Listing All Maximal Cliques in Sparse Graphs in Near-Optimal Time - - PowerPoint PPT Presentation

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Listing All Maximal Cliques in Sparse Graphs in Near-Optimal Time - - PowerPoint PPT Presentation

Sep 08, 2023 135 likes •1.75k views

Listing All Maximal Cliques in Sparse Graphs in Near-Optimal Time Darren Strash Department of Computer Science UC Irvine Joint work with David Eppstein and Maarten L offler What is a Maximal Clique? A clique that cannot be extended by

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Listing All Maximal Cliques in Sparse Graphs in Near-Optimal Time

Darren Strash Joint work with David Eppstein and Maarten L¨

  • ffler

Department of Computer Science UC Irvine

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What is a Maximal Clique? A clique that cannot be extended by adding more vertices

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What is a Maximal Clique? A clique that cannot be extended by adding more vertices Maximal Maximal, Maximum Not Maximal Not Clique

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Goal: Design an algorithm to list all maximal cliques

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Goal: Design an algorithm to list all maximal cliques

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Motivation

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Motivation Features in ERGM

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Motivation Features in ERGM Detect structural motifs from similarities between proteins

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Motivation Features in ERGM Detect structural motifs from similarities between proteins Determine the docking regions between biomolecules

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Motivation Features in ERGM Detect structural motifs from similarities between proteins Determine the docking regions between biomolecules Document clustering for information retrieval

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There may be many maximal cliques.

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There may be many maximal cliques.

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There may be many maximal cliques.

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There may be many maximal cliques.

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There may be many maximal cliques.

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There may be many maximal cliques.

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There may be many maximal cliques.

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There may be many maximal cliques. 3 ∗ 3 ∗ 3 ∗ 3

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There may be many maximal cliques. 3n/3 maximal cliques

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There may be many maximal cliques. 3n/3 maximal cliques (Moon–Moser bound)

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Maximal Clique Listing Algorithms Author Year Running Time Bron and Kerbosch 1973 ??? Tsukiyama et al. 1977 O(nmµ) Chiba and Nishizeki 1985 O(αmµ) Makino and Uno 2004 O(∆4µ) n = number of vertices m = number of edges µ = number of maximal cliques ∆ = maximum degree of the graph α = arboricity

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Tomita et al. (2006)

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Tomita et al. (2006) Worst-case optimal running time O(3n/3)

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Tomita et al. (2006) Worst-case optimal running time O(3n/3) Computational experiments:

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Tomita et al. (2006) Worst-case optimal running time O(3n/3) Computational experiments: fast! slow

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The Bron–Kerbosch Algorithm Its variations work well in practice. Easy to implement Confirmed through computational experiments Johnston (1976), Koch (2001), Baum (2003) There are many heuristics, which make it faster Easy to understand One variation is worst-case optimal (O(3n/3) time) Tomita et al. (2006)

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The Bron–Kerbosch Algorithm Its variations work well in practice. Easy to implement Confirmed through computational experiments Johnston (1976), Koch (2001), Baum (2003) There are many heuristics, which make it faster Easy to understand One variation is worst-case optimal (O(3n/3) time) Tomita et al. (2006)

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Finding one maximal clique

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Finding one maximal clique

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Finding one maximal clique

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Finding one maximal clique

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Finding one maximal clique

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Finding one maximal clique

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Finding one maximal clique

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The Bron–Kerbosch Algorithm

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The Bron–Kerbosch Algorithm

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The Bron–Kerbosch Algorithm

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The Bron–Kerbosch Algorithm

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The Bron–Kerbosch Algorithm

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The Bron–Kerbosch Algorithm

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The Bron–Kerbosch Algorithm

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The Bron–Kerbosch Algorithm

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The Bron–Kerbosch Algorithm

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The Bron–Kerbosch Algorithm

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The Bron–Kerbosch Algorithm

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The Bron–Kerbosch Algorithm

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The Bron–Kerbosch Algorithm

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The Bron–Kerbosch Algorithm

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The Bron–Kerbosch Algorithm

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The Bron–Kerbosch Algorithm proc BronKerbosch(P, R, X)

1: if P ∪ X = ∅ then 2:

report R as a maximal clique

3: end if 4: for each vertex v ∈ P do 5:

BronKerbosch(P ∩ Γ(v), R ∪ {v}, X ∩ Γ(v))

6:

P ← P \ {v}

7:

X ← X ∪ {v}

8: end for

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The Bron–Kerbosch Algorithm

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The Bron–Kerbosch Algorithm

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The Bron–Kerbosch Algorithm

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The Bron–Kerbosch Algorithm

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The Bron–Kerbosch Algorithm

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The Bron–Kerbosch Algorithm

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The Bron–Kerbosch Algorithm

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The Bron–Kerbosch Algorithm with Pivoting proc BronKerboschPivot(P, R, X)

1: if P ∪ X = ∅ then 2:

report R as a maximal clique

3: end if 4: choose a pivot u ∈ P ∪ X 5: for each vertex v ∈ P \ Γ(u) do 6:

BronKerboschPivot(P ∩ Γ(v), R ∪ {v}, X ∩ Γ(v))

7:

P ← P \ {v}

8:

X ← X ∪ {v}

9: end for

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The Bron–Kerbosch Algorithm with Pivoting proc BronKerboschPivot(P, R, X)

1: if P ∪ X = ∅ then 2:

report R as a maximal clique

3: end if 4: choose a pivot u ∈ P ∪ X to minimize |P \ Γ(u)| 5: for each vertex v ∈ P \ Γ(u) do 6:

BronKerboschPivot(P ∩ Γ(v), R ∪ {v}, X ∩ Γ(v))

7:

P ← P \ {v}

8:

X ← X ∪ {v}

9: end for

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The Bron–Kerbosch Algorithm with Pivoting proc BronKerboschPivot(P, R, X)

1: if P ∪ X = ∅ then 2:

report R as a maximal clique

3: end if 4: choose a pivot u ∈ P ∪ X to maximize |P ∩ Γ(u)| 5: for each vertex v ∈ P \ Γ(u) do 6:

BronKerboschPivot(P ∩ Γ(v), R ∪ {v}, X ∩ Γ(v))

7:

P ← P \ {v}

8:

X ← X ∪ {v}

9: end for

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The Bron–Kerbosch Algorithm with Pivoting T(n) ≤ max

k {kT(n − k)} + O(n2)

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The Bron–Kerbosch Algorithm with Pivoting T(n) ≤ max

k {kT(n − k)} + O(n2)

T(n) = O(3n/3)

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The Bron–Kerbosch Algorithm

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The Bron–Kerbosch Algorithm

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The Bron–Kerbosch Algorithm All cliques in planar graphs may be listed in time O(n)

Chiba and Nishizeki (1985), Chrobak and Eppstein (1991)

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The Bron–Kerbosch Algorithm Want to characterize the running time with a parameter. Let p be our parameter of choice. An algorithm is fixed-parameter tractable with parameter p if it has running time f(p)nO(1) The key is to avoid things like np.

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Parameterize on Sparsity

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Parameterize on Sparsity degeneracy:

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Parameterize on Sparsity degeneracy: The minimum integer d such that every subgraph of G has a vertex of degree d or less.

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Degeneracy

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Degeneracy d = 1

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degeneracy: The minimum integer d such that there is an

  • rdering of the vertices where each vertex has at

most d neighbors later in the ordering. Degeneracy

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h

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h

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h d = 1

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Planar graphs have degeneracy at most 5

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Degeneracy is easy to compute

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d-degenerate graphs...

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d-degenerate graphs... cannot contain cliques with more than d + 1 vertices

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d-degenerate graphs... cannot contain cliques with more than d + 1 vertices

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d-degenerate graphs... cannot contain cliques with more than d + 1 vertices

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d-degenerate graphs... cannot contain cliques with more than d + 1 vertices > d later neighbors.

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d-degenerate graphs... cannot contain cliques with more than d + 1 vertices > d later neighbors.

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d-degenerate graphs...

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d-degenerate graphs... have fewer than dn edges.

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d-degenerate graphs... have fewer than dn edges. ≤ d later neighbors.

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A few more facts about degeneracy... Degeneracy is within a constant factor of other popular sparsity measures.

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A few more facts about degeneracy... Degeneracy is within a constant factor of other popular sparsity measures. Graphs generated by the preferential attachment mechanism of Barab´ asi and Albert have low degeneracy.

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proc BronKerboschDegeneracy(V , E)

1: for each vertex vi in a degeneracy ordering v0, v1, v2, . . . of (V, E)

do

2:

P ← vi’s later neighbors

3:

X ← vi’s earlier neighbors

4:

BronKerboschPivot(P, {vi}, X)

5: end for

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|P|≤ d X

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Computing the pivot

X P

Pick u ∈ X ∪ P that maximizes |P ∩ Γ(u)|.

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X P

Computing the pivot Pick u ∈ X ∪ P that maximizes |P ∩ Γ(u)|.

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X P

O(|P|(|X| + |P|)) Computing the pivot Pick u ∈ X ∪ P that maximizes |P ∩ Γ(u)|.

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proc BronKerboschPivot(P, R, X)

1: if P ∪ X = ∅ then 2:

report R as a maximal clique

3: end if 4: choose a pivot u ∈ P ∪ X to maximize |P ∩ Γ(u)| 5: for each vertex v ∈ P \ Γ(u) do 6:

BronKerboschPivot(P ∩ Γ(v), R ∪ {v}, X ∩ Γ(v))

7:

P ← P \ {v}

8:

X ← X ∪ {v}

9: end for

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proc BronKerboschPivot(P, R, X)

1: if P ∪ X = ∅ then 2:

report R as a maximal clique

3: end if 4: choose a pivot u ∈ P ∪ X to maximize |P ∩ Γ(u)| 5: for each vertex v ∈ P \ Γ(u) do 6:

BronKerboschPivot(P ∩ Γ(v), R ∪ {v}, X ∩ Γ(v))

7:

P ← P \ {v}

8:

X ← X ∪ {v}

9: end for

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X P

Find subgraph induced by v’s neighbors.

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X P

v Find subgraph induced by v’s neighbors.

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X P

v Find subgraph induced by v’s neighbors.

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X ∩ Γ(v) P ∩Γ(v)

Find subgraph induced by v’s neighbors.

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X ∩ Γ(v) P ∩Γ(v)

O(|P|(|X| + |P|)) Find subgraph induced by v’s neighbors.

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proc BronKerboschPivot(P, R, X)

1: if P ∪ X = ∅ then 2:

report R as a maximal clique

3: end if 4: choose a pivot u ∈ P ∪ X to maximize |P ∩ Γ(u)| 5: for each vertex v ∈ P \ Γ(u) do 6:

BronKerboschPivot(P ∩ Γ(v), R ∪ {v}, X ∩ Γ(v))

7:

P ← P \ {v}

8:

X ← X ∪ {v}

9: end for

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proc BronKerboschPivot(P, R, X)

1: if P ∪ X = ∅ then 2:

report R as a maximal clique

3: end if 4: choose a pivot u ∈ P ∪ X to maximize |P ∩ Γ(u)| 5: for each vertex v ∈ P \ Γ(u) do 6:

BronKerboschPivot(P ∩ Γ(v), R ∪ {v}, X ∩ Γ(v))

7:

P ← P \ {v}

8:

X ← X ∪ {v}

9: end for

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X P

v Remove v from P and add it to X.

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X P

v Remove v from P and add it to X.

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X P

v Remove v from P and add it to X.

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X P

v O(|P|(|X| + |P|)) Remove v from P and add it to X.

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proc BronKerboschPivot(P, R, X)

1: if P ∪ X = ∅ then 2:

report R as a maximal clique

3: end if 4: choose a pivot u ∈ P ∪ X to maximize |P ∩ Γ(u)| 5: for each vertex v ∈ P \ Γ(u) do 6:

BronKerboschPivot(P ∩ Γ(v), R ∪ {v}, X ∩ Γ(v))

7:

P ← P \ {v}

8:

X ← X ∪ {v}

9: end for

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proc BronKerboschPivot(P, R, X)

1: if P ∪ X = ∅ then 2:

report R as a maximal clique

3: end if 4: choose a pivot u ∈ P ∪ X to maximize |P ∩ Γ(u)| 5: for each vertex v ∈ P \ Γ(u) do 6:

BronKerboschPivot(P ∩ Γ(v), R ∪ {v}, X ∩ Γ(v))

7:

P ← P \ {v}

8:

X ← X ∪ {v}

9: end for

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O(|P|2(|X| + |P|))

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T(n) ≤ max

k {kT(n − k)} + O(n2)

= O(3n/3)

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T(n) ≤ max

k {kT(n − k)} + O(n2)

D(p, x) ≤ max

k {kD(p − k, x)} + O(p2(p + x))

= O(3n/3)

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T(n) ≤ max

k {kT(n − k)} + O(n2)

D(p, x) ≤ max

k {kD(p − k, x)} + O(p2(p + x))

≤ (d + x)

  • maxk
  • kD(p−k,x)

d+x

  • + O(p2)
  • = O(3n/3)
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T(n) ≤ max

k {kT(n − k)} + O(n2)

D(p, x) ≤ max

k {kD(p − k, x)} + O(p2(p + x))

≤ (d + x)

  • maxk
  • kD(p−k,x)

d+x

  • + O(p2)
  • ≤ (d + x)
  • maxk{kT(p − k)} + O(p2)
  • = O(3n/3)
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T(n) ≤ max

k {kT(n − k)} + O(n2)

D(p, x) ≤ max

k {kD(p − k, x)} + O(p2(p + x))

≤ (d + x)

  • maxk
  • kD(p−k,x)

d+x

  • + O(p2)
  • ≤ (d + x)
  • maxk{kT(p − k)} + O(p2)
  • = O((d + x)3p/3)

= O(3n/3)

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T(n) ≤ max

k {kT(n − k)} + O(n2)

D(p, x) ≤ max

k {kD(p − k, x)} + O(p2(p + x))

≤ (d + x)

  • maxk
  • kD(p−k,x)

d+x

  • + O(p2)
  • ≤ (d + x)
  • maxk{kT(p − k)} + O(p2)
  • = O((d + x)3p/3)

= O(3n/3) = O((d + x)3d/3)

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  • v∈V

O(d + |Xv|)3d/3)

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  • v∈V

O(d + |Xv|)3d/3) = O((dn + m)3d/3)

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  • v∈V

O(d + |Xv|)3d/3) = O((dn + m)3d/3) = O(dn3d/3)

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  • v∈V

O(d + |Xv|)3d/3) = O((dn + m)3d/3) = O(dn3d/3) where f(d) = d3d/3 = O(f(d)n)

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Our running time: O(dn3d/3)

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Our running time: O(dn3d/3) Worst-case output size: O(d(n − d)3d/3)

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Our running time: O(dn3d/3) Worst-case output size: O(d(n − d)3d/3) When n − d = Ω(n), our algorithm is worst-case optimal.

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≤ d later neighbors. An upper bound

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≤ d later neighbors. An upper bound

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≤ d later neighbors. An upper bound

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≤ d later neighbors. at most O(3d/3) maximal cliques An upper bound

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≤ d later neighbors. at most O(3d/3) maximal cliques An upper bound

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n − d − 3 vertices d + 3 vertices An upper bound

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n − d − 3 vertices d + 3 vertices (n − d − 3)3d/3 3

d+3 3

An upper bound

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n − d − 3 vertices d + 3 vertices (n − d − 3)3d/3 3

d+3 3

at most (n − d)3d/3 maximal cliques An upper bound

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Kn−d,3,3,3,... A lower bound

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Kn−d,3,3,3,... n − d d d d . . . A lower bound

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Kn−d,3,3,3,... n − d d d d . . . A lower bound

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Kn−d,3,3,3,... n − d d d d . . . (n − d)3d/3 maximal cliques A lower bound

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Kn−d,3,3,3,... n − d d d d . . . (n − d)3d/3 maximal cliques has degeneracy d when (n − d) ≥ 3 A lower bound

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at most (n − d)3d/3 maximal cliques each clique is of size at most d + 1 O(d(n − d)3d/3) worst-case output size.

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The Bron–Kerbosch Algorithm

proc BronKerbosch(P, R, X)

1: if P ∪ X = ∅ then 2:

report R as a maximal clique

3: end if 4: for each vertex v ∈ P do 5:

BronKerbosch(P ∩ Γ(v), R ∪ {v}, X ∩ Γ(v))

6:

P ← P \ {v}

7:

X ← X ∪ {v}

8: end for

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The Bron–Kerbosch Algorithm

proc BronKerbosch(P, R, X)

1: if P ∪ X = ∅ then 2:

report R as a maximal clique

3: end if 4: for each vertex v ∈ P (in degeneracy order) do 5:

BronKerbosch(P ∩ Γ(v), R ∪ {v}, X ∩ Γ(v))

6:

P ← P \ {v}

7:

X ← X ∪ {v}

8: end for

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The Bron–Kerbosch Algorithm

proc BronKerbosch(P, R, X)

1: if P ∪ X = ∅ then 2:

report R as a maximal clique

3: end if 4: for each vertex v ∈ P (in degeneracy order) do 5:

BronKerboschPivot(P ∩ Γ(v), R ∪ {v}, X ∩ Γ(v))

6:

P ← P \ {v}

7:

X ← X ∪ {v}

8: end for

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Experiments Linux Workstation: 3.00Ghz Pentium 4, 1GB RAM

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Experimental results for UCI data sets

graph d BK BK-pivot BK-hybrid BK-degen Uno karate 4 < 1sec < 1sec < 1sec < 1sec <1sec∗ dolphins 4 < 1sec < 1sec < 1sec < 1sec < 1sec power 5 < 1sec < 1sec < 1sec < 1sec < 1sec polbooks 6 < 1sec < 1sec < 1sec < 1sec < 1sec adjnoun 6 < 1sec < 1sec < 1sec < 1sec < 1sec football 8 < 1sec < 1sec < 1sec < 1sec < 1sec lesmis 9 < 1sec < 1sec < 1sec < 1sec < 1sec celegens 9 < 1sec < 1sec < 1sec < 1sec

  • seg. fault∗

netscience 19 2.8sec < 1sec < 1sec < 1sec < 1sec internet 25 19.4sec 10.3sec < 1sec < 1 sec <1sec∗ condmat 29 > 3min 65sec 1.6sec 2.61sec < 1sec polblogs 36 > 3min 2sec 1.5sec 1.2sec 1.8sec astro 56 > 3min 12.3sec 1.4sec 3.14sec < 1sec

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Experimental results for BIOGRID data sets (PPI Networks)

graph d BK BK-pivot BK-hybrid BK-degen Uno mouse 6 < 1sec < 1sec < 1sec < 1sec < 1sec worm 10 < 1sec < 1sec < 1sec < 1sec < 1sec plant 12 < 1sec < 1sec < 1sec < 1sec < 1sec fruitfly 12 < 1sec 2.2sec < 1sec < 1sec < 1sec human 12 1.4sec 3.3sec < 1sec < 1sec < 1sec fission-yeast 34 2.8sec 1.1sec < 1sec < 1sec < 1sec yeast 64 > 3min 81sec 44.3sec 20.5sec 121.1sec∗

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Experimental results for Pajek data sets

graph d BK BK-pivot BK-hybrid BK-degen Uno foldoc 12 4.2sec 9sec < 1sec 1sec < 1sec patents 24 > 5min > 5min 4.3sec 5.3sec 2.2sec eatRS 34 19.8sec 53sec 12.3sec 9.12sec 14.9sec hep-th 37 > 5min 69.6sec 22.6sec 17.2sec 41.5sec∗ days-all 73 > 5min 379.1sec 206.5sec 51.4sec 10min 25sec ND-www 155 > 5min > 5min 27.8sec 41.11sec 9.7sec∗

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Thank you!