Combinatorial Benders’ Cuts
for Mixed-Integer Linear Programming
Gianni Codato and Matteo Fischetti
DEI, University of Padova, Italy
e-mail: matteo.fischetti@unipd.it
May 24, 2004; Revised, February 16, 2005
Abstract
Mixed-Integer Programs (MIP’s) involving logical implications mod-
elled through big-M coefficients, are notoriously among the hardest to
solve. In this paper we propose and analyze computationally an au-
tomatic problem reformulation of quite general applicability, aimed at
removing the model dependency on the big-M co efficients. Our solu-
tion scheme defines a master Integer Linear Problem (ILP) with no
continuous variables, which contains combinatorial information on the
feasible integer variable combinations that can be “distilled” from the
original MIP model. The master solutions are sent to a slave Linear
Program (LP), which validates them and possibly returns combinato-
rial inequalities to be added to the current master ILP. The inequal-
ities are associated to minimal (or irreducible) infeasible subsystems
of a certain linear system, and can be separated efficiently in case the
master solution is integer. The overall solution mechanism resembles
closely the Benders’ one, but the cuts we pro duce are purely com-
binatorial and do not depend on the big-M values used in the MIP
formulation. This produces an LP relaxation of the master problem
which can be considerably tighter than the one associated with origi-
nal MIP formulation. Computational results on two specific classes of
hard-to-solve MIP’s indicate the new method produces a reformulation
which can be solved some orders of magnitude faster than the original
MIP model.
Key words: Mixed-Integer Programs, Benders’ Decomposition, Branch
and Cut, Computational Analysis.
1
1 Introduction
We first introduce the basic idea underlying combinatorial Benders’ cuts—
more elaborated versions will be discussed in the sequel.
Suppose one has a basic 0-1 ILP of the form
min{c
T
x : F x ≤ g, x ∈ {0, 1}
n
} (1)
amended by a set of “conditional” linear constraints involving additional
continuous variables y, of the form
x
j(i)
= 1 ⇒ a
T
i
y ≥ b
i
, for all i ∈ I (2)
plus a (possibly empty) set of “unconditional” constraints on the continuous
variables y, namely
Dy ≥ e (3)
Note that the continuous variables y do not appear in the objective
function—they are only introduced to force some feasibility properties of
the x’s.
A familiar example of a problem of this type is the classical Asymmetric
Travelling Salesman Problem with time windows. Here the binary variables
x
ij
are the usual arc variables, and the continuous variables y
i
give the
arrival time at city i. Implications (2) are of the form
x
ij
= 1 ⇒ y
j
≥ y
i
+ travel time(i, j) (4)
whereas (3) bound the arrival time at each city i
early arrival time(i) ≤ y
i
≤ late arrival time(i). (5)
Another example is the map labelling problem [29], where the binary vari-
ables are associated to the relative position of two labels to be placed on
a map, the continuous variables give their placement coordinates, and the
conditional constraints impose non-overlapping conditions of the type“if la-
bel i is placed on the right of label j, then the placement coordinates of
i and j must obey a certain linear inequality giving a suitable separation
condition”.
The usual way implications (2) are modelled within the MIP framework
is to use the (in)famous big-M method, where large positive coefficients M
i
are introduced to activate/deactivate the conditional constraints as in:
a
T
i
y ≥ b
i
− M
i
(1 − x
j(i)
) for all i ∈ I (6)
This yields a (often large) mixed-integer model involving both x and
y variables—whereas, in principle, y variables are just artificial variables.
2
Even more imp ortantly, due to the presence of the big-M coefficients, the
LP relaxation of the MIP model is typically very poor. As a matter of
fact, the x solutions of the LP relaxation are only marginally affected by
the addition of the y variables and of the associated constraints. In a sense,
the MIP solver is “carrying on its shoulders” the burden of all additional
constraints and variables in (2)-(3) at all branch-decision nodes, while they
becomes relevant only when the corresponding x
j(i)
attains value 1 (typically,
because of branching).
Of course, we can get rid of the y variables by using Benders’ decom-
position [5], but the resulting cuts are weak and still depend on the big-M
values. As a matter of fact, the classical Benders’ approach can be viewed
as a tool to speed-up the solution of the LP relaxation, but not to improve
its quality.
The idea behind “combinatorial” Benders’ cuts is to work on the space
of the x-variables only, as in the classical Benders’s approach. However, we
model the additional constraints (2)-(3) through the following Combinatorial
Benders’ (CB) cuts:
X
i∈C
x
j(i)
≤ |C| − 1 (7)
where C ⊆ I induces a Minimal (or Irreducible) Infeasible Subsystem (MIS,
or IIS, for short) of (2)-(3), i.e., any inclusion-minimal set of row-indices of
system (2) such that the linear subsystem
SLAV E(C) :=
(
a
T
i
y ≥ b
i
, for all i ∈ C
Dy ≥ e
has no feasible (continuous) solution y.
A CB cut is violated by a given x
∗
∈ [0, 1]
n
if and only if
P
i∈C
(1−x
∗
j(i)
) <
1. Hence the corresponding separation problem essentially consists of the
following steps: (i) weigh each conditional constraint a
T
i
y ≤ b
i
in (2) by
1 − x
∗
j(i)
; (ii) weigh each unconditional constraint in (3) by 0; and (iii) look
for a minimum-weight MIS of the resulting weighted system—a NP-hard
problem [1, 17].
A simple polynomial-time heuristic for CB-cut separation is as follows.
Given the (fractional or integer) point x
∗
to be separated, start with C :=
{i ∈ I : x
∗
j(i)
= 1}, verify the infeasibility of the corresponding linear subsys-
tem SLAV E(C) by classical LP tools, and then make C inclusion-minimal
in a greedy way. Though extremely simple, this efficient separation turns
out to be exact when x
∗
is integer.
The discussion above suggests the following exact Branch & Cut solution
scheme. We work explicitly with the integer variables x only. At each
branching node, the LP relaxation of a master problem (namely, problem
(1) amended by the CB cuts generated so far) is solved, and the heuristic
3
CB separation is called so as to generate new violated CB cuts (and to assert
the feasibility of x
∗
, if integer).
The new approach automatically produces a sequence of CB cuts, which
try to express in a purely-combinatorial way the feasibility requirement in
the x space—the CB cut generator acting as an automatic device to distill
more and more combinatorial information from the input model. As a con-
sequence of the interaction among the generated CB cuts, other classes of
combinatorial cuts are likely to b e violated, hence allowing other cut sep-
arators to obtain a further improvement. We found that the {0,
1
2
}–cuts
addressed in [9, 2] fit particularly well in this framework, and contribute
substantially to the overall efficacy of the approach.
It is worth noting that, using the new technique, the role of the big-M
terms in the MIP model vanishes–only implications (2) are relevant, no
matter the way they are modelled. Actually, the approach suggests an
extension of the MIP modelling language where logical implications of the
type (2) can be stated explicitly in the model, as in Hooker and Osorio [21].
In this paper we aim at investigating whether the above method can be
useful to approach certain types of MIP’s which are notoriously very hard
to solve. As shown in the computational section, this is indeed the case:
even in its simpler implementation, on some classes of instances the new
approach allows for a speed-up of some orders of magnitude with respect to
ILOG-Cplex 8.1, one of the best MIP solvers on the market.
Our technique is based on Hooker’s idea of deriving Benders’ cuts from
minimal sets of inconsistencies, as proposed in [19]. In this respect, our main
contributions have been (a) to present a separation heuristic that finds a min-
imal Benders’ cuts for the special case of conditional constraints with linear
implications, and (b) to test these cuts computationally on some hard MIP
problems. This is an important special case because conditional constraints
of this form are a very useful mo deling device.
The paper is organized as follows. In Section 2 we present the new
approach in a more general context, whereas previous literature on the sub-
ject is reviewed in Section 3. As already stated, our CB cut separator
requires a fast determination of MIS’s; this important topic is addressed in
Section 4, where an approach particularly suited to our application is de-
scribed. Computational results are presented in Section 5, with the aim of
verifying whether a simple implementation of the new method can already
produce improved performance with respect to the application of a sophisti-
cated MIP solver such as ILOG-Cplex 8.1 (at least, on some problem classes
which fit particularly well in our scheme). Finally, some conclusions are
drawn in Section 6.
The present paper is based on the master thesis of the first author [12],
which was awarded the 2003 Camerini-Carraresi prize by the Italian Oper-
ation Research association (AIRO). Moreover, the paper was presented at
the IPCO X meeting held in New York, June 2004.
4
2 Combinatorial Benders’ cuts
Let P be a MIP problem with the following structure:
P : z
∗
:= min c
T
x + d
T
y (8)
s.t. F x ≤ g (9)
Mx + Ay ≥ b (10)
Dy ≥ e (11)
x
j
∈ {0, 1} for j ∈ B (12)
x
j
integer for j ∈ G (13)
where x is a vector of integer variables, y is a vector of continuous variables,
G and B are the (possibly empty) index sets of the general-integer and
binary variables, respectively, and M is a matrix with exactly one nonzero
element for every row i, namely the one indexed by column j(i) ∈ B. In
other words, we assume the linking between the integer variables x and the
continuous variables y is only due to a set of constraints of the type
m
i,j(i)
x
j(i)
+ a
T
i
y ≥ b
i
for all i ∈ I (14)
where variables x
j(i)
are binary for all i ∈ I.
We consider the case d = 0 first, i.e., we assume the MIP objective
function does not depend on the continuous variables, and leave case d 6= 0
for a later analysis. In this situation, we can split problem P into two
sub-problems:
• MASTER:
z
∗
= min c
T
x (15)
s.t. F x ≤ g (16)
x
j
∈ {0, 1} for j ∈ B (17)
x
j
integer for j ∈ G (18)
• SLAVE(˜x), a linear system parametrized by ˜x:
Ay ≥ b − M ˜x (19)
Dy ≥ e (20)
Let us solve the master problem at integrality. If this problem turns out
to be infeasible, then P also is. Otherwise, let x
∗
be an optimal solution (we
exclude the unbounded case here, under the mild assumption that, e.g., the
general-integer variables are bounded).
5