ2, Π2, …
  • Analogue(s) of polynomial hierarchy
  • Certificate size matters much more
slide-21
SLIDE 21

Reduction-based perspective to relationships between natural problems?

  • MST: O(1)
  • MIS: O(log log Δ)
  • Triangle: O(n0.157)
  • APSP: O(n1/3)
  • k-subgraph: O(n1-2/k)

(subpolynomial vs. polynomial)

slide-22
SLIDE 22

Highlight 3:

Fine-grained Complexity

fine-grained complexity is a useful tool for understanding polynomial complexities

slide-23
SLIDE 23

Highlight 3:

Fine-grained Complexity

δ(P) = inf { δ : P ∈ CLIQUE(nδ) }

fine-grained complexity is a useful tool for understanding polynomial complexities

slide-24
SLIDE 24

Highlight 3:

Fine-grained Complexity

δ(P) = inf { δ : P ∈ CLIQUE(nδ) } δ(Ring-MM) ≤ 0.157… δ(APSP) ≤ 1/3 δ(k-IS) ≤ 1−2/k δ(k-COL) ≤ 1

fine-grained complexity is a useful tool for understanding polynomial complexities

slide-25
SLIDE 25

Highlight 3:

Fine-grained Complexity

δ(P) = inf { δ : P ∈ CLIQUE(nδ) } δ(Ring-MM) ≤ 0.157… δ(APSP) ≤ 1/3 δ(k-IS) ≤ 1−2/k δ(k-COL) ≤ 1

  • Proving relationships of form

δ(P) ≤ δ(Q) via subpolynomial reductions

fine-grained complexity is a useful tool for understanding polynomial complexities

slide-26
SLIDE 26

APSP

w/d

0.2096

k-IS APSP

uw/ud

Transitive closure Boolean MM (min,+) MM Ring MM APSP

w/ud

APSP

w/ud/(2-ε)

APSP

uw/d

APSP

w/ud/(1+ε)

size 3

subgraph

Triangle/ 3-IS k-cycle size k

subgraph

k-DS

1-2/ω 1/3 1-2/k 1-1/k

BFS tree SSSP

uw/ud

SSSP

w/ud

SSSP

w/d

SSSP

w/ud/(1+ε)

SSSP

uw/d

MaxIS MinVC k-COL

1

Semiring MM

slide-27
SLIDE 27

APSP

w/d

0.2096

k-IS APSP

uw/ud

Transitive closure Boolean MM (min,+) MM Ring MM APSP

w/ud

APSP

w/ud/(2-ε)

APSP

uw/d

APSP

w/ud/(1+ε)

size 3

subgraph

Triangle/ 3-IS k-cycle size k

subgraph

k-DS

1-2/ω 1/3 1-2/k 1-1/k

BFS tree SSSP

uw/ud

SSSP

w/ud

SSSP

w/d

SSSP

w/ud/(1+ε)

SSSP

uw/d

MaxIS MinVC k-COL

1

Semiring MM

slide-28
SLIDE 28

APSP

w/d

0.2096

k-IS APSP

uw/ud

Transitive closure Boolean MM (min,+) MM Ring MM APSP

w/ud

APSP

w/ud/(2-ε)

APSP

uw/d

APSP

w/ud/(1+ε)

size 3

subgraph

Triangle/ 3-IS k-cycle size k

subgraph

k-DS

1-2/ω 1/3 1-2/k 1-1/k

BFS tree SSSP

uw/ud

SSSP

w/ud

SSSP

w/d

SSSP

w/ud/(1+ε)

SSSP

uw/d

MaxIS MinVC k-COL

1

Semiring MM

slide-29
SLIDE 29

APSP

w/d

0.2096

k-IS APSP

uw/ud

Transitive closure Boolean MM (min,+) MM Ring MM APSP

w/ud

APSP

w/ud/(2-ε)

APSP

uw/d

APSP

w/ud/(1+ε)

size 3

subgraph

Triangle/ 3-IS k-cycle size k

subgraph

k-DS

1-2/ω 1/3 1-2/k 1-1/k

BFS tree SSSP

uw/ud

SSSP

w/ud

SSSP

w/d

SSSP

w/ud/(1+ε)

SSSP

uw/d

MaxIS MinVC k-COL

1

Semiring MM

slide-30
SLIDE 30

APSP

w/d

0.2096

k-IS APSP

uw/ud

Transitive closure Boolean MM (min,+) MM Ring MM APSP

w/ud

APSP

w/ud/(2-ε)

APSP

uw/d

APSP

w/ud/(1+ε)

size 3

subgraph

Triangle/ 3-IS k-cycle size k

subgraph

k-DS

1-2/ω 1/3 1-2/k 1-1/k

BFS tree SSSP

uw/ud

SSSP

w/ud

SSSP

w/d

SSSP

w/ud/(1+ε)

SSSP

uw/d

MaxIS MinVC k-COL

1

Semiring MM

slide-31
SLIDE 31

APSP

w/d

0.2096

k-IS APSP

uw/ud

Transitive closure Boolean MM (min,+) MM Ring MM APSP

w/ud

APSP

w/ud/(2-ε)

APSP

uw/d

APSP

w/ud/(1+ε)

size 3

subgraph

Triangle/ 3-IS k-cycle size k

subgraph

k-DS

1-2/ω 1/3 1-2/k 1-1/k

BFS tree SSSP

uw/ud

SSSP

w/ud

SSSP

w/d

SSSP

w/ud/(1+ε)

SSSP

uw/d

MaxIS MinVC k-COL

1

Semiring MM

k-vertex cover: O(k) rounds

slide-32
SLIDE 32

Closing remarks:

Beyond the congested clique?

clique message passing
 models (k-machine model, BSP, MapReduce, MPC,…)

slide-33
SLIDE 33

clique message passing
 models n vertices in input n processors n vertices in input p < n processors

Closing remarks:

Beyond the congested clique?

slide-34
SLIDE 34

clique message passing
 models n vertices in input n processors n vertices in input p < n processors

Closing remarks:

Beyond the congested clique?

slide-35
SLIDE 35

Thanks! Questions?

(arXiv:1705.03284)