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ROUND-UP PROPERTY. James B. ORLIN ... This paper extends the work of Baum and Trotter [1] concerning certain round-up properties of .... 2, x - e is optimal for C(A, w - ri)implying that f(w - ri) = f(w)l - 1,contrary to the above. Conversely ...
Mathematical Programming 22 (1982) 231-235. North-Holland Publishing Company

SHORT COMMUNICATION A POLYNOMIAL ALGORITHM FOR INTEGER PROGRAMMING COVERING PROBLEMS SATISFYING THE INTEGER ROUND-UP PROPERTY James B. ORLIN

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Sloan School of Management, Massachusetts Institute of Technology, Cambridge, MA, U.S.A.

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Received 3 November 1980 Revised manuscript received 2 June 1981 Let A be a non-negative matrix with integer entries and no zero column. The integer round-up property holds for A if for every integral vector w the optimum objective value of the generalized covering problem min{1 y: yA > w, y > 0 integer} is obtained by rounding up to the nearest integer the optimum objective value of the corresponding linear program. A polynomial time algorithm is presented that does the following: given any generalized covering problem with constraint matrix A and right hand side vector w, the algorithm either finds an optimum solution vector for the covering problem or else it reveals that matrix A does not have the integer round-up property. Key words: Integer Programming, Covering.

This paper extends the work of Baum and Trotter [1] concerning certain round-up properties of matrices. The notation in this paper is designed to be compatible with that of [1]. Matrix A will denote an m n matrix with non-

negative entries and no column or row of all zeroes. For each integral n-vector w we define the following linear and integer programs: C(A, w) D(A, w)

min {1 y: yA > w, y 0}, min{1 y: yA > w, y 0 integral}.

We let 1 denote the vector of all ones. Henceforth, let w+ denote the nonnegative part of vector w. Thus wt = max(0, w).

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Let the optimum objective values of C(A, w) and D(A, w) be denoted respectively as fA(w) and gA(w). The integer round up property (IRU) is said to hold for A if fA()l = gA(w) for all non-negative integral vectors w. Baum and Trotter [1] show that if matrix A does not have the IRU property then there is an integer n-vector w such that w c 1A and fA(w)l • gA(w) -

. Therefore, it is possible to

check to see if A has the IRU property by solving an exponential number of

linear and integer programs. This note presents a polynomial time algorithm which solves D(A, w) whenever A has the IRU property. If A does not have the IRU property then the 231

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J.B. Orlin/ Solving IP's satisfying the IRU property

232

algorithm either solves D(A, w) or else it demonstrates that A does not have the IRU property. The algorithm relies on a characterization of those constraint

matrices not having the IRU property. Before proceeding to the characterization of matrices that do not have the IRU property, we give some well-known elementary properties of the functions

fA(-) and

gA(')

fA(U) + fA(v) > fA(u + v),

(ii)

gA(U) + gA(V) > gA(U + V).

Lemma 2. Suppose that y is optimal for C(A, w) (resp., D(A, w)) and that x is a non-negative integral vector such that x • y. Then y - x is optimal for C(A, w -

xA) (resp., D(A, w - xA)). Proof. Consider the case that y is optimal for C(A, w), as the other case is analogous. Then vectors x and y - x are feasible for C(A, xA) and C(A, w - xA).

By Lemma 1, lx + (y - x) > fA(xA) + fA(w - xA) Ž fA(w) = 1y. -

xA) = 1(y - x).

In the following, the rows of A will be denoted r, ... , rm. Theorem 1. Matrix A does not have the IRU property if and only if there exists an integer vector w with at least one positive component such that w • 1A and fA(W - ri)l = rfA(W)1 for i = 1,..., m. Proof. that A vector for all [f(w -

In the following, we drop the index A for functions f and g. Suppose first does not have the IRU property. Let w be a minimal non-negative integer such that r f(w)l sg(w) - 1. It follows that w # Oand rf(w)l = f(w - ri)l i. To see this, suppose that such were not the case. Then for some i, rl s Ff(w)l - 1. Furthermore, by Lemma 1, g(w) g(w - r) + 1. There-

fore, [f(w - ri)l -< f(w) Let w' = (w - r)+. Then

- 1

Ff(w')1l

g(w)- 2 - w'- W.

itO Extensions to the integer round down property Consider the integer programming packing problem: Q(A, w)

.

max{ly: yA c w, y > 0 integer}

and let P(A, w) denote the associated linear program. Matrix A is said to have the integer round-down property (IRD) is for every integral vector w the optimal objective value for Q(A, w) is obtained by rounding down to the nearest integer the optimal objective value of P(A, w). The IRD property is analogous to the IRU property, and the preceding algorithm can be modified slightly so as to either solve Q(A, w) or else prove that A does not have the IRD property. It

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suffices to reinterpret f(w) in the algorithm as the optimal objective value of P(A, w) and to change the function L J in steps 7 and 8 to L . The proof is a straightforward reinterpretation of the proofs in this paper.

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J.B. Orlin/ISolving IP's satisfying the IRU property

235

Acknowledgment

Lnd in each I gratefully acknowledge the helpful comments of Les Trotter. I also thank

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two anonymous referees for their helpful suggestions.

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the current References [1] S.P. Baum and L.E. Trotter, Jr., "Finite checkability for integer rounding properties in combinatorial programming problems", Mathematical Programming, to appear. [2] L.G. Khachian, "A polynomial algorithm for linear programming", Doklady Akademii Nauk 244 (1979); translated in Soviet Mathematics Doklady 20 (1579) 191-194.

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in step 4. 5, and let i = (w - ri) + .

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