AbstractWe consider combinatorial optimization problems with a feasible solution set S⊆{0,1}n specified by a system of linear constraints in 0–1 variables. Additionally, several cost functions c1,…,cp are given. The max-linear objective function is defined by f(x):=max{c1x,…,cpx: x∈S}; where cq:=(cq1,…,cqn) is for q=1,…,p an integer row vector in Rn.The problem of minimizing f(x) over S is called the max-linear combinatorial optimization (MLCO) problem.We will show that MLCO is NP-hard even for the simplest case of S⊆{0,1}n and p=2, and strongly NP-hard for general p. We discuss the relation to multi-criteria optimization and develop some bounds for MLCO

Chung, Sung-Jin

Hamacher, Horst W.

Maffioli, Francesco

Murty, Katta G.

We consider combinatorial optimization problems with a feasible solution set S[subset of or equal to]{0,1}n specified by a system of linear constraints in 0-1 variables. Additionally, several cost functions c1,...,cp are given. The max-linear objective function is defined by f(x):=max{c1x,...,cpx: x[set membership, variant]S}; where cq:=(cq1,...,cqn) is for q=1,...,p an integer row vector in n.The problem of minimizing f(x) over S is called the max-linear combinatorial optimization (MLCO) problem.We will show that MLCO is NP-hard even for the simplest case of S[subset of or equal to]{0,1}n and p=2, and strongly NP-hard for general p. We discuss the relation to multi-criteria optimization and develop some bounds for MLCO.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/30833/1/0000495.pd

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Note on combinatorial optimization with max-linear objective functions

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Note on combinatorial optimization with max-linear objective functions 

Discrete Applied Mathematics 42 (1993) 139-145 
North-Holland 
139 
Note on combinatorial optimization 
with max-linear objective functions 
Sung-Jin Chung 
Department ofIndustrial Engineering, Seoul National University. Seoul, South Korea; and Department of 
Industrial and Operations Engineering, University of Michigan, Ann Arbor, MI, USA 
Horst W. Hamacher” 
Fachbereich Mathemhtik, Universittit Kaiserslautern, Kaiserslautern, Germany 
Francesco Maffioli” 
Dipartimento di Elettronica, Politecnico di Milano. Milano, Ital) 
Katta G. Murty 
Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, MI, USA 
Received 15 August 1990 
Revised 1 March 1991 
Abstract 
Chung, S.-J., H.W. Hamacher, F. Maffioli and K.G. Murty, Note on combinatorial optimization with 
max-linear objective functions, Discrete Applied Mathematics 42 (1993) 139-145. 
We consider combinatorial optimization problems with a feasible solution set SC(O,I )” specified by a 
system of linear constraints in O-l variables. Additionally, several cost functions c,,., ,,c~ are given. The 
max-linear objective function is defined by flx):=max(c’x,...+!‘x: YES} where c?=(c~,.,,,c:) is for 
q=l,...,p an integer row vector in Iw”. 
The problem of minimizingflx) over S is called the max-linear combinatorial optimization (MLCO) 
problem. 
We will show that MLCO is NP-hard even for the simplest case of S={O,l}” andp=2, and strongly 
NP-hard for generalp. We discuss the relation to multi-criteria optimization and develop some bounds 
for MLCO. 
Correspondence to: Professor S-J. Chung, Department of Industrial Engineering, Seoul National Universi- 
ty, Seoul, South Korea. 
* Supported by NATO Grant RG85/0240. 
0166-218X/93/$06.00 0 1993 - Elsevier Science Publishers B.V. All rights reserved 
140 S.J. Chin et al. 
1. Introduction 
We consider combinatorial optimization problems with a feasible solution set 
S c (0, I}” specified by a system of linear constraints in O-l variables. Additionally, 
several cost functions cl, . . . , cp are given. The max-linear objective function is 
defined by 
f(x):=max{clx,...,cPx:xES} 
where cq:=(cy,..., c,Q) is an integer row vector in R”, q = 1, . . . ,p. (1.1) 
The problem of minimizing f(x) over S is called the max-linear combinatorial 
optimization (MLCO) problem. 
MLCO can always be modeled as an integer program by standard techniques 
(Nemhauser and Wolsey [15]). In the problem which we study in this paper the set 
S always has a special structure so that a single linear objective function can be 
optimized over it efficiently, i.e., in polynomial time. The focus of our investigation 
will be MLCO problems with p? 2 over such sets S. 
MLCO plays a significant role in the assembly of printed circuit boards (see 
Drezner and Nof [6]). There, S is the set of all incidence vectors of maximum 
cardinality matchings in a bipartite graph. Other applications include partition 
problems (Garey and Johnson [S]), multi-processor scheduling problems and certain 
stochastic optimization problems (Granot and Zang [lo]). 
A special case of this problem is the ML matroid problem. The NP-completeness 
of this problem has been proved by Warburton [17] who also analyzes worst-case 
performances of some Greedy heuristics. Granot [9] introduces Lagrangean duals 
for the problem. 
In Section 2 of this paper we show that even the case S= (0, 1)” (the uncon- 
strained MLCO) with p = 2 is NP-hard and strongly NP-hard for general p. Section 
3 deals with the relation of MLCO and discrete multi-criteria optimization 
problems. Section 4 contains some remarks on branch and bound strategies for 
MLCO. 
2. Complexity results 
Elegant methods are available for minimizing convex functions over convex sets 
(see Fletcher [7], Luenberger [2]). However, this problem becomes hard even for 
simple discrete sets as the following example taken from Murty and Kabaldi [14] 
shows. 
Let do, d,, . . . , d, be given positive integers. Then the subset-sum problem is 
that of checking whether there exists a Boolean vector XE (0, l}” such that 
dlxl + ... + d,,x,, = do. This problem is well known to be NP-complete (Garey and 
Johnson [S]). If we define the convex quadratic functionf(x) := (dlxl + ... +d,x,-d~J~, 
then the subset-sum problem is equivalent to checking whether the optima1 value in 
min{f(x): XE (0, 1)“) is 0 or strictly greater than 0. 
In this section we show that even the unconstrained MLCO problem defined with 
the simplest convex functions-max-linear functions defined by two linear func- 
tions c1 and c*-leads to an NP-hard optimization problem. 
Theorem 2.1. The unconstrained MLCO with respect to two linear functions is 
NP-hard. 
Proof. Consider an instance of the interval subset-sum (ISS) problem: Let a,, . . . , a, 
be positive integral weights, and let u and d be positive integers with odd. The ISS 
problem asks for a subset N of { 1, . . . , m} such that the sum of integral weights in- 
dexed by the elements of N is contained in the interval [u, d]. Since the subset-sum 
problem is a special case of the ISS problem (with o = d), ISS is NP-complete. We 
reduce ISS to the unconstrained MLCO, such that an instance of ISS has a feasible 
solution if and only if the optimum objective value in the corresponding instance 
of MLCO is strictly less than 0. 
Set n:=m+l, cf:=ai, ~::=a;, i=l,...,m, &+,:=-d-l and ci+i:=u-1. 
Given any x E { 0, 1 } @+ ‘) the following two cases can occur. 
Case 1: x,+l= 0. Then the objective value of x in the unconstrained MLCO is 
greater than or equal to 0, since ai>O, i= 1, . . . , m. 
Case 2: x,+l = 1. The objective value of x in the unconstrained MLCO is less 
than 0 if and only if 
C ai<d+l and 
ieN 
iF&(-ai)<-(u-l) 
where N={i: lrilm and X,=1}. 
In this case, N is a feasible solution to the given instance of the interval subset- 
sum problem. q 
Theorem 2.2. The unconstrained MLCO problem with p cost functions c,, . . . , cp is 
strongly NP-hard. 
Proof. Let A be a (0,l) matrix with m rows and n columns and let e be the vector 
with each of its m components being equal to 1. The set-partitioningproblem (SPP) 
is the problem to find some XE (0, l}” such that Ax= e. This problem is strongly 
NP-hard (see Garey and Johnson [8]). 0 
In order to reduce SPP to MLCO we denote with Ai the ith row of matrix A, 
define p = 2m, and introduce n + 1 variables xi, . . . ,x,,, x,, + 1, and p cost vectors cq , 
each with n + 1 components, defined by 
(-A,, 0) for q=l,...,m, 
cq = 
(4-m, -2) for q=m+l, . . ..2m. 
142 S.J. Chin et al 
Then it can easily be verified that SPP has a feasible solution if and only if the op- 
timum objective value in this MLCO is strictly less than 0. 
3. Relation to multi-criteria problems 
In multi-criteria optimization (MCO) we also consider several cost functions 
1 c ,..., cp. The goal in MC0 is to find efficient solutions, i.e., solutions XES with 
the following property. 
Ify~Sand~~y<c~xforallq=l,..., p, then none of these inequalities 
is strict. 
Ifx,y~S,cqyIcqxforallq=l ,..., p, and at least one of these inequalities is strict, 
then we call x dominated by y. 
Theorem 3.1. For any instance of A4LCO there is an optimum solution which is ef- 
ficient with respect o the cost functions c’, . . ..cp. 
Proof. Suppose x is an optimum solution to a given MLCO, and let x be dominated 
by YES. Then c’ylcqx for all q= 1, . . ..p implies 
max{c’y, . . . . cPy}<max{clx,...,c’x}. 
Hence y is also optimal for MLCO. 0 
As a consequence of Theorem 3.1 we can solve MLCO by only considering the 
efficient solutions of the corresponding multi-criteria combinatorial optimization 
problem. Therefore for p = 2 a solution of the following with problem parameters 
CJE (minXEs c2x, maxXcs c2x} will solve MLCO. 
minimize c’x, 
subject to x E S and c2x5 0. 
If S is the set of bases of a matroid, then the latter problem is a matroidal knap- 
sack problem discussed in Camerini and Vercellis [3] and Camerini et al. [2]. These 
papers applied to this particular MLCO problem give an alternative approach to the 
ones taken by Granot [9] or Warburton [ 171. 
4. Branch and bound approach 
We first discuss some general bounding strategies. 
Since the combinatorial optimization problem under consideration can be solved 
in polynomial time for a single objective we can efficiently compute 
Combinatorial opfimization 143 
6q:=min{cqx: XES), q=l,..., p. 
Let x4 be the solution in which a4 is attained. Then 
(4.1) 
L(S)=max{aq: q=l,...,m) (4.2) 
is a lower bound for the MLCO problem. An upper bound is obtained by setting 
U(S):=min(f(xq): q=l,...,p}. (4.3) 
Let y be one of the solutions x4 such that cYq is equal to L(S) and let z be one 
of the solutions xr such that f(x’) = U(S). Let T be the union of all variables which 
are equal to 1 either in y or z or both. One of the variables in Twill be selected as 
branching variable: For each t E T let S(t) := {xES: x, =O>. Compute L@(t)) and 
U@(t)). Then take the t with the smallest U(S(t)) -L,@(t)) and x, as the branching 
variable. 
The lower bound (4.2) can be improved by using Lagrangean relaxation: An LP- 
formulation of MLCO is 
minimize z 
subject to z - crxz 0, 
z - C2XZ 0, 
. . . 
z-CPXZO, 
XES, 
z unrestricted. 
(4.4) 
Let 7c1, . .. . rep be nonnegative Lagrange multipliers associated with the constraints 
z-cqxro, q=l,..., p in (4.4). Then a lower bound of MLCO is obtained by maxi- 
mizing over all rcl, . . . , nPcp’ 0 the function 
min{z-7rr(z-c’x)-... - rcp(z - cpx>: z unrestricted, x E S}. (4.5) 
Since (4.5) can be written as 
min{(l-7r,- ~~~-np)z+(nlcl+~~~+~pcP)x: z unrestricted, XES}, (4.6) 
we can restrict ourselves to or, . . . , np satisfying nl + ... + rep = 1. Thus we get the 
following result. 
Theorem 4.1. 
L,(S) := max min (nrc’ + 
n,+...+rr,=l XGS 
0.. + 7cpcP)x 
is a lower bound for the MLCO. However L, improves the bound of (4.2), i.e., 
L(S)iL,(S). 
Proof. L*(S) is a lower bound since it is the optimal objective value of the 
144 S.J. Chin et al. 
Lagrangean dual of LP (4.4). Since rr with rcq = 1 for exactly one q E { 1, . . . , p> is 
feasible the result follows. 0 
If the set S is specified by a unimodular system of linear constraints in (O,l)- 
variables it can be solved through LP techniques. In this case (4.6) is a piecewise 
linear concave function over the set of all nonnegative rcq, q = 1, . . . , p, and can be 
computed efficiently by using techniques of nondifferentiable concave program- 
ming (see, for instance, Shapiro [16]). For the casep =2 one can use algorithms for 
solving parametric combinatorial optimization problems with respect to a single 
parameter (see Carstensen [4,5], Hamacher and Foulds [II], etc.) or use efficient 
approximation techniques for its solution (see Burkard et al. [l]). 
IfalineardescriptionS={x:Ax=b,xj=Oorx~=1,j=1,...,n}ofSisgiven,it 
is well known (see, for instance, Murty [13]) that the bound L,(S) can be further 
improved by replacing in Theorem 4.1 the set S by Siin := {x: Ax= b, O<XjsO, 
j=l 3 a**, n}. Hence 
L,(S):=L,(S):= max min (nr cl + .a. + 7+)x 
ZTl+.“+Rp=l XtZSli, 
is a lower bound such that the optimal solution x* of MLCO satisfies 
L(s)IL,(s)IL,(s)If(x*)_( U(S). (4.7) 
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Graz, Graz (1987). 
[2] P. Camerini, F. Maffioli and C. Vercellis, Multi-constrained matroidal knapsack problems, Math. 
Programming 45 (1989) 211-231. 
[3] P. Camerini and C. Vercellis, The matroidal knapsack: A class of (often) well-solvable problems, 
Oper. Res. Lett. 3 (1984) 157-162. 
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