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Briand et al. (2012b) proposed an efficient integer
linear program formulation for this problem (Briand
et al., 2012b).
Another important application to the network opti-
mization is the well-known Network Flow theory
(Ford and Fulkerson, 1958). Several algorithms have
been developed in order to find a maximum flow in
a network. Ford and Fulkerson (1956) were the first
to develop a clever algorithm based on the duality
between minimum cut and maximum flow (Ford and
Fulkerson, 1956). Later on, efficiency improvements
were proposed see eg; (Edmunds and Karp, 1972),
(Goldberg and Tarjan, 1986), etc. The minimum cost
maximum flow problem, which is equivalent to the
minimum cost circulation problem, is solvable in
polynomial time (Tardos, 1985).
As regards to social networks, the prediction of
agents’ behavior is of interest. Several papers focus
on games associated with various forms of networks,
see (Tardos and Wexler, 2007) for an overview. In
a recent work, Apt and Markakis (2011) studied the
complexity of finding a Nash Equilibrium for the
multi-agent social networks with multiple products,
in which the agents, influenced by their neighbors,
can choose one out of several alternatives (Apt and
Markakis, 2011).
Specifically, this work considers a transportation
network that involve a set of agents, each one being
in charge of a part of the network. It is assumed
that each agent is able to control the transportation
capacities of its arcs. A lot of features of this work
are inspired by the multi-agent project scheduling
problems, as presented in (Briand et al., 2012a),
especially concerning the reward sharing policy.
In fact, the outcome of an agent depends on its
own strategy and on the satisfaction of a customer,
which depends on the network flow. As proposed
by Fernandez (2012) (Fernandez, 2012), we assume
that the customer gives a reward proportional to the
maximum-flow that can circulate inside the network.
This reward is shared among agents according to
some ratios predefined in the network design phase
(Cachon and Lariviere, 2005).
To the best of our knowledge, the research presented
here is an original way of presenting a transportation
problem using multi-agent network flow with con-
trollable arcs capacities. One important application to
the problem proposed in this paper is the distributed
control of transportation networks, like traffic, water,
where the road of the network are distributed among
several agents which can control the amount of
product or water to circulate on the network.
This paper mainly discusses the complexity of finding
a Nash Equilibrium that maximizes the flow in the
network.
The paper is organized as follows: Section 2 defines
formally the Multi-Agent Minimum-Cost Flow prob-
lem and introduces some important notations. There-
after, Section 3 introduces the duality between effi-
ciency and stability of a strategy and presents some
important definitions and properties. In Section 4 and
5, we illustrate some basic notions for the single agent
and the multi-agent cases, respectively. In Section 6,
an example is provided to illustrate the notions intro-
duced in previous sections. Section 7 deals with the
complexity of the problem of finding a Nash equilib-
rium with bounded flow. Finally, conclusions and fu-
ture directions are drawn in Section 8.
2 PROBLEM STATEMENT AND
NOTATIONS
We focus on a Minimum-Cost Flow problem under
a Multi-Agent context. This problem will be further
referred to as MA-MCF. Considering a transportation
network with limited arc capacities, this problem con-
sists in sending a maximum amount of products from
a source node to a sink node, at minimum cost. In
this work, a major assumption is that arc capacities
are controlled by agents, each arc being assigned to a
specific agent.
2.1 Problem Definition
The MA-MCF problem can be described as a tuple
< G,A,Q,Q,C, π,W >, where:
• G = (V,E) is a flow network. V is the set of nodes,
s,t ∈ V being the source and the sink nodes of the
flow network G, respectively. E is the set of arcs,
each one having its capacity and receiving a flow.
An arc e from node i to node j is denoted by e =
(i, j).
• A is a set of m agents: A = {A
1
,... , A
u
,... , A
m
}.
Arcs are distributed among agents. An agent A
u
owns a set of m
u
arcs, denoted E
u
. Each arc (i, j)
belongs to exactly one agent (i.e., E
u
∩ E
v
=
/
0 for
each agent’s pair (A
u
,A
v
) ∈ A
2
such that u 6= v).
• q
i, j
is the capacity of arc (i, j) which takes value in
an interval [q
i, j
,
q
i, j
]. q
i, j
(resp.
q
i, j
) is the normal
(resp. maximum) arc capacity. Q = (q
i, j
)
(i, j)∈E
and
Q = (q
i, j
)
(i, j)∈E
referred to as the vectors of
normal and maximum arc capacities, respectively.
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