Community Detection by Decomposing a Graph into Relaxed Cliques - - PowerPoint PPT Presentation

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Community Detection by Decomposing a Graph into Relaxed Cliques

Fabio Furini, Timo Gschwind, Stefan Irnich, Roberto Wolfler Calvo

LAMSADE, Université Paris-Dauphine

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Outline

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Social Network Analysis (SNA)

◮ Relationships between members of a network can be encoded by an

undirected graph G = (V, E)

◮ vertices (V) represent the members of the network ◮ edges (E) indicate the existence of a relationship

◮ Community Detection aims at clustering the members into communities

such that:

◮ relatively few edges are in the cutsets ◮ but relatively many are internal edges.

◮ The clustering is intended to reveal hidden or reproduce known features

• f the network

◮ In this talk we present a framework for community detection when the

internal structure of the community is important

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

First case study: Dolphins – Social Network

Beak Beescratch Bumper CCL Cross DN16 DN21 DN63 Double Feather Fish Five Fork Gallatin Grin Haecksel Hook Jet Jonah Knit Kringel MN105 MN23 MN60 MN83 Mus Notch Number1 Oscar Patchback PL Quasi Ripplefluke Scabs Shmuddel SMN5 SN100 SN4 SN63 SN89 SN9 SN90 SN96 Stripes Thumper Topless TR120 TR77 TR82 TR88 TR99 Trigger TSN103 TSN83 Upbang Vau Wave Web Whitetip Zap Zig Zipfel

◮ 62 bottlenose dolphins living in Doubtful

Sound, New Zealand

◮ An edge indicates that a pair of dolphins were

seen together more often than expected (love? )

◮ After dolphin SN100 left the place for some

time, the dolphins separated into two groups indicated with the two colors

◮ Question 1: where did SN100 go? ◮ Question 2: is it possible to predict this split?

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

First case study: Dolphins – Social Network

Beak Beescratch Bumper CCL Cross DN16 DN21 DN63 Double Feather Fish Five Fork Gallatin Grin Haecksel Hook Jet Jonah Knit Kringel MN105 MN23 MN60 MN83 Mus Notch Number1 Oscar Patchback PL Quasi Ripplefluke Scabs Shmuddel SMN5 SN100 SN4 SN63 SN89 SN9 SN90 SN96 Stripes Thumper Topless TR120 TR77 TR82 TR88 TR99 Trigger TSN103 TSN83 Upbang Vau Wave Web Whitetip Zap Zig Zipfel

◮ 62 bottlenose dolphins living in Doubtful

Sound, New Zealand

◮ An edge indicates that a pair of dolphins were

seen together more often than expected (love? )

◮ After dolphin SN100 left the place for some

time, the dolphins separated into two groups indicated with the two colors

◮ Question 1: where did SN100 go? ◮ Question 2: is it possible to predict this split?

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

First case study: Dolphins – Social Network

Beak Beescratch Bumper CCL Cross DN16 DN21 DN63 Double Feather Fish Five Fork Gallatin Grin Haecksel Hook Jet Jonah Knit Kringel MN105 MN23 MN60 MN83 Mus Notch Number1 Oscar Patchback PL Quasi Ripplefluke Scabs Shmuddel SMN5 SN100 SN4 SN63 SN89 SN9 SN90 SN96 Stripes Thumper Topless TR120 TR77 TR82 TR88 TR99 Trigger TSN103 TSN83 Upbang Vau Wave Web Whitetip Zap Zig Zipfel

◮ 62 bottlenose dolphins living in Doubtful

Sound, New Zealand

◮ An edge indicates that a pair of dolphins were

seen together more often than expected (love? )

◮ After dolphin SN100 left the place for some

time, the dolphins separated into two groups indicated with the two colors

◮ Question 1: where did SN100 go? ◮ Question 2: is it possible to predict this split?

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

First case study: Dolphins – Social Network

Beak Beescratch Bumper CCL Cross DN16 DN21 DN63 Double Feather Fish Five Fork Gallatin Grin Haecksel Hook Jet Jonah Knit Kringel MN105 MN23 MN60 MN83 Mus Notch Number1 Oscar Patchback PL Quasi Ripplefluke Scabs Shmuddel SMN5 SN100 SN4 SN63 SN89 SN9 SN90 SN96 Stripes Thumper Topless TR120 TR77 TR82 TR88 TR99 Trigger TSN103 TSN83 Upbang Vau Wave Web Whitetip Zap Zig Zipfel

◮ 62 bottlenose dolphins living in Doubtful

Sound, New Zealand

◮ An edge indicates that a pair of dolphins were

seen together more often than expected (love? )

◮ After dolphin SN100 left the place for some

time, the dolphins separated into two groups indicated with the two colors

◮ Question 1: where did SN100 go? ◮ Question 2: is it possible to predict this split?

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

First case study: Dolphins – Social Network

Beak Beescratch Bumper CCL Cross DN16 DN21 DN63 Double Feather Fish Five Fork Gallatin Grin Haecksel Hook Jet Jonah Knit Kringel MN105 MN23 MN60 MN83 Mus Notch Number1 Oscar Patchback PL Quasi Ripplefluke Scabs Shmuddel SMN5 SN100 SN4 SN63 SN89 SN9 SN90 SN96 Stripes Thumper Topless TR120 TR77 TR82 TR88 TR99 Trigger TSN103 TSN83 Upbang Vau Wave Web Whitetip Zap Zig Zipfel

◮ 62 bottlenose dolphins living in Doubtful

Sound, New Zealand

◮ An edge indicates that a pair of dolphins were

seen together more often than expected (love? )

◮ After dolphin SN100 left the place for some

time, the dolphins separated into two groups indicated with the two colors

◮ Question 1: where did SN100 go? ◮ Question 2: is it possible to predict this split?

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Motivation

Network Analysis: Graphs representing real networks have structures! → community structure or clustering football karate Important applications in many networked systems from biology, sociology, computer science, engineering, economics, politics, linguistics, etc.

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Motivation

Network Analysis:

• 1. Community Detection [Fortunato(2010)]

◮ Partition graphs into vertex subsets ◮ Few edges between subsets, many internal edges ◮ Maximize modularity [Newman and Girvan(2004)]

µ = Q(V1, V2, . . . , Vp) =

p

• i=1

|E(Vi)| |E| − exp(Vi)

• exp(Vi) is the expected fraction of inner-cluster edges

◮ Subsets have no specific structure

• 2. Relaxed Cliques [Pattillo et al.(2013a)]

◮ Subgraphs with a specific structure ◮ Find maximum cardinality/weight relaxed clique

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Motivation

Network Analysis:

• 1. Community Detection [Fortunato(2010)]

◮ Partition graphs into vertex subsets ◮ Few edges between subsets, many internal edges ◮ Maximize modularity [Newman and Girvan(2004)]

µ = Q(V1, V2, . . . , Vp) =

p

• i=1

|E(Vi)| |E| − exp(Vi)

• exp(Vi) is the expected fraction of inner-cluster edges

◮ Subsets have no specific structure

• 2. Relaxed Cliques [Pattillo et al.(2013a)]

◮ Subgraphs with a specific structure ◮ Find maximum cardinality/weight relaxed clique

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Motivation

Network Analysis:

• 1. Community Detection [Fortunato(2010)]

◮ Partition graphs into vertex subsets ◮ Few edges between subsets, many internal edges ◮ Maximize modularity [Newman and Girvan(2004)]

µ = Q(V1, V2, . . . , Vp) =

p

• i=1

|E(Vi)| |E| − exp(Vi)

• exp(Vi) is the expected fraction of inner-cluster edges

◮ Subsets have no specific structure

• 2. Relaxed Cliques [Pattillo et al.(2013a)]

◮ Subgraphs with a specific structure ◮ Find maximum cardinality/weight relaxed clique

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Dolphins case study: Maximizing Modularity

(a) Real-world split: modularity µ(G) = 0.3735; (b) Decomposition with maximum modularity: µ = 0.5285 Partitioning into cliques is also not a good idea!

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Cliques and Clique Relaxations

A clique S forms an extreme subset in the following senses: Degree Every node i ∈ S has maximum degree (=|S| − 1) Distance The distance dist(i, j) between any two nodes i, j ∈ S is minimal (=1) Density G[S] has maximum edge density (=1); Connectivity The vertex connectivity κ(G[S]) is maximum (=|S| − 1) Analysis of large, complex networks:

◮ Cliques can model cohesive substructures, e.g., subgroups. ◮ However, requirements of a clique were found too restrictive!

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Desired properties of communities [Balasundaram et al.(2011)]

◮ We propose a method for the case in which one has a good

understanding of the structural properties that a community must have.

◮ familiarity among members (few strangers) ◮ reachability among members (quick communication) ◮ robustness of the subgroup (not easily destroyable)

In a graph-theoretic description

◮ familiarity concerns vertex degrees (→ k-core/s-plex) ◮ reachability concerns distances (→ s-club/clique) ◮ robustness concerns connectivity (→ k-block/ s-bundle) ◮ Moreover, input data describing a social network may stem from sources

that contain errors (→ s-defective cliques or γ-quasi-cliques).

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Clique Relaxations

Type Definition Relaxation Hereditary k-core δ(G[S]) ≥ k Degree no s-plex δ(G[S]) ≥ |S| − s Degree yes s-clique distG(i, j) ≤ s i, j ∈ S Distance yes s-club distG[S](i, j) ≤ s i, j ∈ S Distance no γ-quasi-clique ρ(G[S]) ≥ γ Density no s-defective clique |E(G[S])| ≥ |S|

2

• − s

Density yes k-block κ(G[S]) ≥ k Connectivity no s-bundle κ(G[S]) ≥ |S| − s Connectivity yes Note: δ(G) minimum degree, dG(i, j) distance, ρ(G) edge density, κ(G) vertex connectivity

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Clique Relaxations: Examples

1-defective clique

‘one missing edge allowed in induced subgraph’

1 2 3 4

⇒ 2-plex

‘one missing edge allowed at each node’

2-plex

1 2 3 4

⇒ 1-defective clique 2-club

‘distance between nodes in S at most 2 in induced subgraph’

1 2 3 4 5

⇒ 2-clique

‘distance between nodes of S at most 2 in graph’

2-clique

1 2 3 4 5

⇒ 2-club

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Clique Relaxations: Examples

1-defective clique

‘one missing edge allowed in induced subgraph’

1 2 3 4

⇒ 2-plex

‘one missing edge allowed at each node’

2-plex

1 2 3 4

⇒ 1-defective clique 2-club

‘distance between nodes in S at most 2 in induced subgraph’

1 2 3 4 5

⇒ 2-clique

‘distance between nodes of S at most 2 in graph’

2-clique

1 2 3 4 5

⇒ 2-club

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Clique Relaxations (Π): Properties

Weakly-Heredity

S satisfies Π ⇒ any S′ ⊂ S satisfies Π. Weakly-Hereditary RELAXED CLIQUES: s-plex, s-clique, s-defective clique, s-bundle Non-hereditary RELAXED CLIQUES: k-core, s-club, γ-quasi-clique, k-block

1 2 3 4 5

2-club

1 2 3 4 5

no 2-club

Connectivity

For all i, j ∈ S there exists a path between i and j in G[S]. Connected RELAXED CLIQUES: s-club, k-block Non-connected RELAXED CLIQUES: k-core, s-plex, s-clique, s-bundle, γ-quasi-clique, s-defective clique

1 2 3

2-defective clique Note:

◮ Non-connected communities may not

be reasonable → Connected RELAXED CLIQUES

◮ Connectivity is non-hereditary

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Clique Relaxations (Π): Properties

Weakly-Heredity

S satisfies Π ⇒ any S′ ⊂ S satisfies Π. Weakly-Hereditary RELAXED CLIQUES: s-plex, s-clique, s-defective clique, s-bundle Non-hereditary RELAXED CLIQUES: k-core, s-club, γ-quasi-clique, k-block

1 2 3 4 5

2-club

1 2 3 4 5

no 2-club

Connectivity

For all i, j ∈ S there exists a path between i and j in G[S]. Connected RELAXED CLIQUES: s-club, k-block Non-connected RELAXED CLIQUES: k-core, s-plex, s-clique, s-bundle, γ-quasi-clique, s-defective clique

1 2 3

2-defective clique Note:

◮ Non-connected communities may not

be reasonable → Connected RELAXED CLIQUES

◮ Connectivity is non-hereditary

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Large RELAXED CLIQUES

The optimization-related literature related to RELAXED CLIQUES is (as far as we know) exclusively on finding maximum/inclusion maximal relaxed cliques.

Type of relaxation Exact Approach and Reference k-core

• polynom. solvable, see [Kosub(2004)]

s-plex B&C: [Balasundaram et al.(2011)], B&B: [Trukhanov et al.(2013)][Gschwind et al.(2015)] s-clique clique in the sth power graph s-club B&C: [Almeida and Carvalho(2012)] [Almeida and Carvalho(2013)], B&B: [Bourjolly et al.(2002)] [Mahdavi Pajouh and Balasundaram(2012)], MIP: [Bourjolly et al.(2000)] [Veremyev and Boginski(2012)], SAT: [Wotzlaw(2014)] γ-quasi-clique MIP: [Pattillo et al.(2013b)], B&B: [Pajouh et al.(2014)] s-defective clique B&C: [Sherali and Smith(2006)], B&B: [Trukhanov et al.(2013)][Gschwind et al.(2015)] k-block

• polynom. solvable, see [Kammer and Täubig(2005)]

s-bundle B&B: [Gschwind et al.(2015)]

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Large RELAXED CLIQUES

◮ Different formulations for the MC-RC and MW-RC variants have been suggested

in the literature

◮ All of them have vertex and/or edge variables

Generic MIP-formulation:

◮ xi ∈ {0, 1} indicates if vertex i ∈ V is in the RELAXED CLIQUE S ◮ ye ∈ {0, 1} indicates if G[S] contains edge e ∈ E

max

• i∈V

xi (0.1) s.t. (x, y) = (xi, yij) ∈ F(G) (0.2) Note: F(G) is a polytope such that (x, y) ∈ F(G) if and only if G[S] with S = {i ∈ V : x : i = 1} satisfies Π.

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Partitioning and Covering a Graph with RELAXED CLIQUES

The problems we consider in the following are partitioning and covering a graph with a minimum number of RELAXED CLIQUES: Generic compact formulation – very weak!:

◮ Let ¯

rc(G) be an upper bound on the min number of RC

◮ Index set H = {1, ...,¯

rc(G)} to refer to the individual RELAXED CLIQUES in the partitioning/covering

◮ Introduce indicator variables zh ∈{0, 1}, h ∈ H ◮ Duplicate xi and ye variables and constraints for each h ∈ H

min

• h∈H

zh (0.1) s.t.

• h∈H

xh

i =(≥) 1

i ∈ V (0.2) zh ≥ xh

i

i ∈ V, h ∈ H (0.3) (xh

i , yh ij ) ∈ F(G)

h ∈ H (0.4)

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Covering vs Partitioning with Relaxed Cliques

Covering and partitioning into a minimum number of 2-clubs.

3 2 4 1 5 6 7 9 8

Figure 1: Two 2-clubs.

3 2 4 1 5 6 7 9 8

Figure 2: Three 2-clubs.

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Partitioning into Connected vs Disconnected Relaxed Cliques

1 2 3 4 5 7 6 8 9

11 10 12 13 14 15

1 2 3 4 5 7 6 8 9

11 10 12 13 14 15

Four general 2-cliques Five connected 2-cliques Note that S = {13, 14, 15} induces a disconnected subgraph G[S] = (S, ∅).

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Partitioning and Covering a Graph with RELAXED CLIQUES

Interesting problem variants:

General With connectivity required Type of relaxation Partitioning Covering Partitioning Covering k-core ✗ ✗ ✗ ✗ s-plex ✔ ✗ ✔ ✔ s-clique ✗ ✗ ✔ ✔ s-club ✔ ✔ ✗ ✗ γ-quasi-clique ✔ ✔ ✔ ✔ s-defective clique ✔ ✗ ✔ ✔ k-block ✗ ✗ ✗ ✗ s-bundle ✔ ✗ ✔ ✔

Note: ✗ Some vertices may have a degree smaller than k ✗ Determination of k-connected components [Kammer and Täubig(2005)] ✗ Clique cover in sth power graph ✗ Partitioning = covering for hereditary Π ✗ s-Club is always connected

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Compact MIP Formulation

Theorem

For the polyhedra F 1(G) (clique and s-clique), F 2(G) (s-plex), F 3(G) (s-defective clique), F 4(G) and F 5(G) (γ-quasi-clique), F 6(G) (s-club), and F 7(G) (s-bundle), let lp(G) be the value of the linear relaxation of the compact formulation. (a) For every solution (ˆ x, ˆ y, ˆ z) to the linear relaxation, there always exists an equivalent perfectly symmetric solution with x1

i = x2 i = · · · = x ¯ rc(G) i

=

1 ¯ rc(G)

¯

rc(G) h=1 ˆ

xh

i for each i ∈ V,

y 1

e = y 2 e = · · · = y ¯ rc(G) e

=

1 ¯ rc(G)

¯

rc(G) h=1 ˆ

y h

e for each e ∈ E, and

z1 = z2 = · · · = z¯

rc(G) = 1 ¯ rc(G)

¯

rc(G) h=1 ˆ

zh. (b) For ¯ rc(G) = 1, the linear relaxation is tight, i.e., lp(G) = rc(G). (c) For ¯ rc(G) ≥ 2, the linear relaxation is not tight and lp(G) = 1.

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Dantzig-Wolfe decomposition

Master problem derived from Dantzig-Wolfe decomposition of the generic compact formulation and subsequent aggregation:

◮ Ω the set of all S satisfying Π ◮ Indicators aiS for i ∈ V, S ∈ Ω with aiS = 1 if i ∈ S, and 0 otherwise

min

• S∈Ω

λS s.t.

• S∈Ω

aiSλS = 1 (or ≥ 1) for all i ∈ V (dual πi) λ ≥ 0 (∈ R|Ω|)

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Pricing Problem

Reduced Costs (rdc) of a RELAXED CLIQUE S: ˜ cS = 1 −

• i∈S

πi Pricing Problem: 1 − ˜ cS = max

• i∈V

πixi s.t. (xi, yij) ∈ F(G) This is a maximum weight RELAXED CLIQUE problem (MW-RC)

◮ Generalization of the maximum (cardinality) RELAXED CLIQUE problem ◮ For most types of RELAXED CLIQUES this has not been studied in the literature so

far

◮ MIP-solver (+ cutting plane algorithm) ◮ We developed new combinatorial B&B and Russian Doll Search Algorithm ◮ Weights πi ∈ R can be negative in case of partitioning

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Pricing Problem

Reduced Costs (rdc) of a RELAXED CLIQUE S: ˜ cS = 1 −

• i∈S

πi Pricing Problem: 1 − ˜ cS = max

• i∈V

πixi s.t. (xi, yij) ∈ F(G) This is a maximum weight RELAXED CLIQUE problem (MW-RC)

◮ Generalization of the maximum (cardinality) RELAXED CLIQUE problem ◮ For most types of RELAXED CLIQUES this has not been studied in the literature so

far

◮ MIP-solver (+ cutting plane algorithm) ◮ We developed new combinatorial B&B and Russian Doll Search Algorithm ◮ Weights πi ∈ R can be negative in case of partitioning

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Support graph

◮ Let λ be a solution of the RMP

.

◮ The support graph of λ is the weighted undirected graph Gλ = (V, Eλ)

defined by Eλ := {{i, j} : i, j ∈ V, i = j, f λ

ij > 0}

with f λ

ij := S∈Ω:i,j∈S λS. ◮ It follows f λ ij ≥ 0 and, for partitioning, also f λ ij ≤ 1. ◮ Based on the Support graph we derive a Generic Branching Rule (GBR)

for Relaxed Clique Partitioning

◮ It consists of removing edges from the graph G.

◮ it does not require any additional constraints ◮ it does not change the pricing problems

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Generic Branching Rule (GBR)

◮ Input: Support graph Gλ and weights f λ ij

(1) Determine the connected components S1, S2, . . . , Sp of Gλ. (2) Identify one component S for which G[S] does not fulfill Π. (3) If no such component exists, then stop (the solution is already a union of feasible relaxed cliques). (4) Determine a maximum-weight spanning tree T = (S, E(T)) of Gλ[S] using the weights f λ

ij . ◮ Output: E(T), the set of edges to eliminate one by one

Proposition

The GBR is a complete rule for partitioning into RELAXED CLIQUES with hereditary Π and connectivity constraints.

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Branching: the Support Graph

1 2 3 4 5 6 7 1 2 3 4 5 6 7

1 1/2 1/2 1/2

1 2 3 4 5 6 7 1 1 1 1 1 1 1/2 1/2 1/2

(a) (b) (c)

◮ Partitioning with (possibly disconnected) 2-plexes

(a) the graph G (b) a fractional solution: λ{1,2,3,4} = 1, λ{5,6} = λ{5,7} = λ{6,7} = 0.5 (c) the Support Graph having binary f λ

ij for all {i, j} ∈ E

The figure shows that the connectivity requirement is crucial!

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Branching for RELAXED CLIQUE Partitioning

Branching schemes: (Pricing problem non-preserving) [Ryan and Foster(1981)] branching Branching on whether two nodes i, j ∈ V are in the same (xi = xj) or different (xi + xj ≤ 1) RELAXED CLIQUES Pros

• 1. Complete
• 2. Easy to understand
• 3. Effective for set partitioning in general
• 4. Simple to handle in MIP-based pricing algorithms

Cons

• 1. Destroys structure of MW-RC pricing problem
• 2. Substantially complicates combinatorial B&B-based

pricing algorithms

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Branching for RELAXED CLIQUE Partitioning

Branching schemes: (Pricing problem non-preserving) [Ryan and Foster(1981)] branching Branching on whether two nodes i, j ∈ V are in the same (xi = xj) or different (xi + xj ≤ 1) RELAXED CLIQUES Pros

• 1. Complete
• 2. Easy to understand
• 3. Effective for set partitioning in general
• 4. Simple to handle in MIP-based pricing algorithms

Cons

• 1. Destroys structure of MW-RC pricing problem
• 2. Substantially complicates combinatorial B&B-based

pricing algorithms

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Dolphins case study: Covering with connected 5-cliques

(a) Real-world split (b) Covering with connected 5-cliques, two clusters

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Computational Results

Zachary’s Karate Club ([Zachary(1977)], also: 10th DIMACS challenge)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

‘Real’ solution [Zachary(1977)]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

3-club partitioning

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Computational Results

Zachary’s Karate Club ([Zachary(1977)], also: 10th DIMACS challenge)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

‘Real’ solution [Zachary(1977)]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34

3-club partitioning

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Scalability analysis

200 400 600 800 1,000 100 200 300

Number of vertices n Computation time in seconds

degavg = 5 degavg = 10 degavg = 15 degavg = 20 degavg = 25

Computation time for linear relaxation for 2-plex partitioning

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Scalability analysis

200 400 600 800 1,000 100 200 300 400

Number of vertices n Computation time in seconds

degavg = 5 degavg = 10 degavg = 15 degavg = 20 degavg = 25

Computation time for linear relaxation for 3-plex partitioning

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Precision analysis

200 400 600 800 1,000 10 20 30 40 50

Number of vertices n Number of instances (max. 50)

• pt

gap < 2 gap < 3

Quality of integer solutions for 2-plex (1 hour)

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Precision analysis

200 400 600 800 1,000 10 20 30 40 50

Number of vertices n Number of instances (max. 50)

• pt

gap < 2 gap < 3

Quality of integer solutions for 3-plex (1 hour)

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Computational Results

Branch-and-Price insights:

• 1. Excellent lower bounds
• 2. Practical hardness increases with s (decreases with γ) and the density of the

graph and depends on the type of relaxed clique

• 3. Covering vs. partitioning

◮ Covering slightly easier than partitioning for solving LP-relaxation of the

master program

◮ Covering much harder when it comes to branching

• 4. Branching

◮ Ryan/Foster branching seems to be much more effective ◮ Pricing problems get harder when using Ryan/Foster branching

• 5. Subproblem solution

◮ Combinatorial B&Bs much faster than MIPs for solving the pricing problem ◮ Ryan/Foster branching complicates the pricing problems for the

combinatorial B&Bs

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Conclusions and Outlook

Conclusions:

◮ Introduced the RELAXED CLIQUE covering/partitioning problem ◮ New approach for community detection ◮ Interesting components of branch-and-price

◮ Branching ◮ Solution of maximum weight RELAXED CLIQUE pricing problem

A draft on this topic can be found here: http: //wiwi.uni-mainz.de/Papers/Discussion_Paper_1520.pdf Outlook:

◮ Finding a complete pricing problem preserving branching scheme ◮ Heuristics and metaheuristics can accelerate pricing

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Conclusions and Outlook

Conclusions:

◮ Introduced the RELAXED CLIQUE covering/partitioning problem ◮ New approach for community detection ◮ Interesting components of branch-and-price

◮ Branching ◮ Solution of maximum weight RELAXED CLIQUE pricing problem

A draft on this topic can be found here: http: //wiwi.uni-mainz.de/Papers/Discussion_Paper_1520.pdf Outlook:

◮ Finding a complete pricing problem preserving branching scheme ◮ Heuristics and metaheuristics can accelerate pricing

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Literature Review

[Almeida and Carvalho(2012)] Almeida, M. T. and Carvalho, F. D. (2012). Integer models and upper bounds for the 3-club problem. Networks, 60(3), 155–166. [Almeida and Carvalho(2013)] Almeida, M. T. and Carvalho, F. D. (2013). An analytical comparison of the LP relaxations of integer models for the k-club problem. European Journal of Operational Research, (0). [Balasundaram et al.(2011)] Balasundaram, B., Butenko, S., and Hicks, I. V. (2011). Clique relaxations in social network analysis: The maximum k-plex problem. Operations Research, 59(1), 133–142. [Bourjolly et al.(2000)] Bourjolly, J.-M., Laporte, G., and Pesant, G. (2000). Heuristics for finding k-clubs in an undirected graph. Computers & Operations Research, 27(6), 559–569. [Bourjolly et al.(2002)] Bourjolly, J.-M., Laporte, G., and Pesant, G. (2002). An exact algorithm for the maximum k-club problem in an undirected graph. European Journal of Operational Research, 138(1), 21–28. [Brandes and Erlebach(2005)] Brandes, U. and Erlebach, T., editors (2005). Network Analysis: Methodological Foundations [outcome of a Dagstuhl seminar, 13-16 April 2004], volume 3418 of Lecture Notes in Computer Science. Springer. [Fortunato(2010)] Fortunato, S. (2010). Community detection in graphs. Physics Reports, 486(3–5), 75–174. [Girvan and Newman(2002)] Girvan, M. and Newman, M. E. J. (2002). Community structure in social and biological networks. Proceedings of the National Academy of Sciences, 99(12), 7821–7826.

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Literature Review

[Gschwind et al.(2015)] Gschwind, T., Irnich, S., and Podlinski, I. (2015). Maximum weight relaxed cliques and russian doll search revisited. Technical Report LM-2015-02, Chair of Logistics Management, Gutenberg School of Management and Economics, Johannes Gutenberg University Mainz, Mainz, Germany. [Kammer and Täubig(2005)] Kammer, F. and Täubig, H. (2005). Connectivity. In [Brandes and Erlebach(2005)], pages 143–177. [Kosub(2004)] Kosub, S. (2004). Local density. In [Brandes and Erlebach(2005)], pages 112–142. [Luce and Perry(1949)] Luce, R. D. and Perry, A. D. (1949). A method of matrix analysis of group structure. Psychometrika, 14(2), 95–116. [Mahdavi Pajouh and Balasundaram(2012)] Mahdavi Pajouh, F. and Balasundaram, B. (2012). On inclusionwise maximal and maximum cardinality -clubs in graphs. Discrete Optimization, 9(2), 84–97. [Newman and Girvan(2004)] Newman, M. E. J. and Girvan, M. (2004). Finding and evaluating community structure in networks. Physical Review E, 69, 026113. [Pajouh et al.(2014)] Pajouh, F. M., Miao, Z., and Balasundaram, B. (2014). A branch-and-bound approach for maximum quasi-cliques. 216(1), 145–161. [Pattillo et al.(2013a)] Pattillo, J., Youssef, N., and Butenko, S. (2013a). On clique relaxation models in network analysis. European Journal of Operational Research, 226(1), 9–18.

General context Relaxed Clique Definitions Unified Branch-and-Price Framework Computational Results

Literature Review

[Pattillo et al.(2013b)] Pattillo, J., Veremyev, A., Butenko, S., and Boginski, V. (2013b). On the maximum quasi-clique problem. Discrete Applied Mathematics, 161(1–2), 244–257. [Ryan and Foster(1981)] Ryan, D. and Foster, B. (1981). An integer programming approach to scheduling. In A. Wren, editor, Computer Scheduling of Public Transport: Urban Passenger Vehicle and Crew Scheduling, chapter 17, pages 269–280. Elsevier, North-Holland. [Sherali and Smith(2006)] Sherali, H. D. and Smith, J. C. (2006). A polyhedral study of the generalized vertex packing problem. Mathematical Programming, 107(3), 367–390. [Trukhanov et al.(2013)] Trukhanov, S., Balasubramaniam, C., Balasundaram, B., and Butenko, S. (2013). Algorithms for detecting optimal hereditary structures in graphs, with application to clique relaxations. Computational Optimization and Applications, 56(1), 113–130. [Veremyev and Boginski(2012)] Veremyev, A. and Boginski, V. (2012). Identifying large robust network clusters via new compact formulations of maximum k-club problems. European Journal of Operational Research, 218(2), 316–326. [Wotzlaw(2014)] Wotzlaw, A. (2014). On solving the maximum k-club problem. Technical Report arXiv:1403.5111v2, Institut für Informatik, Universität zu Köln, Köln, Germany. [Zachary(1977)] Zachary, W. W. (1977). An information flow model for conflict and fission in small groups. Journal of Anthropological Research, 33(4), 452–473.