Clique Relaxation Models in Networks: Theory, Algorithms, and Applications

Clique Relaxation Models in Networks:
Theory, Algorithms, and Applications
Sergiy Butenko (butenko@tamu.edu)
Department of Industrial and Systems Engineering
Texas A&M University
College Station, TX 77843-3131

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Outline
Introduction
Graph Theory Basics

Taxonomy of Clique Relaxations

Theory
Complexity Issues
Analytical bounds
Mathematical programming formulations

Algorithms
Node Deletion Problems with Heredity
Scale Reduction Approaches

Conclusion and References
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Outline
Introduction
Graph Theory Basics


Theory
Complexity Issues
Analytical bounds

Algorithms

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Graphs
Graph is the mathematical term for a “network” - often visualized as
vertices (points, nodes) connected by edges (lines, arcs)

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Graph theory basics

The origins of graph theory are attributed to the Seven Bridges of
K¨nigsberg problem solved by Leonhard Euler in 1735.
o

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Graph theory basics
A simple, undirected graph is a pair G = (V , E ), where V is a ﬁnite set of
vertices and E ⊆ V × V is a set of edges, with each edge deﬁned on a pair
of vertices.
6

5 1

4 2

3

V = {1, 2, 3, 4, 5, 6}

E = {(1, 2), (1, 6), (2, 3), (2, 4), (3, 4), (4, 5), (5, 6)}
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Graph theory basics

A graph can be represented in several ways.
We can represent a graph G = (V , E ) by its |V | × |V |
adjacency matrix AG = [aij ], such that

1, if (vi , vj ) ∈ E ,
aij =
0, otherwise.

Another way of representing a graph is by its adjacency lists,
where for each vertex v ∈ V we record the set of vertices
adjacent to it.

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Graph theory basics

A multigraph is a graph with repeated edges.
A directed graph, or digraph, is a graph with directions assigned
to its edges.

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Graph theory basics

If G = (V , E ) is a graph and e = (u, v ) ∈ E , we say that u is
adjacent to v and vice-versa. We also say that u and v are
neighbors.

Neighborhood N(v ) of a vertex v is the set of all its neighbors
in G : N(v ) = {j : (i, j) ∈ E }.

If G = (V , E ) is a graph and e = (u, v ) ∈ E , we say that e is
incident to u and v .

The degree of a vertex v is the number of its incident edges.

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Graph theory basics

G = (V , E ) is a simple undirected graph, V = {1, 2, . . . , n}.

G = (V , E ), is the complement graph of G = (V , E ), where
E = {(i, j) | i, j ∈ V , i = j and (i, j) ∈ E }.
/

For S ⊆ V , G (S) = (S, E ∩ S × S) the subgraph induced by S.

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Cliques and independent sets

A subset of vertices C ⊆ V is called a clique if G (C ) is a
complete graph.

A subset I ⊆ V is called an independent set (stable set, vertex
packing) if G (I ) has no edges.

¯
C is a clique in G if and only if C is an independent set in G .

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A clique (independent set) is said to be
– maximal, if it is not a subset of any larger clique (independent
set);
– maximum, if there is no larger clique (independent set) in the
graph.

ω(G ) – the clique number of G .

α(G ) – the independence (stability) number of G .

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G 1 1 G
Complement
5 2 5 2

4 3 4 3

{1,2,5} : maximal clique {1,2,5} : maximal
independent set
{1,4} : maximal
{1,4} : maximal clique
independent set

{2,3,4,5} : maximum clique {2,3,4,5} : maximum
independent set

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Social networks

A social network is described by G = (V , E ) where V is the set of
“actors” and E is the set of “ties”.

actors are people and a tie exists if two people know each other.
actors are wire transfer database records and a tie exists if two
records have the same matching ﬁeld.
actors are telephone numbers and a tie exists if calls were made
between them.

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Social network analysis

“Popular” social networks:
Kevin Bacon Number
Erd¨s Number
o
Six degrees of separation and small world phenomenon in
acquaintance networks

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Cohesive subgroups

Cohesive subgroups are “tightly knit groups” in a social
network.
Social cohesion is often used to explain and develop sociological
theories.
Members of a cohesive subgroup are believed to share
information, have homogeneity of thought, identity, beliefs,
behavior, even food habits and illnesses.

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Applications

Acquaintance Networks - criminal network analysis
Wire Transfer Database Networks - detecting money
laundering
Call Networks - organized crime detection
Protein Interaction Networks - predicting protein functions
Gene Co-expression Networks - detecting network motifs
Metabolic Networks - identifying metabolic pathways
Stock Market Networks - stock portfolios
Internet Graphs - information search and retrieval
Wireless and telecommunication networks - clustering and
routing
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Random graph models

Uniform random graphs (Erdos Renyi):
G (n, p): n vertices, each edge exists with a probability p;
G (n, M): all graphs with n vertices and M edges are assigned
the same probability
Power-law random graphs:
the probability that a vertex has a degree k is proportional to
k −β
Random graphs with a given degree sequence

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Scale-free graphs

Scale-free nature : A property observed in many natural and
man-made networks - biological networks, social networks,
internet graphs
Degree distribution follows a power law - P(k) ∝ k −β
Fundamentally diﬀerent from classical random graph model of
Erd¨s and Renyi
o
Principle of preferential attachment underlies the growth and
evolution of such networks (i.e. the rich get richer!)

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Science - co-authorship network
Image source:
http://www.jeﬀkennedyassociates.com:16080/connections/concept/image.html

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Internet colored by IP addresses
Image source:
http://www.jeﬀkennedyassociates.com:16080/connections/concept/image.html

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H. Pylori - protein interactions
Image generated using Graphviz

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Properties of cohesive subgroups

Some desirable properties of a cohesive subgroup are:
Familiarity (degree);
Reachability (distance, diameter);
Robustness (connectivity);
Density (edge density).
Earliest models of cohesive subgroups were cliques (Luce and Perry,
1949), representing an “ideal” model of a cohesive subgroup.

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Clique relaxations: k-cliques and k-clubs

A k-clique is a subset of vertices C such that for every i, j ∈ C ,
d(i, j) ≤ k.

A k-club is a subset of vertices D such that diam(G [D]) ≤ k.
6

5 1

4 2

3

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d(i, j) ≤ k.

6 {2,3,4} is a 1-club ... the
“regular” clique
5 1

4 2

3

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d(i, j) ≤ k.

6 {2,3,4} is a 1-club ... the
5 1
{1,2,4,5,6} is a 2-club

4 2

3

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d(i, j) ≤ k.

6 {2,3,4} is a 1-club ... the
5 1
{1,2,4,5,6} is a 2-club
{1,2,3,4,5} is a 2-clique but
4 2 NOT a 2-club

3

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d(i, j) ≤ k.

6 {2,3,4} is a 1-club ... the
5 1
{1,2,4,5,6} is a 2-club
{1,2,3,4,5} is a 2-clique but
4 2 NOT a 2-club
maximality of a 2-club is harder
3 to test

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k-plex

Deﬁnition
A subset of vertices S is said to be a k-plex if the minimum degree
in the induced subgraph δ(G [S]) ≥ |S| − k
i.e. every vertex in G [S] has degree at least |S| − k.

2 1

6 3

5 4

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k-plex

Deﬁnition
2 1 {3,4,5,6} is a 1-plex ... the
6 3

5 4

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k-plex

Deﬁnition
2 1 {3,4,5,6} is a 1-plex ... the
6 3 {1,3,4,5,6} is a 2-plex
(and NOT a 1-plex)

5 4

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k-plex

Deﬁnition

2 1
{3,4,5,6} is a 1-plex ... the
6 3 {1,3,4,5,6} is a 2-plex
(and NOT a 1-plex)
{1,2,3,4,5,6} is a 3-plex
5 4
(and NOT a 2-plex)

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co-k-plex
Deﬁnition
A subset of vertices S is a co-k-plex if the maximum degree in the
induced subgraph ∆(G [S]) ≤ k − 1.
i.e. degree of every vertex in G [S] is at most k − 1.
S is a co-k-plex in G if and only if S is a k-plex in the complement
¯
graph G .

2 1 2 1

6 3 6 3

5 4 5 4

3-plex Co-3-plex
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Structural Properties of a k-Plex

If G is a k-plex then
1. Every subgraph of G is a k-plex;
n+2
2. If k 2 then diam(G ) ≤ 2;
3. κ(G ) ≥ n − 2k + 2.

k-plexes for “small” k values, guarantee reachability and
connectivity while relaxing familiarity.

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Known clique relaxation models

k-clique (distance)
k-club (diameter)
k-plex (degree)
k-core (degree)
k-connected subgraph (connectivity)
γ-quasi-clique (density)
(γ, λ)-quasi-clique (density, degree)
R-robust k-club (connectivity, diameter)

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Outline
Introduction
Graph Theory Basics


Theory
Complexity Issues
Analytical bounds

Algorithms

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Alternative clique deﬁnitions

Distance-based
Vertices are distance one away from each other.

Diameter-based
Vertices induce a subgraph of diameter one.

Domination-based
Every one vertex forms a dominating set.

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Alternative clique deﬁnitions

Degree-based
Each vertex is connected to all other vertices.

Connectivity-based
All vertices need to be removed to disconnect the induced subgraph.

Density-based
Vertices induce a subgraph that has all possible edges.

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Order of a clique relaxation

We introduce a hierarchy of clique relaxation models by defining
their order as follows.
Clique is the only clique relaxation of order 0.
Clique relaxations of order 1 relax one of the elementary
properties (distance, diameter, ...) used to define clique.
Clique relaxations of order 2 relax two of the elementary
clique-defining properties.
...

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Nature of a clique relaxation

Parametric clique relaxations
relax a parameter (“one”, “all”) describing an elementary
clique-deﬁning property.

Partial clique relaxations
require only a (high) fraction of elements to satisfy an elementary
clique-deﬁning property or a parametric clique-relaxation property.

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Types of parametric clique relaxations

Parametric clique relaxation of type 1
Replace “one” with “(at most) k”, k 1, in a definition of clique.

Parametric clique relaxation of type 2
Replace “one” with “(at most) all but k”, k 1, in a definition of clique.

Parametric clique relaxation of type 1a
Replace “all” with “(at least) k”, k 1, in a definition of clique.

Parametric clique relaxation of type 2a
Replace “all” with “(at least) all but k”, k 1, in a definition of clique.

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First-order parametric
clique relaxations of type 1

Replace “one” with “(at most) k”, k 1.

Distance-based: k-clique
Vertices are distance at most k away from each other.

Diameter-based: k-club
Vertices induce a subgraph of diameter at most k.

Domination-based: k-plex
Every k vertices form a dominating set.

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clique relaxations of type 2

Replace “one” with “(at most) all but k”, k 1

Distance-based: N/A
Vertices in S are distance at most |S| − k away from each other.

Diameter-based: N/A
Vertices induce a subgraph G [S] of diameter at most |S| − k.

Domination-based: N/A
Every |S| − k vertices form a dominating set.

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clique relaxations of type 1a

Replace “all” with “(at least) k”, k 1

Degree-based: k-core
Each vertex is connected to at least k other vertices.

Connectivity-based: k-connected subgraph
At least k vertices need to be removed to disconnect the induced
subgraph.

Density-based: N/A
Vertices induce a subgraph that has at least k edges.

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clique relaxations of type 2a

Replace “all” with “(at least) all but k”, k 1.

Degree-based: k-plex
Each vertex is connected to at least all but k other vertices.

Connectivity-based: N/A
At least all but k vertices need to be removed to disconnect the
induced subgraph.

Density-based: N/A
Vertices induce a subgraph that has at least all but k edges.

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Outline
Introduction
Graph Theory Basics


Theory
Complexity Issues
Analytical bounds

Algorithms

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Basics of the complexity theory

Given a combinatorial optimization problem, a natural question
is:
is this problem “easy” or “hard”?
Methods for easy and hard problems are fundamentally
diﬀerent.

How do we distinguish between “easy” and “hard” problems?

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“Easy” problems

By easy or tractable problems we mean the problems that can
be solved in time polynomial with respect to their size.

We also call such problems polynomially solvable and denote
the class of polynomially solvable problems by P.
Sorting
Minimum weight spanning tree
Linear programming

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Defining “hard” problems

How do we define “hard” problems?

How about defining hard problems as all problems that are not
easy, i.e., not in P?

Then some of the problems in such a class could be TOO hard
– we cannot even hope to be able to solve them.

We want to define a class of hard problems that we may be
able to solve, if we are lucky (say, we may be able to guess the
solution and check that it is indeed correct).

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Defining “hard” problems

Let us take the maximum clique problem as an example and try
to guess a solution.

We can pick some random clique, compute its size, but how do
we know if this solution is optimal?

Indeed, this is as difficult to verify as to solve the maximum
clique problem – it is unlikely that we can count on luck in this
case...

How about transforming our optimization problem to such a
form, where instead of finding an optimal solution the question
would be “is there a solution with objective value ≥ s, where s
is a given integer?
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Three versions of optimization problems

Consider a problem

min f (x) subject to x ∈ X .

Optimization version: ﬁnd x from X that maximizes f (x);
Answer: x ∗ maximizes f (x)

Evaluation version: ﬁnd the largest possible f (x);
Answer: the largest possible value for f (x) is f ∗

Recognition version: Given f ∗ , does there exist an x such that
f (x) ≥ f ∗ ?
Answer: “yes” or “no”

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Recognition problems

Recognition problems can still be undecidable.
Halting problem: Given a computer program with its
input, will it ever halt?

Recall the “luck factor”: if we pick a random feasible solution
and it happens to give “yes” answer, then we solved the
problem in polynomial time.
Max clique: Randomly pick a solution (a clique). If its
size is ≥ s (which we can verify in polynomial time),
then obviously the answer is “yes”. This clique can be
viewed as a certiﬁcate proving that this is indeed a yes
instance of max clique.

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Class N P

We only consider problems for which any yes instance there
exists a concise (polynomial-size) certiﬁcate that can be veriﬁed
in polynomial time.

We call this class of problems nondeterministic polynomial and
denote it by N P.

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P vs N P

Note that any problem from P is also in N P (i.e., P ⊆ N P),
so there are easy problems in N P.

So, are there “hard” problems in N P, and if there are, how do
we deﬁne them?

We don’t know if P = NP, but “most” people believe that
P = NP.
An “easy” way to make $1,000,000!
http://www.claymath.org/millennium/P vs NP/

We can call a problem hard if the fact that we can solve this
problem would mean that we can solve any other problem in
comparable amount of time.
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Polynomial reducibility

Reduce π1 to π2 : if we can solve π2 fast, then we can solve π1
fast, given that the reduction is “easy”.
Polynomial reduction from π1 to π2 requires existence of
polynomial-time algorithms
1. A1 converts an input for π1 into an input for π2 ;
2. A2 converts an output for π2 into output for π1 .

Transitivity: If π1 is polynomially reducible to π2 and π2 is
polynomially reducible to π3 then π1 is polynomially reducible
to π3 .

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NP-complete problems
A problem π is called NP-complete if
1. π ∈ NP;
2. Any problem from NP can be reduced to π in polynomial time.
A problem π is called NP-hard if any problem from NP can be
reduced to π in polynomial time. (no π ∈ NP requirement)

Due to transitivity of polynomial reducibility, in order to show
that a problem π is N P-complete, it is suﬃcient to show that
1. π ∈ NP;
2. There is an NP-complete problem π that can be reduced to π
in polynomial time.

To use this observation, we need to know at least one
N P-complete problem...
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Satisﬁability (SAT) problem

A Boolean variable x is a variable that can assume only the values
true and false.
Boolean variables can be combined to form Boolean formulas using
the following logical operations:
1. Logical AND (∧ or ·) -conjunction
2. Logical OR (∨ or +) - disjunction
3. Logical NOT (¯)
x
kj
A clause is Cj = yjp , where a literal yjp is xr or xr for some r .
¯
p=1
m
Conjunctive normal form (CNF):F = Cj , where Cj is a clause.
j=1

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Satisfiability problem

A CNF F is called satisfiable if there is an assignment of
variables such that F = 1 (TRUE ).
Satisfiability (SAT) problem: Given m clauses C1 , . . . , Cm
involving the variables x1 , . . . , xn , is the CNF
m
F = Cj ,
j=1

satisfiable?

Theorem (Cook, 1971)
SAT is NP-complete.

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“Best” approximation algorithms and
heuristics

For some problems there are hardness of approximation results
stating that the problem is hard to approximate within a certain
factor.
For example, the k-center problem is hard to approximate
within a factor better than 2.
Then any polynomial-time algorithm approximating the
k-center problem within the factor of 2 can be considered the
“best” approximation algorithm for this problem.

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“Best” approximation algorithms and
heuristics

However, some problems are even harder to approximate. For
example, the maximum clique is hard to approximate within a
factor n1− for any positive .
In this case, by the “best” heuristic we could mean a heuristic
that cannot be provably outperformed by any other
polynomial-time algorithm (unless P = NP).

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Recognizing the gap between k-club and
l -club numbers

Theorem
Let positive integer constants k and l, l k be given. The problem
of checking whether ωl (G ) = ωk (G ) is NP-hard.
¯ ¯

Note that
ω(G ) ≤ ∆(G ) + 1 ≤ ωk (G )
¯
and observe that we can easily check whether ω(G ) = ∆(G ) + 1.
Hence, it is NP-hard to check whether ωk (G ) = ∆(G ) + 1.
¯

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“Best” heuristics for k-club/clique

Corollary
Let k be a ﬁxed integer, k ≥ 2. Unless P = NP, there cannot be a
polynomial time algorithm that ﬁnds a k-club of size greater than
∆(G ) + 1 whenever such a k-club exists in the graph.

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Protein Interaction Networks

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A max 2-club/clique of S. Cerevisiae.

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A max 2-club/clique of H. Pylori.

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A max 3-clique/club of S. Cerevisiae

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Outline
Introduction
Graph Theory Basics


Theory
Complexity Issues
Analytical bounds

Algorithms

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Size of clique relaxations in a graph

Question: How large could a k-clique (k-club, k-plex) be for a
graph with a small clique number?

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Answers:
k-club: ω(K1,n ) = 2, but ωk (K1,n ) = n + 1, could be arbitrary
¯
large
k-clique: same as k-club

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Answers:
k-club: ω(K1,n ) = 2, but ωk (K1,n ) = n + 1, could be arbitrary
¯
large
k-clique: same as k-club
k-plex:
Lemma
For any graph G and a positive integer k,

ωk (G ) ≤ kω(G ) and αk (G ) ≤ kα(G )

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Upper bound on the quasi-clique number

Proposition
The γ-clique number ωγ (G ) of a graph G with n vertices and m
edges satisﬁes the following inequality:

γ+ γ 2 + 8γm
ωγ (G ) ≤ . (1)
2γ
Moreover, if a graph G is connected then

γ+2+ (γ + 2)2 + 8(m − n)γ
ωγ (G ) ≤ . (2)
2γ

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Relation between ωγ (G ) and ω(G )

Proposition
The γ-clique number ωγ (G ) and the clique number ω(G ) of graph
G satisfy the following inequalities:

ω(G ) − 1 ωγ (G ) − 1 1 ω(G ) − 1
≤ ≤ . (3)
ω(G ) ωγ (G ) γ ω(G )

Corollary
1
If γ 1 − ω(G ) then

ω(G )γ
ωγ (G ) ≤ . (4)
1 − ω(G ) + ω(G )γ

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Relation between omegaγ (G ) and ω(G )

Table: The value of upper bound (4) on γ-clique number with
γ = 0.95, 0.9, 0.85 for graphs with small clique number.
1
ω(G ) 1 − ω(G ) 0.95 0.9 0.85
3 0.667 3.35 3.86 4.64
4 0.75 4.75 6 8.5
5 0.8 6.33 9 17
6 0.83 8.14 13.5 51
7 0.86 10.23 21 –
8 0.88 12.67 36 –
9 0.89 15.55 81 –
10 0.9 19 – –

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Outline
Introduction
Graph Theory Basics


Theory
Complexity Issues
Analytical bounds

Algorithms

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Math programming formulations

Consider a simple undirected graph G = (V , E ) with n vertices
Let A = [aij ]n
i,j=1 be the adjacency matrix of G
Let x = (x1 , . . . , xn ) be a 0-1 vector with xi = 1 if node i
belongs to Gs , and xi = 0 otherwise.

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GS is a γ-clique if it has at least γ|GS |(|GS | − 1)/2 edges
n
We have: |GS | = xi
i=1
This number of edges can be expressed in terms of vector x as:
n n n n
1
2γ xi xi − 1 = 1γ
2 xi xj − xi
i=1 i=1 i,j=1 i=1
n n n n
1 1
= 2γ xi xj + xi2 − xi = 2γ xi xj
i,j=1;i=j i=1 i=1 i,j=1;i=j
n
1
The number of edges in GS is 2 aij xi xj
i,j=1

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We obtain the following 0-1 problem with one quadratic constraint:
n
max xi
i=1

subject to:
n n
aij xi xj ≥ γ xi xj
i,j=1 i,j=1;i=j

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Deﬁne wij = xi xj . The constraint wij = xi xj is equivalent to

wij ≤ xi , wij ≤ xj , wij ≥ xi + xj − 1 .
n
Linearized formulation: max xi
i=1
n n
subjectto : aij wij ≥ γ wij
i,j=1 i,j=1;i=j

wij ≤ xi , wij ≤ xj , wij ≥ xi + xj − 1 .
wij , xi ∈ {0, 1}, ∀i j = 1, . . . , n
O(n2 ) 0-1 variables, O(n2 ) constraints

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Outline
Introduction
Graph Theory Basics


Theory
Complexity Issues
Analytical bounds

Algorithms

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Outline
Introduction
Graph Theory Basics


Theory
Complexity Issues
Analytical bounds

Algorithms

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Clique relaxation and node deletion problems

Node deletion problems

Clique Node deletion
relaxation problems
problems with heredity

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Node deletion problems

A graph property Π is said to be hereditary on induced
subgraphs, if G is a graph with property Π and the deletion of
any subset of vertices does not produce a graph violating Π
Π is said to be nontrivial if it is true for a single vertex graph
and is not satisﬁed by every graph
Π is said to be interesting if there are arbitrarily large graphs
satisfying Π

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Yannakakis theorem

The maximum Π problem is to ﬁnd the largest order induced
subgraph that does not violate property Π

Theorem (Yannakakis, 1978)
The maximum Π problem for nontrivial, interesting graph properties
that are hereditary on induced subgraphs is NP-hard.

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Max weight node deletion problem

Given
a simple, undirected graph G = (V , E ),
positive weights w (v ) for each v ∈ V ,
and a nontrivial, interesting and hereditary (on induced
subgraphs) property Π.
Find

ν(G ) = max{w (P) : P ⊆ V , G [P] satisﬁes Π},
where w (P) = v ∈P w (v ) and G [P] is the subgraph induced by P.

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Generalizing max clique algorithms

Some of the most practically eﬀective algorithms for the maximum
clique problem rely on the fact that clique is hereditary on induced
subgraphs.
Carraghan and Pardalos (1990) - used as DIMACS benchmark
¨
Osterg˚rd (2002)
a
We use this observation to develop a generalized algorithm for the
minimum weight node deletion problem (GAMWNDP).

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GAMWNDP

Order vertices V = {v1 , v2 , · · · , vn }, deﬁne Si = {vi , vi+1 , · · · , vn }
We compute the function c(i) that is the weight of the maximum
induced subgraph with property Π in G [Si ].
Obviously, c(n) = w (vn ) and c(1) = ν(G ).
For the unweighted case with w (vi ) = 1, i = 1, . . . , n we have

c(i + 1) + 1, if the solution must contain vi
c(i) =
c(i + 1), otherwise

For the weighted case, c(i) c(i + 1) implies that vi belongs to
every optimal solution, and c(i) ≤ c(i + 1) + w (vi ).

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GAMWNDP

We compute the value of c(i) starting from c(n) and down to
c(1), and in each major iteration work with a current feasible
solution P, a candidate set C , and an incumbent solution S.
Pruning occurs when
w (C ) + w (P) w (S) or
c(i) + w (P) w (S), where i = min{j : vj ∈ C }.
Problem-speciﬁc features:
candidate set generation;
vertex ordering.

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Outline
Introduction
Graph Theory Basics


Theory
Complexity Issues
Analytical bounds

Algorithms

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Outline
Introduction
Graph Theory Basics


Theory
Complexity Issues
Analytical bounds

Algorithms

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k-Core and k-community

k-Core
A sub-graph G of G such that each node has degree greater than or
equal to k.

k-Community
A subgraph G of G such that each (i, j) ∈ E if (i, j) ∈ E and i, j
have at least k common neighbors in G .

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Example - k-Core k-Core example

(a) Original Graph (b) Max 3-core

Each node in the max 3-core is connected to at least 3 other
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vertices.

Example - k-Community k-Community example

(c) Original Graph (d) Max 2-community

Any two vertices in the max 2-community are connected only if 83/93
they have at least 2 mutual neighbors.

Scale reduction approaches

Reduce problem size by ignoring certain parts of the graph
which we are certain would not be in the graph.
Most commonly used technique: Peeling
1. Remove all the vertices with degree ≤ k − 1.
2. If no vertices removed, stop. Else, redo step 1 using the
obtained graph.
Peeling is nothing but ﬁnding the k-core!

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Scale reduction approaches

Property 1
A clique of size k is both a (k − 1)-core and (k − 2)-community.

Property 2
If the k-Core of G is empty, then ω(G ) k + 1.

Property 3
If the k-Community of G is empty, then ω(G ) k + 2.

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k-Core Upper Bound
Find smallest k1 such that k1 -Core is empty. UB = k1 = 5.k-Core upper bound

(e) Original Graph (f) 4-core

(g) Original Graph (h) 5-core
Anurag Verma, Sergiy Butenko, Texas AM University 10/ 20

Find smallest k1 such that k1 -Core is empty. UB = k1 = 5.
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k-Community Upper Bound
k-Community upper bound
Find smallest k2 such that k2 -Comm is empty.

(i) Original Graph (j) Intermediate (k) 2-comm

Find smallest k2 such that k2 -Comm 1 = 4.
3-comm is empty. Thus UB = k2 + is empty. UB = k2 + 1 = 4.

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Finding the least k2

A simple binary search strategy:
Step 0 Set ku = n − 2, kl = 0, k2 = n − 2
Step 1 If k2 -Comm(G) is empty, ku = k2 . Else, kl = k2 .
Step 2 If ku − kl ≤ 1, set k2 = ku , STOP. Else, set
k2 = (ku + kl )/2, go to Step 1.

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The call graph

Abello, Pardalos, and Resende, 1999:
20-day period: 290 million vertices and 4 billion edges.
one-day call graph: 53,767,087 vertices and over 170 millions of
edges.
3,667,448 connected components; 302,468 of them with more
than 3 vertices.
A giant connected component with 44,989,297 vertices.

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Upper Bounds obtained bounds on SNAP
Upper on SNAP Graphs database

Graph n m k-Core UB k-Comm UB
UB n’ UB n”
WikiTalk 2394385 4659565 132 700 52 237
cit-Patents 3774768 16518947 65 106 35 83
Email-EuAll 265214 364481 38 292 19 62
Cit-HepPh 34546 420877 31 40 24 36
Cit-HepTh 27770 352285 38 52 29 48
Slashdot0811 77360 469180 55 129 34 87
Slashdot0902 82168 504230 56 134 35 96
soc-Epinions1 75879 405740 68 486 32 61
Wiki-Vote 7115 100762 54 336 22 50
p2p-Gnutella31 62586 147892 7 1004 4 57
p2p-Gnutella04 10876 39994 8 365 4 12
p2p-Gnutella24 26518 65369 6 7480 4 41
p2p-Gnutella25 22687 54705 6 6091 4 25
p2p-Gnutella30 36682 88328 8 14 4 42
web-Stanford 281903 1992636 72 387 61 126
web-NotreDame 325729 1090108 156 1367 155 155
web-Google 875713 4322051 45 48 44 48
web-BerkStan 685230 6649470 202 392 201 392
Amazon0601 403394 2443408 11 32886 11 5361
Amazon0505 410236 2439437 11 32632 11 4878
Amazon0302 262111 899792 7 286 7 105
Amazon0312 400727 2349869 11 27046 11 4534

Comparison of the upper bounds obtained by a k-core scheme vs a 90/93

k-comm scheme.

Upper Bounds obtained on SNAPon SNAP
Upper bounds Graphs database

Graph n m k-Core UB k-Comm UB
UB n’ UB n”
WikiTalk 2394385 4659565 132 700 52 237
cit-Patents 3774768 16518947 65 106 35 83
Email-EuAll 265214 364481 38 292 19 62
Cit-HepPh 34546 420877 31 40 24 36
Cit-HepTh 27770 352285 38 52 29 48
Slashdot0811 77360 469180 55 129 34 87
Slashdot0902 82168 504230 56 134 35 96
soc-Epinions1 75879 405740 68 486 32 61
Wiki-Vote 7115 100762 54 336 22 50
p2p-Gnutella31 62586 147892 7 1004 4 57
p2p-Gnutella04 10876 39994 8 365 4 12
p2p-Gnutella24 26518 65369 6 7480 4 41
p2p-Gnutella25 22687 54705 6 6091 4 25
p2p-Gnutella30 36682 88328 8 14 4 42
web-Stanford 281903 1992636 72 387 61 126
web-NotreDame 325729 1090108 156 1367 155 155
web-Google 875713 4322051 45 48 44 48
web-BerkStan 685230 6649470 202 392 201 392
Amazon0601 403394 2443408 11 32886 11 5361
Amazon0505 410236 2439437 11 32632 11 4878
Amazon0302 262111 899792 7 286 7 105
Amazon0312 400727 2349869 11 27046 11 4534

Comparison of the number of nodes in the residual graph after the 91/93

scale reduction.

Upper bounds on random graphs

Uniform random graphs with 1000 nodes.
Bollobas-Erd¨ s estimate: 2 log(n)/ log(1/p).
o
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Outline
Introduction
Graph Theory Basics


Theory
Complexity Issues
Analytical bounds

Algorithms

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Conclusion

We propose a taxonomy of clique relaxation models
opens new avenues for systematic study of clique relaxations
covers known clique relaxation models
unveils new models of potential interest
An eﬀective algorithm is provided for node deletion problems
that are nontrivial, interesting, and hereditary on induced
subgraphs
state-of-the-art exact approach to max k-plex
can be used to solve many other important problems
A scale-reduction approach based on the concept of
k-community is proposed
outperforms the peeling approach based on k-core
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References
B. Balasundaram, S. Butenko, I. V. Hicks.
Clique relaxations in social network analysis: The maximum k-plex
problem.
Operations Research, 59: 133–142, 2011.
S. Trukhanov, B. Balasundaram, and S. Butenko.
Exact algorithms for hard node deletion problems in networks.
Submitted.
S. Butenko, J. Pattillo, and N. Youssef.
A taxonomy of clique relaxation models in networks.
In preparation.
A. Verma and S. Butenko.
A new scale-reduction technique for the maximum clique and related
problems.
In preparation.
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References

J. Pattillo, A. Veremyev, S. Butenko, and V. Boginski.
On the maximum quasi-clique problem.
Submitted.
S. Butenko, S. Kahruman, and O. Prokopyev.
On “provably best” heuristics for hard optimization problems.
In preparation.

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Clique Relaxation Models in Networks: Theory, Algorithms, and Applications

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Clique Relaxation Models in Networks: Theory, Algorithms, and Applications