A Branch and Price Algorithm for Coalition Structure

Generation over Graphs

Emanuel Florentin Olariu

, Cristian Fr

asinaru

and Policiuc Abel Albert

Department of Computer Science, ”Alexandru Ioan Cuza” University, Bd. Carol, Ias¸i, Romania

Keywords:

Coalition Structure Generation, Linear/Quadratic Programming, Column Generation, Branch and Price.

Abstract:

This paper presents an integer linear programming approach for the coalition structure generation (CSG)

problem over graphs. Forming such structures is a major problem in areas like artiﬁcial intelligence an multi-

agent systems. The problem asks to partition a given set of agents into coalitions in order to maximize their

social well-fare - the agents being vertices in a given graph and their communication links being the edges.

We give a truncated branch and price algorithm using valuation functions for which this problem is proven

to be computationally hard. We consider three cases: ﬁrst when the value of a coalition is the sum of the

weights of its edges, second when the value takes account of both inter- and intra-coalitional disagreements

and agreements, respectively, and another one when the value takes account of the pairs of adjacent agents

which have common neighbors outside. The experimental results cover sets of up to ﬁfty agents. Our approach

prove that an off the shelf optimization solver can be used to solve CSG problem over graphs for some of the

most used valuation functions. We prove also that for the coordination valuation the corresponding decision

problem is NP-complete when the number of coalitions must be two.

1 INTRODUCTION

Coalition formation is one of the main approaches in

multi-agent systems when cooperation among agents

is required, especially if the agents have different ob-

jectives and collaboration skills. Coalition structure

generation (CSG) is one of the steps in the coalition

formation process (Sandholm et al., 2009) which in-

cludes also optimizing the coalitions performance and

rewarding the coalitions value among the members.

Coalition structure generation is a major problem

in artiﬁcial intelligence (Voice et al., 2012a), multi-

agent systems (Bistaffa et al., 2014), (Voice et al.,

2012a) communication networks, cooperative game

theory (Deng and Papadimitriou, 1994), (Flammini

et al., 2018), (Ueda et al., 2018), scheduling (Hoff-

man and Padberg, 1993), combinatorial auctions etc.

Given a set of agents V = {1,2,.. .,n}, and a valu-

ation function v : 2

→ R assigning a value to any

coalition of agents, the problem is to partition the set

of agents into pairwise disjoint classes (coalitions)

such that the sum of their values is maximized.

https://orcid.org/0000-0001-5218-0312

https://orcid.org/0000-0002-5246-7396

https://orcid.org/0000-0002-7111-0026

CSG comes from real-world applications: con-

sider a set of agents who can cooperate by working

in groups. Some of them work better together while

others ﬁnd difﬁcult to cooperate. The problem is to

maximize the so called social well-fare or the total

value of the designated coalitions. Cooperative game

theory usually uses a super-additive valuation func-

tion that values better a merged coalition than the sum

of the values of the component coalitions; this leads

to the great coalition formation which can be costly

to coordinate or manipulate. Besides this there are

natural constraints on possible coalitions, hence the

abstractions like super-additivity are not always ap-

propriate for modeling the coalitions values and the

agents should be divided into smaller coalitions.

The problem is computationally difﬁcult: the in-

put speciﬁcation for all 2

|V |

possible coalitions is in-

tractable even for reasonable values of |V | and the

computational difﬁculty maintains even under quite

restrictive assumptions; hence implicit representa-

tions of the valuation function are not practical. The

literature presents a very large number of approaches

to CSG problem like (Rahwan et al., 2015): dy-

namic programming, meta heuristic methods, branch

and bound algorithms based on dividing the searching

space, anytime algorithms that maintain a monotoni-

390

Olariu, E., Fr

asinaru, C. and Albert, P.

A Branch and Price Algorithm for Coalition Structure Generation over Graphs.

DOI: 10.5220/0010316303900397

In Proceedings of the 13th International Conference on Agents and Artiﬁcial Intelligence (ICAART 2021) - Volume 1, pages 390-397

ISBN: 978-989-758-484-8

cally improving feasible solution etc.

The classic CSG framework assumes no particular

structure of the set of agents, while in many real world

applications the agents are the nodes of a network.

This is the background of the following version of the

CSG problem: to generate a coalition structure when

the agents are placed in the vertices of a given graph

and the edges represent the communication links be-

tween them; each such link having a capacity that in-

dicates the probability that the link works, the amount

of information that can be changed along it, or the

level of trust between the two agents. In such a con-

text the value of a group of nodes must depend on the

capacities of the edges between them and the group

members must share a certain level of connectedness

in order to ensure the spread of the information.

In this paper we provide a truncated branch and

price algorithm which proved to be effective for some

of the most used coalition valuations to form coali-

tions among agents embedded in a graph. Our algo-

rithm is independent of the structure of the underlying

graph and works for up to 50 agents. Our setting sug-

gests that we can deal with CSG over graphs as long

as the value of a coalition can be expressed as a poly-

nomial of the characteristic vector of the coalition.

The remaining of the paper is organized as fol-

lows: Section 2 describes our setting and the related

work, Section 3 describes the LP model and the col-

umn generation method, Section 4 introduces the val-

uation functions, the corresponding sub-problems and

their equivalent formulations, Section 5 describes the

branch and price algorithm, and the last Sections con-

tain the numerical results and the conclusions.

2 BACKGROUND

Various approaches were employed for this problem

ranging from constraint optimization to integer lin-

ear programming (ILP). In (Voice et al., 2012a) a

dynamic programming (DP) algorithm is proposed

for valuations having the independent disconnected

members property, other DP approaches can be found

in (Rahwan et al., 2009a) and (Michalak et al., 2016).

Mixed integer linear programming (MILP) tech-

nique is used in (Ueda et al., 2018) and (Tran–Thanh

et al., 2013). In (Tran–Thanh et al., 2013) a coali-

tional skill vector model is introduced and the value

of a coalition is measured as a function of the distance

between the corresponding coalition’s skill vector and

a set of goals, while (Ueda et al., 2018) introduced a

mixed integer programming formulation based on a

set of rules. Other approaches are based on concise

representations of the searching space and include

branch and bound algorithms (Bistaffa et al., 2014),

(Rahwan et al., 2009b), and constraint optimization

methods (Ueda et al., 2018). Sometimes in order to

reduce the execution time of the algorithm a truncat-

ing techniques are employed like in the case of any-

time algorithms that can offer at any moment a sub-

optimal solution to the problem (Bistaffa et al., 2014),

(Rahwan et al., 2009b), (Sandholm et al., 2009).

The CSG problem over graphs has been also stud-

ied for graphs with a particular structure, e. g. syner-

gistic graphs (Voice et al., 2012b) and (Bistaffa et al.,

2014), graphs with bounded tree-width in (Voice

et al., 2012a)). To a limited extent the only related

algorithms to ours are the MIlP formulations (Ueda

et al., 2018) which require that the valuation function

to have a rules-based representation or (Tran–Thanh

et al., 2013) which uses a vector-based representation

that assigns a skill vector to each agent, and a coali-

tion’s skill level will be expressed as an aggregation of

the composing skill vectors. Hence we cannot com-

pare our approach against these two approaches.

Our setting is that of a (mixed) integer lin-

ear/quadratic programming and is based on the clas-

sical model of the set partitioning problem in a graph.

Given a graph G = (V,E) the vertices are agents

and the edges represent connections between agents,

therefore we require that the coalitions induce con-

nected sub-graphs.

We study three types of valuations: the edge sum,

the correlation, and the coordination functions. The

edge sum valuation of a coalition (Deng and Papadim-

itriou, 1994) is the sum of the weights of its edges,

this function frequently occurs in communication net-

works and cooperative game theory. The correla-

tion valuation function (Bansal et al., 2004) occurs in

the clustering framework and it takes account of the

agreements from inside and the disagreements from

outside the structure - an edge being labeled with a

plus or a minus depending on whether the involved

agents are similar or different. The coordination valu-

ation accounts for all 3-cliques of three agents, two in-

side the coalition and one outside; this valuation aims

to privileges coalitions having agents with common

outside neighbors.

Our approach is based on solving an ILP prob-

lem, which is equivalent with CSG, using a branch

and bound algorithm in which the problems in nodes

are built by means of column generation method.

This approach works as long as the involved sub-

problem can be solved. We proved that this can

be done for edge sum, correlation, and coordina-

tion valuation functions since the sub-problem be-

comes a quadratic knapsack problem or a mixed inte-

ger quadratically constrained programming problem

A Branch and Price Algorithm for Coalition Structure Generation over Graphs

391

with forbidden conﬁgurations. These problems can

be solved in practice using the corresponding mixed

integer quadratic programming model or the MILP

equivalent model.

Our numerical results show that we can approach

by this method sets of up to 45 agents depending on

the magnitude of the weight function for edge sum

valuation function, up to 50 agents for the correlation

valuation function, and up to 35 agents for the coor-

dination valuation function using just a regular PC.

3 LP MODEL AND COLUMN

GENERATION

Consider V = {1, 2,.. .,n} to be a set of n agents and

v : 2

→ R a valuation function on the power set of V .

A coalition structure is a collection of pairwise dis-

joint exhaustive subsets (coalitions) V

,.. .,V

, i.e.

∩V

= ∅ for all 1 ≤ i < j ≤ p and

[

i=1

= V . The

problem of ﬁnding a coalition structure of maximum

value is equivalent with the set partitioning problem

(SPP) (Hoffman and Padberg, 1993):

max

−1

∑

j=1

v(C

(1)

−1

∑

j=1

i j

= 1,∀i ∈ V (2)

∈ {0,1},∀ j = 1,2

− 1 (3)

where {C

,.. .,C

−1

} is an enumeration of 2

\ {∅}

and (a

i j

)

i∈V

is the characteristic vector of C

, that is

i j



1, if i ∈ C

0, otherwise

, for each j = 1,2

− 1.

We relax (3) by replacing them with

≥ 0,∀ j ∈ {1,2,. .., 2

− 1}

(3’)

We write the relaxed problem as a minimum one by

replacing (1) by

min

−1

∑

j=1

−v(C

(1’)

The dual of the problem (1’), (2) , (3’) is

max

∑

j=1

(4)

∑

i=1

i j

≤ −v(C

),∀ j = 1,2

− 1 (5)

∈ R,∀i ∈ V (6)

We will use the column generation method (Lubbecke

and Desrosiers, 2005) for dealing with the huge num-

ber of variables in the primal problem. Let C

= { j},

for any j ∈ {1,2,. .., n}. An initial feasible basic solu-

tion to the problem (1’), (2), (3’) could be that corre-

sponding to the coalition structure ({1},{2},... {n}),

i. e., x

,.. .,x

. We restrict the problem to a

smaller number of variables including x

,.. .,x

and we get the restricted master problem (RMP):

min

∑

j∈J

−v(C

(7)

∑

j∈J

i j

= 1,∀i ∈ V (8)

≥ 0,∀ j ∈ J (9)

where J =

{

1,2,. .., m

}

∪ J

⊆ {1,2,. .., 2

− 1}.

The dual of the RMP is

max

∑

i=1

(10)

∑

i=1

i j

≤ −v(C

),∀ j ∈ J (11)

∈ R,∀1 ≤ i ≤ n (12)

Let (x,π) be an optimum primal-dual solution for

this pair of problems; we look for a non-basic vari-

able (column) with the minimum negative reduced

cost (Dantzig rule) - by solving a corresponding sub-

problem - that would be added to the current restricted

master problem. When such a variable does not exist

we can stop: we have an optimum solution to the pri-

mal problem. The sub-problem is to ﬁnd

= argmax

j∈{0,1,...,2

−1}\J

v(C

) +

∑

i=1

i j

(13)

The arising question is: how can we solve problem

(13)? The answer depends on the form of the valua-

tion function v(·). In the following sections we will

analyze this sub-problem for two of the most frequent

used valuation functions deﬁned for coalition struc-

ture over graphs: the edge-sum and the correlation,

and for another one the coordination valuation.

4 COALITION STRUCTURE

GENERATION OVER GRAPHS

4.1 The Edge-Sum Valuation

Let G = (V,E) be graph and w : E → R a weight func-

tion on its edges. The corresponding edge-sum coali-

tion valuation function is

v : P (V ) → R, v(C) =

∑

i j∈E:i, j∈C

i j

,∀C ⊆ V.

ICAART 2021 - 13th International Conference on Agents and Artiﬁcial Intelligence

392

This function was extensively studied in the context

of cooperative game theory (Deng and Papadimitriou,

1994) and the corresponding CSG problem is NP-

complete even for planar graphs (Voice et al., 2012a).

If v is the characteristic vector of the generic coali-

tion C, then the sub-problem becomes:

max

v∈{0,1}

,v6=χ

,∀ j∈J

∑

i j∈E

i j

∑

i∈V

> 0

(14)

That is, the sub-problem is a quadratic knapsack prob-

lem with forbidden conﬁgurations (QKPf). We can

ﬁnd a new column to add to the restricted master prob-

lem if and only if this quadratic knapsack problem has

a strictly positive optimal objective value.

QKPf could be a computationally difﬁcult prob-

lem since the quadratic knapsack problem (QKP)

is known to be NP-hard (being a generalization of

Clique problem). By rephrasing it we get:

max

∑

i j∈E

i j

∑

i∈V

(15)

∑

i/∈C

∑

i∈C

(1 − v

) ≥ 1,∀ j ∈ J (16)

∈ {0,1},∀i ∈ V (17)

We cast the constraints (16) in order to avoid the al-

ready added coalitions/variables to the original prob-

lem and the already revealed unconnected coalitions

(J is the set of indexes of these variables).

The problem (15) - (17) has a quadratic objective

but only linear constraints: it is an Integer Quadratic

Programming (IQP) problem. Such a problem can be

solved by using a mathematical optimization solver as

such or by linearizing it ﬁrst in order to transform it

into a MILP problem. We note here that the known

methods used for solving such a problem are not ap-

plicable here since the usual requirements are to have

non-negative integer or positive coefﬁcients for the

quadratic (some times non-diagonal) terms (Caprara

et al., 1999), (Gallo et al., 1980), (Pisinger, 2007).

Hence we had to settle to solve it as a general IQP

problem using a mathematical optimization solver.

4.2 The Correlation Valuation

Suppose we have a function e : E → {+,−} that as-

signs to each edge i j ∈ E a label + or − depending

on whether agents i and j are similar or different (this

function arises in clustering frameworks). A valuation

function that takes account of both inter- and intra-

coalitional similarities can be deﬁned in the following

way (Bansal et al., 2004): we want to minimize the

number of mistakes: a positive mistake occurs when

e(i j) = −, with i and j belonging to the same coali-

tion, a negative one occurs when e(i j) = +, but i and

j belong to different coalitions. Or, equivalently, we

maximize the number of agreements; ﬁrst deﬁne, for

a coalition C ⊆ V , the intra- and inter-coalitional con-

nections

Intra

Inter

−

The correlation valuation function is

v : 2

→ R, v(C) = Intra

(C)+Inter

−

(C)/2, ∀C ⊆V.

It was known that the corresponding CSG problem

is NP-complete (Bansal et al., 2004). We deﬁne two

weight functions on the set of edges of G, w

−

E → {0,1}, by

(i j) = w

i j



1, if e(i j) = +

0, if e(i j) = −

−

(i j) = w

−

i j



1, if e(i j) = −

0, if e(i j) = +

,∀i j ∈ E.

Let v be the characteristic vector of the generic coali-

tion C, then

v(C) =

∑

i∈V

∑

j∈V

i j

∑

i∈V

∑

j∈V

−

i j

(1 − v

The sub-problem (13) becomes the following QKPf

problem:

max

∑

i∈V

∑

j∈V

i j

∑

i∈V

∑

j∈V

−

i j

(1 − v

) +

∑

i∈V

∑

i/∈C

∑

i∈C

(1 − v

) ≥ 1, ∀ j ∈ J

∈ {0,1},∀i ∈ V

4.3 The Coordination Valuation

Let G = (V,E) be graph and C ⊆ V be a coalition, we

deﬁne

The coordination coalition valuation function (Voice

et al., 2012a) is

v : 2

→ R,v(C) =

∑

i∈C

(C), ∀C ⊆ V.

This function accounts for all cliques of three agents,

two of them being inside the coalition, while the third

A Branch and Price Algorithm for Coalition Structure Generation over Graphs

393

is outside. It is designed to offer coalitions that in-

clude agents that have common neighbors from out-

side. We will generalize this deﬁnition by includ-

ing the weight of the 3-clique as the product of the

weights on its edges. Hence, we consider ﬁrst w :

E → R

to be a weight on the edges of G. Second, if

v = (v

,.. .,v

) is the characteristic vector for the

generic coalition C, then its coordination value is

∑

i=1

∑

j=1

∑

k=1

i j

(1 − v

We get the original coordination valuation function by

taking w to be the edge characteristic function (i. e.,

i j

= 1, if i j ∈ E, and 0 otherwise).

We note here that the generalization may have the

following interpretation: consider a graph that have

probabilities on its edges representing the reliability

of the corresponding communication links - suppos-

ing that the edges works independently. By multiply-

ing these probabilities for the edges of a 3-clique one

gets the probability that the members of the group can

communicate to each other.

The sub-problem (13) becomes:

max

∑

i=1

∑

j=1

∑

k=1

i j

(1 − v

) +

∑

i∈V

∑

i/∈C

∑

i∈C

(1 − v

) ≥ 1,∀ j ∈ J

∈ {0, 1}, ∀i ∈ V

Using the method of Glover and Wolsey (1974, 1975),

we will transfom the cubic objective function into a

linear one; by introducing the variables

= v

∑

j=1

∑

k=1

i j

(1 − v

)

,∀i ∈ V.

the above problem becomes

max

∑

i=1

∑

i=1

∑

i/∈C

∑

i∈C

(1 − v

) ≥ 1, ∀ j ∈ J,

∑

j=1

∑

k=1

i j

(1 − v

)

+ β

− z

≤ β

,∀i ∈ V

∑

j=1

∑

k=1

i j

(1 − v

)

− z

≥ 0,∀i ∈ V

− z

≥ 0,∀i ∈ V

∈ {0,1},∀i ∈ V

∈ R,∀i ∈ V

where β

∑

j=1





∑

k:w

i j





5 BRANCH AND PRICE

ALGORITHM

In this section we give the details of our branch and

price algorithm. At each node of the branching tree

we ﬁrst build the current LP relaxation in two steps:

(1) we reduce the variables number by taking account

of the branching variables along the path to the root,

and (2) we add the necessary variables to the parent

node LP relaxation by the means of column genera-

tion method.

Step (1) is implemented by effectively ﬁxing to 0

the variables x

such that C

∩U 6= ∅ (or, equivalently,

∩C

6= ∅, for some branching variable x

set to 1),

and removing the branching variables x

set to 0. The

ﬁxing procedure can be achieved by looking in each

equation (2) that contains a branching variable set to

1. In this way any branching variable set to 1, that cor-

responds to a medium sized coalition, has the effect

of drastically reducing the size of the corresponding

mathematical programming model.

While step (1) is basically the same for both valu-

ation functions, step (2) depends on the speciﬁc sub-

problem. Step (2) consists in repeatedly solving the

corresponding sub-problem while the optimum ob-

jective function value is strictly positive. The imple-

mentation, however, requires this value to be positive

within some tolerance. If this step would not have

been subjected to numerical restriction, then the algo-

rithm would have been an exact method.

Our branching rule chooses a variable, x

, from

the optimal solution in the current node of the branch-

ing tree having a value around 0.5 (e. g. x

∈

(0.35,0.65) - if possible). Our branch and bound does

not need upper bounds because in almost all cases a

feasible solution occurs very early (for the edge-sum

and the correlation functions) - mostly after perform-

ing one of the ﬁrst type (1) steps. The overall effect is

that the branching tree has a medium size: we limited

the number of branching tree size to 40, but this bound

was hardly reached for the ﬁrst two valuations. Sup-

pose that the current node in the branching tree has

a subset of already covered agents, U =

[

j∈J:x

and U

= V \U. For the three valuation functions the

sub-problems become:

ICAART 2021 - 13th International Conference on Agents and Artiﬁcial Intelligence

394

max

∑

i∈U

∑

j∈U

i j

∑

i∈U

∑

i/∈C

∑

i∈C

(1 − v

) ≥ 1, if C

⊆ U

∈ {0,1},∀i ∈ U

max

∑

i∈U

∑

j∈U

i j

− w

−

i j

∑

i∈U

∑

j∈V

−

i j

+ π

∑

i/∈C

∑

i∈C

(1 − v

) ≥ 1, if C

⊆ U

∈ {0,1},∀i ∈ U

and

max

∑

i∈U

∑

i∈U

∑

i/∈C

∑

i∈C

(1 − v

) ≥ 1, if C

⊆ U

∑

j∈U

∑

k=1

i j

−

∑

j∈U

∑

k∈U

i j

+β

− z

≤ β

,∀i ∈ U

∑

j∈U

∑

k=1

i j

−

∑

j∈U

∑

k∈U

i j

−

−z

≥ 0,∀i ∈ U

− z

≥ 0,∀i ∈ U

∈ {0,1},∀i ∈ U

∈ R,∀i ∈ U

6 NUMERICAL RESULTS

In this section we evaluate the performance of our al-

gorithm. A major part of the running time of our algo-

rithm is concentrated in the root node of the branching

tree, where step (2) adds hundreds of new variables to

the root model, hence hundreds of sub-problems to

solve (but the solver quickly ﬁnds optimal solutions

to these QKPf problems), while for the other nodes

the number of sub-problems drastically reduces.

The algorithm has been written in Java and run on

an Intel(R) Core (TM) i5-7500 CPU 3.40GHz com-

puter with 8GB RAM, under Ubuntu 18.04.4 LTS.

The linear and quadratic programming problems were

solved using Gurobi 9.0 under an Academic License.

Since there are no benchmarks in the literature for

edge-sum, correlation or coordination valuation func-

tions, the algorithm has been tested on randomly gen-

erated problems. Our test problems were built using

the Gilbert model (each edge has a ﬁxed probability of

being present in the graph, independently of any other

edges - probability 1 gives a complete graph). The

weights are independently generated using a Gaus-

sian distribution N(0,0.2) (a certain similar valuation

function occurs in (Rahwan et al., 2009a). The pa-

rameters of the benchmark ﬁle indicate the probabil-

ity that an edge belongs to the graph (”p”), the number

of agents (”n”), and the instance number (”s”).

Table 1: Solutions and computational times for different in-

stances (Edge-sum valuation).

Instance

s0 s1 s2 s3 s4

Benchmark p = 0.8, n = 40

LP solution 17.87 21.49 20.10 19.32 18.16

ILP solution 17.86 21.35 20.10 19.29 18.13

Overall time (s) 744 1,684 673 756 850

Root time (s) 529 924 673 615 557

Per node (s) 248 122 673 151 179

# of nodes 3 15 1 5 5

# of integer sol. 1 1 1 1 1

# of variables 318 856 292 320 319

Gap 0.05% 0.65% 0.00% 0.19% 0.16%

Benchmark p = 0.8, n = 45

LP solution 22.60 22.58 21.48 23.88 21.50

ILP solution 22.41 22.49 21.46 23.88 21.50

Overall time (s) 6,372 4,381 2,898 2,726 4,345

Root time (s) 3,488 2,875 1,779 2,726 4,345

Per node (s) 159 398 193 2,726 4,345

# of nodes 40 11 15 1 1

# of integer sol. 5 7 2 1 1

# of variables 1,600 2,222 801 1,114 1,399

Gap 0.84% 0.39% 0.09% 0.00% 0.00%

Benchmark p = 1.0, n = 35

LP solution 17.15 16.42 19.40 15.50 19.06

ILP solution 17.15 16.38 19.25 15.38 19.01

Overall time (s) 323 425 1,603 594 937

Root time (s) 323 325 561 375 397

Per node (s) 323 85 45 84 104

# of nodes 1 5 35 7 9

# of integer sol. 1 1 2 2 1

# of variables 204 274 1,232 335 516

Gap 0.00% 0.24% 0.77% 0.77% 0.26%

Benchmark p = 1.0, n = 40

LP solution 25.28 23.99 22.40 21.94 18.34

ILP solution 25.28 23.78 22.40 21.94 18.25

Overall time (s) 3,874 2,988 2,444 3,264 3,395

Root time (s) 3,874 143 2,444 3,264 1,620

Per node (s) 3,874 332 2,444 3,264 147

# of nodes 1 9 1 1 23

# of integer sol. 1 3 1 1 4

# of variables 299 536 264 276 870

Gap 0.00% 0.88% 0.00% 0.00% 0.49%

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Table 2: Solutions and computational times for different in-

stances (Correlation valuation).

Instance s0 s1 s2 s3 s4

Benchmark p = 0.4, pSign = 0.6,n = 50

LP solution 321.0 339.4 324.0 332.0 336.2

ILP solution 321 339 324 332 336

Overall time (s) 2,893 7,108 4,197 4,458 6,487

Root time (s) 2,893 2,947 4,197 4,458 3,284

Per node (s) 2,893 473.8 4,197 4,458 1,297

# of nodes 1 15 1 1 5

# of integer sol. 1 1 1 1 2

# of variables 1,072 3,071 1,168 1,139 4,399

# of disc. coal. 239 554 6 131 3,207

Gap 0.00% 0.11% 0.00% 0.00% 0.20%

Benchmark p = 0.6, pSign = 0.6,n = 40

LP solution 291.0 298.0 289.0 311.0 313.5

ILP solution 291 298 289 311 313

Overall time (s) 880 1,558 1,379 1,039 3,289

Root time (s) 880 1,558 1,379 1,039 2,352

Per node (s) 293 1,558 1,379 1,039 657

# of nodes 3 1 1 1 5

# of integer sol. 1 1 1 1 1

# of variables 843 1,031 527 531 1,241

# of disc. coal. 78 64 7 9 127

Gap 0.00% 0.00% 0.00% 0.00% 0.16%

Benchmark p = 0.6, pSign = 0.6,n = 45

LP solution 369.0 374.0 383.0 380.0 378.0

ILP solution 369 374 383 380 378

Overall time (s) 6,646 8,009 6,140 5,241 6,176

Root time (s) 5,105 8,009 6,140 5,241 6,176

Per node (s) 2,215 8,009 6,140 5,241 6,176

# of nodes 3 1 1 1 1

# of integer sol. 1 1 1 1 1

# of disc. coal. 19 13 3 20 3

# of variables 1,380 1,230 846 1,264 657

Gap 0.00% 0.00% 0.00% 0.00% 0.00%

Benchmark p = 0.8, pSign = 0.6,n = 35

LP solution 304.0 294.5 299.0 307.0 298.0

ILP solution 304 294 299 307 298

Overall time (s) 988 1,254 1,557 1,077 952

Root time (s) 988 808 1,557 1,077 952

Per node (s) 988 139 1,557 1,077 952

# of nodes 1 9 1 1 1

# of integer sol. 1 1 1 1 1

# of variables 634 1,072 705 583 389

# of disc. coal. 4 10 0 1 1

Gap 0.00% 0.16% 0.00% 0.00% 0.00%

Benchmark p = 0.8, pSign = 0.6,n = 40

LP solution 395.0 380.5 385.0 394.0 387.0

ILP solution 395 380 385 394 387

Overall time (s) 4,227 6,533 6,523 9,721 7,775

Root time (s) 4,227 3,782 6,523 9,721 7,775

Per node (s) 4,227 1,306 6,523 9,721 7,775

# of nodes 1 5 1 1 1

# of integer sol. 1 1 1 1 1

# of variables 840 1,728 851 886 735

# of disc. coal. 0 25 2 5 0

Gap 0.00% 0.13% 0.00% 0.00% 0.00%

We slightly generalized the correlation function by

supposing that some edges don’t have signs at all, i.

e., allowing an incomplete underlying graph. The ran-

Table 3: Solutions and computational times for different in-

stances (Coordination valuation).

Instance

s0 s1 s2 s3 s4

Benchmark p = 0.3, n = 35

LP solution 45.32 57.07 56.61 97.68 44.13

ILP solution 35.44 44.88 46.04 76.35 40.43

Total time (s) 1,852 3,234 8,458 6,450 5,119

Root time (s) 826 1,904 3,065 2,734 866

Per node (s) 45.2 78.89 206.31 157.4 124.9

# of nodes 41 41 41 41 41

# of int. sol. 4 3 2 3 2

# of vars. 4,165 2,275 15,566 3,298 2,152

# disc. coal. 3,462 1,710 14,995 2,555 254

Benchmark p = 0.4, n = 30

LP solution 69.08 90.62 79.88 51.72 88.14

ILP solution 54.92 90.34 79.52 51.43 87.92

Total time (s) 1,376 5,925 1,779 2,274 4,419

Root time (s) 727 3,014 1,143 1,202 2,312

Per node (s) 33.6 144.5 53.7 55.5 107.8

# of nodes 41 41 41 41 41

# of int. sol. 3 4 4 2 3

# of vars. 1,294 1,117 1,133 1,237 1,095

# disc. coal. 729 548 650 675 585

dom graphs were generated using the same model and

an edge has a ”+” or a ”−” sign with a prescribed

probability. The parameters of a benchmark ﬁle for

this valuation indicate the probability that an edge be-

longs to the graph (”p”), the probability of the plus

sign (”pSign”), the number of agents (”n”), and the

instance number (”s”).

The parameters of a benchmark ﬁle for coordina-

tion valuation indicate the probability that an edge be-

longs to the graph (”p”), the number of agents (”n”),

and ﬁnally the instance number (”s”). The probabili-

ties (weights) on the edges are randomly generated as

standard uniform variates.

Tables 1, 2, and 3 show the results of our numer-

ical tests; they contain the optimum value in the root

(LP problem), the (most of the time) optimum value

of the ILP problem found by branch and price al-

gorithm, the overall running time, the running time

for ﬁnding and solving the root LP problem, the run-

ning time per node, the number of nodes of the tree,

the total number of variables added during the exe-

cution, the number of nodes fathomed by integrality

(where the best integral solution improves the objec-

tive function), the number of disconnected coalitions

(which must be bypassed), and the gap between the

LP and the ILP optimum values where is the case.

For the edge-sum valuation the instances generated

with p = 0.8 or 1.0 the algorithm found no variables

corresponding to disconnected coalitions. The run-

ning time increases as the edge probability (hence the

density) of the graph increases, but the number of dis-

connected coalitions decreases.

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7 CONCLUSIONS

We addressed the problem of coalition structure gen-

eration over graphs for three valuations functions

where the set of feasible coalitions is restrained to

those that induce connected sub-graphs. Our goal was

to test the efﬁciency of the column generation method

and the branch and bound algorithm (i. e. the branch

and price algorithm) which needed to be truncated at

least for one of the valuations (the coordination).

Our algorithm is one of the ﬁrst to present compet-

itive results for more than 40 agents for some of the

most frequently used valuation functions, using only

a regular PC. For half of the tests the root LP problem

has an integer solution and for all of them the gap is

less than one percent. For the ﬁrst two valuations the

branching trees (with few exceptions) are small and

most of the running time is concentrated in the root

and in the nodes corresponding to variables set to zero

or variables set to one but associated with small sized

coalitions. For the edge-sum and the correlation val-

uations conducting the branching rule to ﬁnd medium

or large sized coalitions variables seems to be of sec-

ondary interest since the resulting trees are small in

size. for the coordination valuation the branching rule

must privilege the variables set to 1 disregarding the

size of the corresponding coalition. One of the con-

clusions of the numerical experiments is that the re-

sults depends on the magnitude of the weights. For

the low density instances the number of disconnected

coalitions could be high since the rejection procedure

for such coalitions could not included in the mathe-

matical programming model.

We listed only the results using the quadratic mod-

els for solving the sub-problem, however the linear

models have a good potential which deserves future

exploring work (Lagrangian relaxation, special treat-

ment of the high degeneracy etc). Future work should

also be directed towards getting rid of the main burden

of our algorithm: ﬁnd speciﬁc methods for solving

QKPf. Another direction of future research could be

a cut generation method (inequalities deﬁning facets

of a subjacent polyhedral relaxation) that tightens the

LP relaxations towards a branch and cut algorithm.

ACKNOWLEDGEMENTS

The ﬁrst author contribution in this this project is

funded by the Ministry of Research and Innovation

within Program 1 – Development of the national

RD system, Subprogram 1.2 – Institutional Perfor-

mance – RDI excellence funding projects, Contract

no.34PFE/19.10.2018.

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