Combinatorial Algorithms for the Maximum k-plex
Problem
Benjamin McClosky, Illya V. Hicks
Department of Computational and Applied Mathematics, Rice University, 6100 Main St - MS 134,
Houston, Texas 77005-1892, USA, {[email protected], [email protected]}
The maximum clique problem provides a classic framework for detecting cohesive subgraphs. However, this approach can fail to detect much of the cohesive structure in a graph.
To address this issue, Seidman and Foster introduced k-plexes as a degree-based relaxation
of graph completeness. More recently, Balasundaram et al. formulated the maximum k-plex
problem as an integer program and designed a branch-and-cut algorithm. This paper derives
a new upper bound on the cardinality of k-plexes and adapts combinatorial clique algorithms
to find maximum k-plexes.
1.
Introduction
All graphs in this paper are finite, simple, and undirected. A complete graph consists of
pairwise adjacent vertices. A maximal complete subgraph defines a clique. The problem of
finding maximum cardinality cliques is a classic NP-complete problem and is of fundamental
importance in combinatorial optimization. The maximum clique problem has applications
in ad hoc wireless networks (Chen et al. 2004), data mining (Washio and Motoda 2003),
and biochemistry and genomics (Butenko and Wilhelm 2006).
Cliques also provide an intuitive approach for detecting cohesion in social networks (Festinger 1949, Luce and Perry 1949). A social network is a graph whose vertices consist of
a set of actors and whose edges indicate relationships between actors. Cohesive subgroups
are subsets of actors among whom there are relatively strong, direct, intense, frequent, or
positive ties (Wasserman and Faust 1994). These subgroups are interesting because they
facilitate the emergence of consensus among the actors (Friedkin 1984). In other words,
members within a cohesive subgroup tend to exhibit homogeneity.
Despite the intuitive appeal of using cliques to model cohesion in social networks, cliques
are in fact rarely appropriate for analysis of real data because the definition is too strict
(Wasserman and Faust 1994). This motivates the study clique relaxations. Researchers
1
have relaxed a variety of clique properties including familiarity, reachability, and robustness
(Balasundaram et al. 2007). In graph theoretic terms, these properties respectively correspond to vertex degree, path length, and connectivity. This paper focuses on a degree-based
relaxation known as a k-plex (Seidman and Foster 1978).
In order to define k-plexes, consider a graph G = (V, E) and vertex v ∈ V . Define
NG (v) := {u ∈ V : uv ∈ E}, degG (v) := |NG (v)|, ∆(G) := maxv∈V degG (v), and δ(G) :=
minv∈V degG (v). Let G[K] denote the subgraph induced by K ⊆ V . Define Ḡ = (V, Ē) to
be the complement of G, where e ∈ Ē ⇔ e $∈ E. In this paper, k ≥ 1 is a positive integer.
Definition 1 (Seidman and Foster 1978). K ⊆ V induces a k-plex if δ(G[K]) ≥ |K| − k.
The term k-plex refers to both the set K and the subgraph G[K]. Notice that 1-plexes
are complete subgraphs. Let ωk (G) denote the cardinality of a largest k-plex in G.
Definition 2 (Seidman and Foster 1978). C ⊆ V induces a co-k-plex if ∆(G[C]) ≤ k − 1.
A co-k-plex in G is a k-plex in Ḡ. Notice that co-1-plexes consist of pairwise nonadjacent
vertices, or stable sets.
Seidman and Foster (1978) introduced k-plexes in the context of social networks. Balasundaram et al. (2006) provided an integer programming formulation for the maximum k-plex
problem, derived inequalities for the k-plex polytope, and established the NP-completeness
of the k-plex decision problem. McClosky and Hicks (2007) studied the co-k-plex polytope.
This paper develops combinatorial algorithms for finding maximum k-plexes. Section 2
describes heuristics for finding upper bounds on ωk (G). Section 3 describes a heuristic
for finding lower bounds on ωk (G). Section 4 contains exact k-plex algorithms. Section 5
summarizes and discusses future research.
Computational Results. Sections 2-4 contain computational results obtained by conducting experiments on two classes of graphs. The first class of graphs consists entirely of
DIMACS (2007) instances. The DIMACS graphs are widely considered the standard testbed
for clique algorithms. The second class of graphs consists of real-life social networks. The
COMP-GEOM-t instances are collaboration networks for computational geometers (Beebe
2002, Jones 2002). Here, two authors are adjacent if they have jointly published more than
t articles. The ERDOS-x-y graphs are collaboration networks of all authors with an Erdös
number at most y as of year x (Grossman et al. 1995). Data for both collaboration networks
can be downloaded from the Pajek data website (Batagelj and Mrvar 2006).
2
Table 1: Test Instances
G
brock200-1
brock200-2
brock200-4
brock400-2
brock400-4
brock800-2
brock800-4
c-fat200-1
c-fat200-2
c-fat200-5
c-fat500-1
c-fat500-2
c-fat500-5
c-fat500-10
hamming6-2
hamming6-4
hamming8-2
hamming8-4
hamming10-2
hamming10-4
johnson8-2-4
johnson8-4-4
keller4
MANN-a9
MANN-a27
MANN-a45
p-hat300-1
p-hat300-2
p-hat300-3
p-hat700-1
p-hat700-2
p-hat700-3
san200-0.7-2
san200-0.9-1
COMP-GEOM-0
COMP-GEOM-1
COMP-GEOM-2
ERDOS-97-1
ERDOS-98-1
ERDOS-99-1
ERDOS-97-2
ERDOS-98-2
ERDOS-99-2
|V |
200
200
200
400
400
800
800
200
200
200
500
500
500
500
64
64
256
256
1024
1024
28
70
171
45
378
1035
300
300
300
700
700
700
200
200
7343
7343
7343
472
485
492
5488
5822
6100
|E|
14834
9876
13089
59786
59765
208166
207643
1534
3235
8473
4459
9139
23191
46627
1824
704
31616
20864
518656
434176
210
1855
9435
918
70551
533115
10933
21928
33390
60999
121728
183010
13930
17910
11898
3939
1976
1314
1381
1417
8972
9505
9939
3
ω2 (G)
[25,53]
[13,24]
[19,41]
[27,133]
[27,133]
[23,253]
[23,252]
12
24
58
14
26
64
126
32
6
128
16
[512,530]
[45,153]
5
14
15
26
236
[662,668]
[9,66]
[28,85]
[43,108]
[10,291]
[50,298]
[73,311]
–
–
22
10
8
8
8
8
8
8
8
ω3 (G)
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
–
22
11
10
9
9
9
9
9
9
Table 1 contains information corresponding to the testbed. The columns ω2 (G) and
ω3 (G) show the largest known k-plex in G. This data is due to Balasundaram et al. (2007).
All values not specified within a range are optimal. There appears to be no known values of
ω4 (G) for these graphs. All implementations were tested using a 2.2 GHz Dual-Core AMD
Opteron processor with 3 GB of memory.
2.
Co-k-plex Coloring
This section develops an upper bound on ωk (G) by generalizing the concept of graph coloring.
A coloring of G is a function cm : V (→ {1, ..., m} such that cm (u) $= cm (v) whenever uv ∈ E.
The chromatic number, χ(G), of G is the smallest integer m for which there exists a valid
coloring cm . Notice that cm (u) $= cm (v) for all u, v ∈ K whenever K ⊆ V induces a complete
subgraph. It follows that the chromatic number is an upper bound for ω(G). That is,
ω(G) ≤ χ(G).
The goal is to find a k-plex analogue of this bound. To that end, suppose the co-k-plexes
C1 , ..., Cm partition the vertex set, and let K be a maximum k-plex in G. The sets C1 , ..., Cm
define a co-k-plex coloring of G. Observe that
ωk (G) = |K| =
m
!
i=1
|K ∩ Ci | ≤
m
!
ωk (G[Ci ]),
(1)
i=1
where the inequality holds because k-plexes are closed under set inclusion (Seidman and
Foster 1978). Let Π be the set of all co-k-plex colorings of G and define the graph invariant
!
χk (G) := min{
ωk (G[C]) : C ∈ Π}.
(2)
C∈C
χk (G) is the co-k-plex chromatic number of G. Notice that (1) reduces to ω(G) ≤ χ(G)
when k = 1 and C1 , ..., Cm is an optimal coloring. It follows that χ1 (G) = χ(G). Moreover,
(1) and (2) together imply the bound
ωk (G) ≤ χk (G).
(3)
In practice, determining the exact value of χk (G) can be computationally prohibitive,
so the remainder of this section is devoted to heuristics for approximating χk (G). These
heuristics fall into two categories: integral and fractional. To see the distinction, let I be
the set of all co-k-plexes in G, and let Iv denote the set of co-k-plexes containing v. Define
"
x(A) := a∈A xa . Consider the following dual pair of integer programs:
4
max{x(V ) : x ∈ {0, 1}, x(C) ≤ ωk (G[C]) for all C ∈ I}
!
min{
ωk (G[C])yC : y ∈ {0, 1}, y(Iv ) ≥ 1 for all v ∈ V }.
(4)
(5)
C∈I
The optimal objective values for (4) and (5) are ωk (G) and χk (G), respectively. Moreover,
by strong duality, the optimal objective values for the linear relaxations are equal. It follows
that any feasible solution to the linear relaxation of (5) produces an upper bound on ωk (G).
Integer Co-k-plex Coloring Heuristics (ICCH) find feasible solutions to (5). Fractional Co-kplex Coloring Heuristics (FCCH) find feasible solutions to the linear relaxation of (5). Both
heuristics make use of the following lemmas which provide analytic bounds on ωk (G).
Lemma 1. Every graph G satisfies ωk (G) ≤ ∆(G) + k.
Proof. If there exists a k-plex K in G such that |K| > ∆(G)+k, then degG[K] (v) ≥ |K|−k >
∆(G) for all v ∈ K, a contradiction.
Lemma 2. Given a graph G and an integer m ≥ 1, let am denote the number of vertices
v ∈ V such that degG (v) ≥ m. If j := max{m : am ≥ k + m}, then ωk (G) ≤ k + j.
Proof. Suppose G contains a k-plex K such that |K| ≥ k + j + 1. By definition of k-plex,
degG[K] (v) ≥ |K| − k ≥ j + 1 for all v ∈ K.
In other words, K contains at least k + j + 1 vertices v such that degG[K] (v) ≥ j + 1. It
follows that aj+1 ≥ k + j + 1, contradicting the definition of j.
Lemma 3 (Balasundaram et al. 2006). If G is a co-k-plex, then ωk (G) ≤ 2k − 2 + k mod 2.
Figure 1 shows an ICCH. Line 4 stores the degree of every vertex in Cm . Line 7 uses Lemmas 1, 2, and 3 to bound ωk (G[Ci ]). Lines 3, 4, 6, and 7 can each be accomplished in linear
time using an adjacency matrix. It follows that this ICCH is an O(|V |2 ) algorithm. Table 2
contains computational results obtained by running the function ICCH on the benchmark
graphs with an arbitrary initial vertex ordering.
The FCCH adapts the fractional coloring procedure of Balas and Xue (1996). More precisely, the FCCH constructs a set of co-k-plexes C1 , ..., Ch ⊆ V such that, after p iterations,
5
G
brock200-1
brock200-2
brock200-4
brock400-2
brock400-4
brock800-2
brock800-4
c-fat200-1
c-fat200-2
c-fat200-5
c-fat500-1
c-fat500-2
c-fat500-5
c-fat500-10
hamming6-2
hamming6-4
hamming8-2
hamming8-4
hamming10-2
hamming10-4
johnson8-2-4
johnson8-4-4
keller4
MANN-a9
MANN-a27
MANN-a45
p-hat300-1
p-hat300-2
p-hat300-3
p-hat700-1
p-hat700-2
p-hat700-3
san200-0.7-2
san200-0.9-1
COMP-GEOM-0
COMP-GEOM-1
COMP-GEOM-2
ERDOS971
ERDOS972
ERDOS981
ERDOS982
ERDOS991
ERDOS992
∗
optimal
Table 2: ICCH Results
χ2 (G)
93
55
78
172
168
248
247
18
34
82
19
36
85
172
32∗
8
128∗
32
512∗
128
12
28
54
38
324
833
39
76
129
76
165
267
105
133
29
22
16
20
19
18
20
18
20
seconds
0.0
0.0
0.0
0.1
0.1
0.3
0.3
0.0
0.0
0.0
0.0
0.0
0.1
0.1
0.0
0.0
0.0
0.0
0.7
0.5
0.0
0.0
0.0
0.0
0.1
1.0
0.0
0.0
0.0
0.1
0.2
0.3
0.0
0.0
10
9.8
10
0.1
5.9
0.1
5.8
0.0
6.3
χ3 (G)
151
95
131
285
286
442
440
20
37
89
23
41
98
191
48
12
192
48
768
192
16
42
100
44
369
1032
69
135
210
135
289
451
147
184
48
37
28
32
36
33
36
34
38
6
seconds
0.0
0.0
0.0
0.1
0.1
0.3
0.3
0.0
0.0
0.0
0.0
0.0
0.0
0.1
0.0
0.0
0.0
0.0
0.7
0.5
0.0
0.0
0.0
0.0
0.1
0.8
0.0
0.0
0.0
0.1
0.1
0.2
0.0
0.0
10
10
11
0.1
4.9
0.1
5.7
0.1
6.5
χ4 (G)
169
118
151
332
330
570
557
21
38
90
24
42
99
192
56
16
224
64
896
256
19
48
128
45
378
1035
90
173
245
184
375
556
159
193
50
37
27
31
36
32
37
34
38
seconds
0.0
0.0
0.0
0.1
0.1
0.3
0.3
0.0
0.0
0.0
0.0
0.0
0.0
0.1
0.0
0.0
0.0
0.0
0.7
0.5
0.0
0.0
0.0
0.0
0.1
0.8
0.0
0.0
0.0
0.1
0.2
0.2
0.0
0.0
11
11
10
0.1
5.1
0.1
5.9
0.1
6.6
every vertex belongs to exactly p distinct co-k-plex sets. These co-k-plexes combine to form
a feasible solution to the linear relaxation of (5) as follows:
h
1!
ȳ :=
yC .
p i=1 i
The feasibility of ȳ implies that
h
ωk (G) ≤
1!
ωk (G[Ci ])ȳCi .
p i=1
Figure 2 shows the FCCH. The set C consists of the co-k-plexes C1 , ..., Ch . At every
iteration, each vertex is either added to an existing Ci ∈ C in Line 7 or to a new partition
set in Line 10. By Line 12, every vertex belongs to exactly p partition sets, so tnew is a valid
upper bound on ωk (G). The FCCH termination condition can be inefficient, so the number
of iterations and the number partition sets in C are both required to be O(|V |).
Theorem 1. If the number of iterations and the number of partition sets are O(|V |), then
FCCH can be executed in O(|V |4 ) time.
Proof. At every iteration, for each vertex v ∈ V , the algorithm tests if Ci ∪{v} is a co-k-plex.
This can be done by counting the number of u ∈ NG (v) ∩ Ci , which requires O(|V |) time.
Since there are O(|V |) partition sets, there can be O(|V |2 ) possible pairs (Ci , v). Thus, after
O(|V |) iterations, this step has complexity O(|V |4 ). Lines 2 and 10 execute the O(|V |2 )
ICCH algorithm. Since there are at most O(|V |) iterations, these steps have complexity
O(|V |3 ). All other operations contribute O(|V |2 ) to the complexity. Therefore, the overall
complexity of FCCH is O(|V |4 ).
function ICCH(V )
1. Ci = ∅ for 1 ≤ i ≤ |V |
2. for all u ∈ V
3.
m = min{i : Ci ∪ {u} is a co-k-plex in G}
4.
Cm = Cm ∪ {u}
5. end
6. Compute ji := max{m : am ≥ k + m} for each Ci
" |
7. bound = |V
i=1 min{ 2k − 2 + k mod 2, k + ji , ∆(G[Ci ]) + k, |Ci |}
8. return bound
Figure 1: An Integer Co-k-plex Coloring Heuristic
7
G
brock200-1
brock200-2
brock200-4
brock400-2
brock400-4
brock800-2
brock800-4
c-fat200-1
c-fat200-2
c-fat200-5
c-fat500-1
c-fat500-2
c-fat500-5
c-fat500-10
hamming6-2
hamming6-4
hamming8-2
hamming8-4
hamming10-2
hamming10-4
johnson8-2-4
johnson8-4-4
keller4
MANN-a9
MANN-a27
MANN-a45
p-hat300-1
p-hat300-2
p-hat300-3
p-hat700-1
p-hat700-2
p-hat700-3
san200-0.7-2
san200-0.9-1
COMP-GEOM-0
COMP-GEOM-1
COMP-GEOM-2
ERDOS971
ERDOS972
ERDOS981
ERDOS982
ERDOS991
ERDOS992
∗
optimal
Table 3: FCCH Results
χ2 (G)
82
48
68
151
151
221
223
16
30
76
18
33
82
166
32∗
8
128∗
32
512∗
128
10
28
45
36
321
803
34
71
115
68
146
243
79
127
23
15
12
14
14
12
14
12
14
seconds
0.0
0.0
0.0
0.2
0.2
1.4
2.4
0.0
0.0
0.0
0.1
0.1
0.2
0.4
0.0
0.0
0.0
0.0
1.3
0.7
0.0
0.0
0.0
0.0
0.2
5.9
0.0
0.1
0.2
0.3
0.7
1.4
0.0
0.0
18
17
17
0.0
9.6
0.1
9.9
0.1
11
χ3 (G)
139
89
122
267
265
401
410
20
37
89
23
41
98
191
48
12
192
48
768
192
16
42
88
44
366
1028
62
129
201
123
272
428
140
177
44
28
22
25
28
25
30
25
28
8
seconds
0.1
0.0
0.0
0.2
0.3
1.7
0.8
0.0
0.0
0.0
0.0
0.0
0.1
0.1
0.0
0.0
0.0
0.0
1.3
0.6
0.0
0.0
0.0
0.0
0.1
1.8
0.0
0.0
0.1
0.3
0.3
0.8
0.0
0.1
18
17
17
0.0
8.7
0.1
9.9
0.0
11
χ4 (G)
164
118
151
320
320
535
537
21
38
90
24
42
99
192
56
16
224
64
896
256
18
46
113
45
378
1035
85
161
240
168
349
532
159
191
43
30
23
29
30
28
29
28
30
seconds
0.0
0.0
0.0
0.2
0.1
1.3
1.2
0.0
0.0
0.0
0.0
0.0
0.1
0.2
0.0
0.0
0.1
0.0
2.0
0.6
0.0
0.0
0.0
0.0
0.1
1.1
0.0
0.1
0.1
0.3
0.5
0.6
0.0
0.1
18
16
17
0.1
8.8
0.1
10
0.1
11
function FCCH(V )
1. told = ∞; p = 1
2. tnew =ICCH(V ); store the partition sets in C
3. while tnew < told
4.
U = V ; told = tnew ; p = p + 1
5.
for all v ∈ U
6.
if ∃ Ci ∈ C such that v $∈ Ci and Ci ∪ {v} is a co-k-plex
7.
Ci = Ci ∪ {v}; U = U \ {v}
8.
end
9.
end
10.
ICCH(U ); append new partition sets in C
11.
Compute j"
i := max{m : am ≥ k + m} for each Ci ∈ C
1
12.
tnew = p · Ci ∈C min{ 2k − 2 + k mod 2, k + ji , ∆(G[Ci ]) + k, |Ci |}
13. end
14. return told
Figure 2: A Fractional Co-k-plex Coloring Heuristic
Table 3 contains computational results obtained by running the function FCCH on the
benchmark graphs with an arbitrary initial vertex. The bound on the number of iterations
was set to 5 · |V |.
3.
A k-plex Heuristic
This section describes a heuristic for finding maximum k-plexes. Feasible k-plexes provide
lower bounds on ωk (G). The heuristic indirectly searches for cohesive subgraphs in G and
extends them to maximal k-plexes.
There has been extensive research on heuristics for finding large complete subgraphs
(Busygin et al. 2002, Feo and Resende 2005, Gendreau 1993, Marchiori 2002). A typical
combinatorial heuristic systematically searches a set of neighborhoods in the feasible solution
space for local optima (Hansen et al. 2004). When a local optimum is obtained, it is
compared to the incumbent solution and stored if necessary. The heuristic then continues
searching in other neighborhoods. Obviously, the solution quality heavily depends on both
the choice of neighborhoods and the local search method.
Recall that if IG denotes the set of all complete subgraphs in G, then IG also denotes
the set of all stable sets in Ḡ. The remainder of this section focuses on finding stable sets
in Ḡ which are extended to maximal k-plexes in G. This approach is valid because every
element in IG is extendible to a maximal k-plex in G. Without loss of generality, assume G
9
#
*
)
+
#*
!
(
&
$
"
%
#'
##
#$
Figure 3: H̄ with root s.
is connected. For if not, simply run the heuristic on each component.
For u, v ∈ V , let d(u, v) be the length of a shortest path from u to v in G. The concept of
neighborhood is based on the parity of shortest path lengths from some root node s. Given
a root s ∈ V , define the following sets:
K0 := {v ∈ V | d(s, v) even} and K1 := {v ∈ V | d(s, v) odd}.
For example, consider the search for k-plexes in some graph H, and suppose that H̄ is
shown in Figure 3. The vertex set V (H) partitions into the sets K0 = {s, 5, 6, 7, 8, 12, 13}
and K1 = {1, 2, 3, 4, 9, 10, 11}. For i ∈ {0, 1}, notice that u, v ∈ Ki and uv ∈ E(H̄)
together imply d(u, s) = d(v, s). Otherwise, d(u, s) and d(v, s) would have different parities.
Therefore, for every v ∈ Ki ,
NH̄ (v) ∩ {u ∈ Ki \ {v} : d(u, s) $= d(v, s)} = ∅.
Hopefully, this property causes Ki to contain large stable sets.
Now Ki ∈
/ IH in general, but there will typically exist many subsets Ki" ⊆ Ki such that
Ki" ∈ IH . In order to examine a variety of these subsets, construct elements in IH from Ki
by removing one end of every edge in H̄[Ki ]. To determine which end of edge uv ∈ E(H̄[Ki ])
to remove, always apply exactly one of the following rules:
Rule 1. If degH̄[Ki ] (v) ≤ degH̄[Ki ] (u), remove u. Otherwise, remove v.
Rule 2. If degH̄ (v) ≤ degH̄ (u), remove u. Otherwise, remove v.
Rule 3. Always remove v.
Rule 4. Always remove u.
Let Kij be the subset obtained from Ki be applying Rule j to every edge in E(H̄[Ki ]).
Rules 1 and 2 are greedy metrics. Rules 3 and 4 are included to diversify the search space.
Now extend each set Kij to a maximal k-plex in H. All k-plexes that can be constructed
from a set Ki in this way constitute a neighborhood. Therefore, the search space is essentially
10
function lbound(R)
1. for all s ∈ R
2.
define K0 and K1 with respect to root s
3.
construct sets Kij ⊆ Ki
4.
extend sets Kij to maximal k-plexes in H
5.
for all j and i
6.
kick(Kij )
7.
end
8.
update incumbent I if necessary
9. end
function kick(K)
10. construct set S := {v ∈ V \ K : |NH̄ (v) ∩ K| ≤ 1}
11. let K = K ∪ S
12. construct sets K j ⊆ K
13. extend sets K j to maximal k-plexes in H
14. end
Figure 4: k-plex heuristic.
a function of the root nodes, and specifying a set of neighborhoods is equivalent to specifying
a set of root nodes R. The k-plex heuristic is shown in Figure 4. The incumbent solution I
is initially empty and stored as a global variable.
To find a k-plex in H, arbitrarily choose a set of vertices to define R. Line 2 builds a
breadth-first-search tree in H̄ rooted at s to determine d(v, s) for all v ∈ V . The breadth-
first-search tree is also used to define degH̄[Ki ] (v) for all v. Line 3 applies Rules 1-4, and
Line 4 uses a greedy heuristic. Line 6 passes the sets Kij to the new function kick. Its
purpose is to help the heuristic escape local optima. Line 12 uses Rules 1-4 for each input
set K. Figure 4 is a basic k-plex heuristic. Table 4 contains computational results obtained
by running lbound on the benchmark graphs. LB1 corresponds to choosing an arbitrary
set of
|V |
40
vertices to define R. LB2 corresponds to setting R = V. The heuristics terminate
after an hour time limit.
4.
Exact Algorithms
This section describes exact algorithms for finding maximum k-plexes in a graph G =
(V, E). The first type is based on a standard clique algorithm (Applegate and Johnson 1993,
Carraghan and Pardalos 1990). The second type adapts an algorithm of Östergård (2002).
11
G
brock200-1
brock200-2
brock200-4
brock400-2
brock400-4
brock800-2
brock800-4
c-fat200-1
c-fat200-2
c-fat200-5
c-fat500-1
c-fat500-2
c-fat500-5
c-fat500-10
hamming6-2
hamming6-4
hamming8-2
hamming8-4
hamming10-2
hamming10-4
johnson8-2-4
johnson8-4-4
keller4
MANN-a9
MANN-a27
MANN-a45
p-hat300-1
p-hat300-2
p-hat300-3
p-hat700-1
p-hat700-2
p-hat700-3
san200-0.7-2
san200-0.9-1
COMP-GEOM-0
COMP-GEOM-1
COMP-GEOM-2
ERDOS971
ERDOS972
ERDOS981
ERDOS982
ERDOS991
ERDOS992
∗ optimal
LB1
ω2 (G)
25
12
18
27
33
22
22
12∗
24∗
58∗
14∗
26∗
64∗
126∗
32∗
4
128∗
16∗
512∗
43
4
14
15∗
22
218
646
9
30
42
10
50
70
26
90
16
8
8
8∗
8∗
8∗
8∗
8∗
8∗
Table 4: lbound Results
sec.
1
1
1
2
2
15
15
2
2
2
20
19
16
13
0
0
0
1
3
12
0
0
1
0
2
35
4
3
1
33
19
8
1
0
1439
1489
1529
18
2506
17
2473
18
2103
LB2
ω2 (G)
25
13∗
19
28
33
23
23
12∗
24∗
58∗
14∗
26∗
64∗
126∗
32∗
4
128∗
16∗
512∗
43
4
14
15∗
22
218
646
9
30
43
12
51
71
26
90
22∗
10∗
8∗
8∗
8∗
8∗
8∗
8∗
8∗
sec.
3
5
4
23
23
299
301
10
9
7
225
210
191
146
0
0
2
8
65
281
0
0
3
0
14
859
28
20
10
537
316
140
3
2
3600
3600
3600
3600
3506
3600
3600
3600
3600
LB1
ω3 (G)
27
15
22
31
33
26
25
12∗
24∗
58∗
14∗
26∗
64∗
126∗
32∗
8∗
128∗
16
512
64
8∗
14
18
30
258
762
11
34
49
13
58
82
36
125∗
16
8
8
9∗
9∗
9∗
9∗
9∗
9∗
12
sec.
1
1
1
2
2
15
15
2
2
2
19
18
16
13
0
0
0
1
3
12
0
0
1
0
3
76
4
3
1
33
19
9
1
0
1551
1597
1543
17
2433
17
2698
18
2319
LB2
ω3 (G)
28
15
23
32
33
26
26
12∗
24∗
58∗
14∗
26∗
64∗
126∗
32∗
8∗
128∗
16
512
64
8∗
14
18
30
260
762
11
34
49
14
58
84
36
125∗
22∗
11∗
8
9∗
9∗
9∗
9∗
9∗
9∗
sec.
3
6
4
23
23
298
301
10
9
7
226
210
185
150
0
0
2
8
67
288
0
0
2
0
29
1748
28
19
10
555
320
140
4
2
3600
3600
3600
3600
3600
3600
3600
3600
3600
LB1
ω4 (G)
31
17
24
35
36
29
29
12∗
24∗
58∗
14∗
26∗
64∗
126∗
32
8
128
16
512
63
9∗
14
20
36∗
250
756
12
39
53
16
65
92
48
125
16
8
8
10
10
10
10
10
10
sec.
1
1
1
2
2
15
15
2
2
2
19
19
16
13
0
0
0
1
3
12
0
0
1
0
2
21
4
3
2
32
19
9
1
0
1545
1531
1468
17
2483
17
2632
18
2746
LB2
ω4 (G)
32
17
25
36
37
30
30
12∗
24∗
58∗
14∗
26∗
64∗
126∗
32
8
128
16
512
64
9∗
14
20
36∗
257
756
13
39
55
16
66
95
48
125
22
11
8
10
10
10
10
10
10
sec.
3
5
4
23
23
299
301
10
10
7
226
212
194
152
0
0
2
8
74
297
0
0
2
0
17
540
28
20
11
529
321
141
4
2
3600
3600
3600
3600
3600
3600
3600
3600
3600
function basicClique(U, K)
1. while U $= ∅
2.
if |K| + |U | ≤ max
3.
return
4.
end
5.
U = U \ {v} for some v ∈ U
6.
basicClique(U ∩ NG (v), K ∪ {v})
7. end
8. if |K| > max
9.
max = |K|
10. end
11. return
Figure 5: Basic Clique Algorithm
4.1.
Algorithm Type 1
Consider the standard clique algorithm shown in Figure 5. At any point, the algorithm is
constructing a complete graph K. The candidate set, U ⊆ V \ K, contains all vertices v such
#
that K ∪ {v} is complete. In other words, U := v∈K NG (v). The global variable max stores
the cardinality of the largest clique found. To find a maximum clique, initialize max = 0
and make the function call basicClique(V, ∅).
This clique algorithm generalizes to find maximum k-plexes. The main difference is that
#
the candidate set U for a k-plex K is no longer v∈K NG (v). It is now defined as
U := {v ∈ V \ K : K ∪ {v} is a k-plex}.
Figure 6 shows the basic k-plex algorithm. To find a maximum k-plex, initialize max = 0 and
make the function call basicPlex(V, ∅). Table 5 contains computational results obtained
by running basicPlex on the benchmark graphs with a one hour time limit.
Without Lines 2-4, the clique algorithm examines every clique in G. Recall that G can
contain an exponential, with respect to |V |, number of cliques (Moon and Moser 1965).
Lines 2-4 attempt to avoid enumeration of an exponential number of subgraphs. This is
known as pruning the search tree. Although there may exist graphs which require the
enumeration of an exponential number of cliques, pruning can reduce the runtime.
The basic clique algorithm has many variants (Régin 2003, Sewell 1998, Tomita and
Seki 2003, Wood 1997). Many researchers focus on improving the pruning strategy using
the coloring bound. A coloring heuristic provides an upper bound on ω(G[U ]) and has the
13
G
brock200-1
brock200-2
brock200-4
brock400-2
brock400-4
brock800-2
brock800-4
c-fat200-1
c-fat200-2
c-fat200-5
c-fat500-1
c-fat500-2
c-fat500-5
c-fat500-10
hamming6-2
hamming6-4
hamming8-2
hamming8-4
hamming10-2
hamming10-4
johnson8-2-4
johnson8-4-4
keller4
MANN-a9
MANN-a27
MANN-a45
p-hat300-1
p-hat300-2
p-hat300-3
p-hat700-1
p-hat700-2
p-hat700-3
san200-0.7-2
san200-0.9-1
COMP-GEOM-0
COMP-GEOM-1
COMP-GEOM-2
ERDOS-97-1
ERDOS-97-2
ERDOS-98-1
ERDOS-98-2
ERDOS-99-1
ERDOS-99-2
∗ optimal
ω2 (G)
25
13∗
20
27
27
23
23
12∗
24∗
58∗
14∗
26∗
64
126
32∗
6∗
128
16
512
32
5∗
14∗
15
26∗
234
660
10∗
29
42
13∗
49
69
24
90
21
10
8
8∗
3
8∗
3
8∗
3
seconds
≥3600
166
≥3600
≥3600
≥3600
≥3600
≥3600
0
0
3112
1
1
≥3600
≥3600
506
0
≥3600
≥3600
≥3600
≥3600
0
110
≥3600
66
≥3600
≥3600
14
≥3600
≥3600
1887
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
10
≥3600
9
≥3600
10
≥3600
Table 5: basicPlex Results
BBN
172822699
9759381
193074734
169761253
160979618
134201916
133857528
975
7308
86024721
2712
31068
84968699
39813170
26461612
4709
39716014
237558610
3595516
146893539
1666
11542436
247583422
5585820
79044110
19339018
665249
167764775
145501695
55769755
116066244
105454118
441219398
107493877
8346
754265
647635
1390
737
1424
650
1463
591
ω3 (G)
28
15
22
31
32
25
24
12∗
24∗
58
14∗
26∗
64
126
32
8∗
128
18
512
43
8∗
18
20
36∗
351
990
12∗
35
51
14
58
81
36
125
22
11
10
9∗
3
9∗
3
9∗
3
14
seconds
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
4
5
≥3600
115
126
≥3600
≥3600
≥3600
1
≥3600
≥3600
≥3600
≥3600
0
≥3600
≥3600
2
≥3600
≥3600
1111
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
1122
≥3600
1250
≥3600
1323
≥ 3600
BBN
182056437
199178608
199289654
162264447
146807899
139748190
144247348
58324
115832
104935293
364617
818322
94102915
45937780
244753572
71069
43138327
222683938
3790553
132802297
12837
350491163
207711375
106834
7146812
1022834
40704167
162883168
145528614
110462323
116785628
105553105
395072520
35590748
4672
313457
423436
198148
480
204612
420
211967
376
ω4 (G)
31
17
25
34
36
28
27
12∗
24∗
58
14
26
64
126
37
10∗
128
22
512
64
9∗
22
22
36∗
351
990
14
41
57
16
65
93
48
125
22
12
11
4
4
4
4
4
4
seconds
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
170
226
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
9
≥3600
≥3600
≥3600
≥3600
0
≥3600
≥3600
278
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥ 3600
≥ 3600
≥ 3600
≥ 3600
≥ 3600
≥ 3600
≥ 3600
≥ 3600
≥ 3600
BBN
180633250
192281927
193677120
155153487
145283598
124918468
127834010
2025883
3827925
108945815
6098258
14272751
92359122
45046647
261840105
849851
50079738
230542048
3877853
54961531
104984
342248079
258859895
25470013
10158168
1283088
128042727
154569677
139965092
98797915
117405227
97553599
330336652
39843163
6269
21132
8674
339906
470
303239
410
292341
369
potential to prune a larger portion of the search tree because χ(G[U ]) ≤ |U |. Figure 7 shows
an algorithm which uses co-k-plex coloring to prune the search tree.
Let k-plex1a and k-plex1b denote the functions obtained by using ICCH and FCCH,
respectively, to execute Line 2 of k-plex1. ICCH and FCCH are discussed in Section 2.
To find a maximum k-plex in G, run LB1 to find an initial value for max and make the
function call to k-plex1a(V, ∅) or k-plex1b(V, ∅). Tables 6 and 7 contain computational
results obtained by running these algorithms on the benchmark graphs with a one hour time
limit.
4.2.
Algorithm Type 2
The algorithm in this subsection is based on the following idea of Östergård (2002). Let
V = {v1 , ..., vn } and Si = {vi , ..., vn }. The algorithm in Figure 5 searches for the largest
clique in S1 containing v1 , the largest clique in S2 containing v2 , and so on. Östergård
suggests reversing this order. That is, search Sn for the largest clique containing vn , Sn−1 for
the largest clique containing vn−1 , and so on. Let c(i) be the size of the largest clique in Si .
Clearly, c(n) = 1 and c(1) = ω(G). Moreover, c(i) ∈ {c(i + 1), c(i + 1) + 1} for i = 1, ..., n − 1.
Figure 8 shows Östergård’s clique algorithm. The search order allows for the following
pruning strategy. Given candidate set U, let i = min{j : vj ∈ U }. Notice that U ⊆ Si and
hence ωk (G[U ]) ≤ c(i). This new bound is used in Line 10 in Figure 8.
Östergård’s algorithm adapts to find maximum k-plexes with two modifications. First,
function basicPlex(U, K)
1. while U $= ∅
2.
if |K| + |U | ≤ max
3.
return
4.
end
5.
K = K ∪ {v}; U = U \ {v} for some v ∈ U
6.
U " := {u ∈ U : K ∪ {u} is a k-plex}
7.
basicPlex(U " , K)
8. end
9. if |K| > max
10.
max = |K|
11. end
12. return
Figure 6: Basic k-plex Algorithm
15
Table 6: k-plex1a Results
G
brock200-1
brock200-2
brock200-4
brock400-2
brock400-4
brock800-2
brock800-4
c-fat200-1
c-fat200-2
c-fat200-5
c-fat500-1
c-fat500-2
c-fat500-5
c-fat500-10
hamming6-2
hamming6-4
hamming8-2
hamming8-4
hamming10-2
hamming10-4
johnson8-2-4
johnson8-4-4
keller4
MANN-a9
MANN-a27
MANN-a45
p-hat300-1
p-hat300-2
p-hat300-3
p-hat700-1
p-hat700-2
p-hat700-3
san200-0.7-2
san200-0.9-1
COMP-GEOM-0
COMP-GEOM-1
COMP-GEOM-2
ERDOS-97-1
ERDOS-97-2
ERDOS-98-1
ERDOS-98-2
ERDOS-99-1
ERDOS-99-2
∗ optimal
ω2 (G)
25
13∗
19
27
33
23
23
12∗
24∗
58
14∗
26∗
64
126
32∗
6∗
128∗
16
512∗
43
5∗
14∗
15
26∗
234
660
10∗
30
42
13∗
50
70
26
90
22∗
10∗
8∗
8∗
8
8∗
8
8∗
8
seconds
≥3600
289
≥3600
≥3600
≥3600
≥3600
≥3600
2
3
≥3600
32
33
≥3600
≥3600
0
0
0
≥3600
1
≥3600
0
138
≥3600
123
≥3600
≥3600
33
≥3600
≥3600
3186
≥3600
≥3600
≥3600
≥3600
1218
1205
1220
313
≥ 3600
427
≥ 3600
406
≥ 3600
BBN
86030174
8663613
102282008
95110220
51472394
94857693
96369941
873
5269
19298000
2293
20601
14631858
5104395
0
3668
0
158903409
0
44661342
1585
7755953
147002319
4111457
78820556
18866263
562727
77146967
73906134
46290951
51487369
42921661
247380139
64534714
3301
2366
1211
954
2698
1016
2359
472
2913
ω3 (G)
28
15
22
31
33
26
25
12∗
24∗
58
14∗
26∗
64
126
32
8∗
128
18
512
64
8∗
18
20
36∗
351
990
12∗
35
50
14
58
82
36
125
22
11
8
9
9
9
9
9
9
16
seconds
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
27
28
≥3600
1580
1617
≥3600
≥3600
≥3600
1
≥3600
≥3600
≥3600
≥3600
0
≥3600
≥3600
6
≥3600
≥3600
2827
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥ 3600
≥ 3600
≥ 3600
≥ 3600
≥ 3600
≥ 3600
BBN
95147581
100542983
106907189
88619566
74724985
81559009
90097273
57730
109070
24247513
357420
787649
21078075
7643111
92535097
59533
14422543
149846604
434296
21254123
12378
191111049
128108327
102896
383569
18029
39631513
84162645
70787196
56967864
56760785
46670755
368232192
5514998
18785
110109
14166
5742
1494
6008
1086
6248
943
ω4 (G)
31
17
25
35
36
29
29
12∗
24∗
58
14
26
64
126
37
10∗
128
22
512
64
9∗
22
22
36∗
351
990
14
41
57
16
65
92
48
125
22
11
8
10
10
10
10
10
10
seconds
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
922
909
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
15
≥3600
≥3600
≥3600
≥3600
1
≥3600
≥3600
739
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥ 3600
≥ 3600
≥ 3600
≥ 3600
≥ 3600
≥ 3600
BBN
97664910
90143057
98966956
81344425
80938337
71154628
72735746
1960167
3385277
28799458
545917
1224327
24930408
9068983
119263227
701425
20007766
157055208
456014
33084666
104804
172931195
169102280
25470013
781334
53068
54618899
84779804
70408792
43437614
61159187
48056462
301494773
6899702
173911
122167
6208
5571
1685
5876
1771
5135
1544
Table 7: k-plex1b Results
G
brock200-1
brock200-2
brock200-4
brock400-2
brock400-4
brock800-2
brock800-4
c-fat200-1
c-fat200-2
c-fat200-5
c-fat500-1
c-fat500-2
c-fat500-5
c-fat500-10
hamming6-2
hamming6-4
hamming8-2
hamming8-4
hamming10-2
hamming10-4
johnson8-2-4
johnson8-4-4
keller4
MANN-a9
MANN-a27
MANN-a45
p-hat300-1
p-hat300-2
p-hat300-3
p-hat700-1
p-hat700-2
p-hat700-3
san200-0.7-2
san200-0.9-1
COMP-GEOM-0
COMP-GEOM-1
COMP-GEOM-2
ERDOS-97-1
ERDOS-97-2
ERDOS-98-1
ERDOS-98-2
ERDOS-99-1
ERDOS-99-2
∗ optimal
ω2 (G)
25
13∗
19
27
33
22
22
12∗
24∗
58∗
14∗
26∗
64∗
126
32∗
6∗
128∗
16
512∗
43
5∗
14∗
15
26∗
234
660
10∗
30
42
12
50
70
26
90
22∗
10∗
8∗
8∗
8
8∗
8
8∗
8
seconds
≥3600
1778
≥3600
≥3600
≥3600
≥3600
≥3600
3
3
836
34
39
1579
≥3600
0
0
0
≥3600
1
≥3600
0
475
≥3600
395
≥3600
≥3600
139
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
2202
2075
2731
370
≥ 3600
414
≥ 3600
438
≥ 3600
BBN
10054924
7722362
12056663
3298972
2879018
847946
846155
802
2050
444241
2060
10177
557081
327032
0
3380
0
9945892
0
323816
1585
6389736
18263136
3240597
3053292
28446
500766
7335988
5279160
2617164
1176955
750652
12473403
9748246
2797
2013
1193
869
574
953
449
989
592
ω3 (G)
28
15
22
31
33
26
25
12∗
24∗
58
14∗
26∗
64
126
32
8∗
128
18
512
64
8∗
18
20
36∗
351
990
12
35
50
13
58
82
36
125
22
11
8
9
9
9
9
9
9
17
seconds
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
41
60
≥3600
1796
2368
≥3600
≥3600
≥3600
3
≥3600
≥3600
≥3600
≥3600
0
≥3600
≥3600
15
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥ 3600
≥ 3600
≥ 3600
≥ 3600
≥ 3600
≥ 3600
BBN
9610060
15728716
11842251
2506819
2769768
687075
708185
57612
96754
3960477
356744
754350
2800459
974543
6160422
58663
636791
9748498
7508
370717
12337
48544486
17714412
100969
78758
7035
12816637
6439794
5693325
2557821
1407568
1007772
16047112
970390
15368
10073
4102
4928
702
4497
492
4379
633
ω4 (G)
31
17
24
35
36
29
29
12∗
24
58
14
26
64
126
36
10∗
128
18
512
64
9∗
22
22
36∗
351
990
14
40
57
16
65
92
48
125
22
11
8
10
10
10
10
10
10
seconds
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
1397
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
33
≥3600
≥3600
≥3600
≥3600
2
≥3600
≥3600
1733
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥ 3600
≥ 3600
≥ 3600
≥ 3600
≥ 3600
≥ 3600
BBN
11595841
15883075
13308843
3084792
3339363
719475
771036
1958425
1586893
5319030
427000
797974
3453142
1193712
44987310
693982
2835717
10199713
29602
236250
104679
52307238
20281149
25470013
253074
20406
12577276
8083792
5957872
2516929
1459008
1001573
16009322
1705032
71325
20237
6028
4961
671
4506
487
4376
615
function k-plex1(U, K)
1. while U $= ∅
2.
Compute χ̃k (G[U ]) ≥ χk (G[U ])
3.
if |K| + χ̃k (G[U ]) ≤ max
4.
return
5.
end
6.
K = K ∪ {v}; U = U \ {v} for some v ∈ U
7.
U " := {u ∈ U : K ∪ {u} is a k-plex}
8.
k-plex1(U " , K)
9. end
10. if |K| > max
11.
max = |K|
12. end
13. return
Figure 7: k-plex Algorithm
define ck (i) = ωk (G[Si ]). Second, the candidate set with respect to K is defined as
U := {v ∈ V \ K : K ∪ {v} is a k-plex}.
Figure 9 shows k-plex2. Table 8 contains computational results obtained by running kplex2 on the benchmark graphs with a one hour time limit.
5.
Conclusions and Future Work
This paper describes combinatorial algorithms for finding maximum k-plexes in a graph.
Section 2 focuses on co-k-plex coloring heuristics which are used as an upper bound on the
k-plex number. Section 3 discusses a heuristic for finding maximum k-plexes. This heuristic
provides a lower bound on the k-plex number. Section 4 describes exact algorithms for finding
maximum k-plexes. Table 9 summarizes the number of instances solved to optimality by
each exact algorithm.
The first three exact algorithms perform similarly within the hour time limit. Although
this type of algorithm appears to benefit from the upper and lower bound heuristics, they
solve a relatively small number of instances to optimality. This suggests that the coloring
heuristics might not produce tight upper bounds, so an interesting avenue for future work
would be to develop stronger coloring heuristics.
18
function OsterClique(U, K)
1. if |U | = 0
2.
if |K| > max
3.
max = |K|
4.
found=true
5.
end
6.
return
7. end
8. while U $= ∅
9.
if |K| + |U | ≤ max
10.
return
11.
end
12.
i = min{j : vj ∈ U }
13.
if |K| + c(i) ≤ max
14.
return
15.
end
16.
U = U \ {vi }
17.
OsterClique(U ∩ NG (vi ), K ∪ {vi })
18.
if found=true
19.
return
20.
end
21. end
22. return
function findClique
23. max = 0
24. for i = n down to 1
25.
f ound = f alse
26.
OsterClique(Si ∩ NG (vi ), {vi })
27. end
28. c(i) = max
29. return
Figure 8: Östergård’s Clique Algorithm
19
function OsterPlex(U, K)
1. if |U | = 0
2.
if |K| > max
3.
max = |K|
4.
found=true
5.
end
6.
return
7. end
8. while U $= ∅
9.
if |K| + |U | ≤ max
10.
return
11.
end
12.
i = min{j : vj ∈ U }
13.
if |K| + ck (i) ≤ max
14.
return
15.
end
16.
K = K ∪ {vi }; U = U \ {vi }
17.
U " := {u ∈ U : K ∪ {u} is a k-plex}.
18.
OsterPlex(U " , K)
19.
if found=true
20.
return
21.
end
22. end
23. return
function k-plex2
24. max = 0
25. for i = n down to 1
26.
f ound = f alse
27.
OsterPlex(Si , {vi })
28. end
29. ck (i) = max
30. return
Figure 9: Östergård’s Algorithm Adapted for k-plexes
A natural approach for improving the co-k-plex coloring heuristics would be to generalize
the DSATUR graph coloring heuristic (Brélaz’s 1979). The DSATUR heuristic dynamically
determines the order in which vertices are colored. More precisely, it maintains the number
of distinct colors adjacent to each uncolored vertex and always colors the vertex with the
highest number of adjacent color classes. The number of adjacent colors classes defines a
vertex’s saturation degree. This idea generalizes to co-k-plex coloring. The saturation degree
of v is redefined as the number of distinct partition sets Ci such that Ci ∪ {v} is not a
20
Table 8: k-plex2 Results
G
brock200-1
brock200-2
brock200-4
brock400-2
brock400-4
brock800-2
brock800-4
c-fat200-1
c-fat200-2
c-fat200-5
c-fat500-1
c-fat500-2
c-fat500-5
c-fat500-10
hamming6-2
hamming6-4
hamming8-2
hamming8-4
hamming10-2
hamming10-4
johnson8-2-4
johnson8-4-4
keller4
MANN-a9
MANN-a27
MANN-a45
p-hat300-1
p-hat300-2
p-hat300-3
p-hat700-1
p-hat700-2
p-hat700-3
san200-0.7-2
san200-0.9-1
COMP-GEOM-0
COMP-GEOM-1
COMP-GEOM-2
ERDOS-97-1
ERDOS-97-2
ERDOS-98-1
ERDOS-98-2
ERDOS-99-1
ERDOS-99-2
∗ optimal
ω2 (G)
23
13∗
19
23
23
21
20
12∗
24∗
58∗
14∗
26∗
64∗
126∗
32∗
6∗
128∗
16∗
512∗
41
5∗
14∗
15∗
26∗
235
661
10∗
29
34
13∗
37
51
24
90
22∗
10∗
8∗
8∗
8∗
8∗
8∗
8∗
8∗
seconds
≥3600
74
≥3600
≥3600
≥3600
≥3600
≥3600
0
0
0
0
0
0
0
0
0
0
47
33
≥3600
0
0
1000
0
≥3600
≥3600
6
≥3600
≥3600
667
≥3600
≥3600
≥3600
≥3600
397
1118
1145
0
1253
0
1514
0
1757
BBN
983266826
19636408
917681365
996101972
989222371
926436143
919220732
3677
1895
760
19733
10081
4195
2141
396
4526
7448
7982728
147919
595181620
2621
40896
284120617
53102
33275514
3326395
1561119
704387512
850772303
90760266
655931022
820473217
1079696544
485362621
2150878
4875730
5034335
25770
12602187
27246
14265207
28035
15748963
ω3 (G)
24
15
20
27
22
20
22
12∗
24∗
58∗
14∗
26∗
64∗
126∗
32∗
8∗
128∗
17
448
46
8∗
18∗
21
36∗
351
990
12∗
29
37
13
39
43
36
125∗
5
4
4
9∗
3
9∗
3
9∗
3
seconds
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
0
0
0
8
2
0
0
1
0
2744
≥3600
≥3600
≥3600
0
39
≥3600
1
≥3600
≥3600
552
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
253
≥3600
≥3600
≥3600
19
≥3600
20
≥3600
21
≥3600
BBN
1061715139
973470713
1078546865
1112675498
1132833870
1077097050
1044131631
123687
30227
6566
1183127
266957
52676
18391
195054
37113
182864232
1053193592
208033053
1058346589
10472
14014988
909167108
375502
23112644
2840051
128854637
909116307
956121775
687249461
804657901
991432587
940350668
64534321
156104820
172334737
162641029
2391684
216016439
2552046
209965048
2632545
205078469
ω4 (G)
27
17
21
29
30
23
24
12∗
24∗
58∗
14∗
26∗
64∗
126∗
40∗
10∗
112
19
430
51
9∗
21
16
36∗
351
990
13
33
36
13
42
44
48
50
4
4
4
11∗
4
11∗
4
11∗
4
seconds
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
19
2
0
1416
91
6
2
872
1
≥3600
≥3600
≥3600
≥3600
0
≥3600
≥3600
129
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
≥3600
1897
≥3600
1675
≥3600
1783
≥3600
BBN
1090413176
1118582507
1158926695
1152442843
1159027166
1142792112
1109841414
4452622
759423
88301
134845416
12812559
956345
221427
226208834
395729
808066329
1045849871
799045131
1042679457
151028
1160638581
1183954679
36511315
93317993
16526869
1001501766
987023800
1050444550
964257683
937295855
1063865458
862621323
1006342152
265372393
267076112
266562165
256331821
331758265
212106816
318616472
223030325
308498817
co-k-plex. This general version of DSATUR was implemented and tested (McClosky 2008).
However, the results did not improve upon ICCH and FCCH.
The final exact algorithm, k-plex2, dominates the Type 1 algorithms with respect to
number of instances solved to optimality. Moreover, k-plex2 converges quickly, when it
converges at all. However, the final solution can be far from optimal when k-plex2 fails
to converge. The algorithm demonstrates this phenomenon on the collaboration networks
for k = 3, 4. The reason for this behavior is that the algorithm can spend much of its time
optimizing over a subset of V . Consequently, if vertices at the end of the vertex ordering are
needed to make large k-plexes, good solutions cannot be found until the entire vertex set is
21
Table 9: Results Summary
Algorithm
basicPlex
k-plex1a
k-plex1b
k-plex2
k=2
16
20
21
28
k=3
11
8
7
18
k=4
5
5
4
14
Total
32
33
32
60
processed. This illustrates the importance of vertex ordering for this type of algorithm.
Type 1 algorithms spend time at each branch and bound node to approximate χk (G[U ])
for the candidate set U . Unfortunately, χk (G) could be an inaccurate bound on ωk (G) in
general. The k-plex2 algorithm spends no time estimating χk (G) but benefits from the
bound obtained using the ck array. In any case, the purpose of these bounds is to prune the
candidate set. The requirement for membership in the candidate set becomes less stringent
as k increases, so the set becomes harder to prune. This contributes to the increase in
runtime as k grows.
When comparing these results to those found in Balasundaram et al. (2006), the combinatorial algorithms outperform branch-and-cut on the DIMACS graphs. On the other hand,
branch-and-cut works better on the larger social network graphs. This suggests that combinatorial methods are superior for graphs on a few hundred vertices, but branch-and-cut
becomes the preferred method as the size of the graphs grow.
Another area for future research is an exact co-k-plex coloring algorithm. The co-k-plex
chromatic number is unknown for most of the benchmark graphs. Therefore, it is difficult
to evaluate the performance of the co-k-plex coloring heuristics in Section 2. It would also
be beneficial to study additional heuristics for both the upper and lower bounds on ωk (G).
Acknowledgments
This research was partially supported by NSF grants DMI-0521209 and DMS-0729251. Svyatoslav Trukhanov has independently generalized Östergård’s clique algorithm to find maximum k-plexes (Trukhanov 2008).
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24