![](bg4.png)
• For all k, l ∈ {1, · · · , n} and for all (i, j) ∈ E, if de-
gree of the node i is equal to 1, then
x
ik
− x
jl
+ f
kl
≤ 1. (21)
Proof: All of these inequalities can be easily ob-
tained by using the fact that |k − l| ≤ |k −m| + |m − l|
(for all triple k, l, m) and x
ik
, f
kl
∈ {0, 1} (for all
i, k, l ∈ {1, · · · , n}).
Definition: For a given graph G, the chromatic
number of G is denoted by χ(G) and is equal to the
minimum number of colors that one needs to color
the nodes of G in such a way that all adjacent nodes
of G get different colors.
The following theorem introduces new valid inequal-
ities that involve the chromatic number of a given
graph G.
Theorem 4 (Chromatic Inequalities): For any
graph G having the chromatic number χ(G) and for
all triple k, l, s, the following inequalities are valid for
(7)-(12):
f
k,l
+ f
l,s
+ f
k,s
≤ χ(G). (22)
Proof: First of all, we note that the left hand side
of (22) is at most equal to 3. If χ(G) ≥ 3, we are
done; otherwise, if χ(G) = 2, then there is (at least)
one node among the nodes having the labels k, l, s that
is not connected to one of the others. Without loss of
generality, let us suppose that the node having the la-
bel k is not connected to the node with the label s; this
means that there will be no flow between the nodes k
and s. Consequently, the nodes k and s can have the
same colors and hence f
k,s
= 0, which completes the
proof.
The results of the theorem 4 can be useful when
χ(G) = 2. This corresponds to some special graphs
such as trees and the graphs without any cycles of odd
lengths. This is rather restricting in using the inequal-
ities (22). Notice that finding the chromatic number
of a given graph is an NP-hard problem. Neverthe-
less, tight upper bound of χ(G) may be found by us-
ing heuristic algorithms.
Definition: For a given graph G, matching is a
subset of edges in which, no two edges are adjacent
to a same node. A maximum matching of G is de-
fined as a matching with the maximum cardinality.
The matching number of G, noted by ν(G), is the size
of a maximum matching.
As in the case of chromatic number, some valid
inequalities depending on the matching number may
be introduced.
Theorem 5 (Matching Inequalities): For any
graph G having the matching number ν(G) and for
all k and “0 ≤ i and k + 2i + 1 ≤ n”; the following
inequality is valid for (7)-(12):
i:k+2i+1≤n
∑
i=1
f
k+2i,k+2i+1
≤ ν(G). (23)
Proof: For an arbitrary (connected) path in G, the
left hand side of (23) is (at most) the number of the
edges in the path such that any couple of edges is
separated by at least one edge. By noting this fact,
one can conclude that the number of these edges
cannot exceed the matching number of G.
4 COMPUTATIONAL
EXPERIMENTS AND
NUMERICAL RESULTS
In this section, we present the preliminary results that
we have obtained by applying the valid inequalities of
the previous section.
The model has been coded in C++ and has been
solved with IBM CPLEX 12.2 in an Intel Core 2 Duo
of 3 GHz and 3.25 GB of RAM. The experiments
have been carried out on some benchmark instances
already used in (Caprara et al., 2010), (Caprara et al.,
2011), and (Schwarz, 2010). Table 1 reports some
characteristics of the instances. In this table, for each
graph, the number of nodes (n), of edges (m), and of
triangles (t) are reported. The absence of triangles can
be useful for chromatic inequalities.
The results are reported in Tables 2 and 3. In Ta-
ble 2, we denote by “Optimal” the known optimal
values in the literature (see (Caprara et al., 2010),
(Caprara et al., 2011), and (Schwarz, 2010)). The
column “Optimal” corresponds to optimal values ob-
tained through exact algorithms, such as Branch-and-
Cut procedures. Concerning our experiments, “LP(2-
3)” is used to denote the optimal value of the relaxed
linear program under the valid inequalities of the the-
orems 2 and 3. Table 3 contains more results on
the smaller sized instances. More precisely, Table 3
presents the optimal value of the relaxed linear pro-
gram under the valid inequalities of the theorems 1-3
(denoted by LP(1-3)) that are compared to the results
of (LP(2-3)). The CPU time (in seconds) of each case
is shown in a side column (i.e., cpu). There is a time
limit of 1200 seconds on CPLEX.
The number of the constraints (10) is huge.
Hence, in our experiments, the constraints (10) have
not been used. We just considered the remaining con-
straints of the model (7)-(12) as well as some of the
rank inequalities. This concerns the valid inequali-
ties that have been introduced in the theorems 1, 2,
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