Finding Facets of General Integer Knapsacks

Finding Facets of General Integer Knapsacks Bruce Davey

Natashia Boland

Peter Stuckey

University of Melbourne e-mail:[email protected] November 29, 1997

1 Introduction One of the more eective techniques applied to solving binary linear programs having general integer constraint coecients is branch-and-cut with lifted knapsack cover cutting planes. (See for example [2, 3, 4].) However there is little in the literature on similar cutting planes for problems with general integer variables. In fact, very little research appears to have been devoted to solving general integer programs compared with binary programs. In this paper we present several results that suggest that the lifted knapsack cover approach for binary problems can be extended to general integer programs. We introduce the notion of a general integer knapsack cover, and corresponding valid inequalities, demonstrably stronger than those previously known. By an appropriate minimality condition, we show that these inequalities are facet1

de ning for the general integer knapsack polytope. We then present another class of inequalities which are facet-de ning under less restrictive conditions. We conclude by presenting extended general integer knapsack cover inequalities and giving conditions under which they may be facet-de ning.

2 Lifting and Finding a Facet Fundamental to the generation of knapsack covers is the concept of lifting. The following theorem, due to Wolsey [5], shows how this may be done.

Theorem 1 Wolsey [5] Given Q = convfx j ax  b, x 2 Z+n , x  ug, where j k ui  abi 8i = 1; : : : ; n, suppose nX ?1 j =1

j xj 

(1)

is a valid inequality for Qn = conv(Q \ fx j xn = k*g). The inequality n X j =1

j xj  + k n

(2)

is valid for Q, where

Zk = maxf

nX ?1 j =1

j xj j

nX ?1 j =1

aj xj  b ? kan ; x 2 Z+n ; x  ug;

k ? ) , and      . Fur = minun k>k ( k??Zkk ) ;  = max0k b. An independent set is a set C such that ai ui  b. A minii2C i2C mally dependent set is a dependent set C such that C nfig is independent for all i 2 C. Note that in the case that u = 1, i.e. P is the binary knapsack polytope, these de nitions are the usual ones.

Proposition 1 Given a minimally dependent cover C and an index i 2 C , the inequality xi  i is both valid and facet-de ning for Pi = P \ fx j xj = uj ; j 2 C n fig and xj = 0; j 2 N n C g

6 7 6 b? P aj uj 7 6 7 where i = 64 j2Cnfaiig 75.

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Proposition 1 and Theorem 1 together suggest the following procedure.

Algorithm 1 1. Choose a cover C 2. Choose a variable i 2 C 3. Start with the constraint xi  i valid and facet-de ning for Pi 4. Apply Theorem 1 sequentially (n-1 times) to nd coecients for the variables in N nfig in turn. The resulting constraint will be a facet for P .

Unfortunately this algorithm requires the solution of a substantial number of knapsack problems. The results below help avoid many of these.

3 General Integer Knapsack Cover Inequalities Algorithm 1 could equally well be applied to the binary knapsack polytope, in which case n ? 1 knapsack problems would need to be solved to obtain a facet. This number can be signi cantly reduced by starting in step 3 with the binary knapsack cover constraint

X

i2C

xi  jC j ? 1

(3)

where C is a minimal cover for the binary knapsack polytope given by P with

u = 1. In this case (3) is both valid and facet-de ning for P \ fx j xj = 0; 8j 2= C g: We now look for ways to generalize (3) to general integers. For any minimal 4

cover we know that not all variables can be at their upper bounds, so X

j 2C

xj 

X

j 2C

uj ? 1

(4)

is valid for the general integer knapsack polytope P . Note that in the binary case (u = 1) this is precisely the constraint (3). Ceria et. al. strengthened (4) in [1] to

 

(5) ui ? a i2C i2C P where  = aj uj ? b and a = maxj2C faj g. We show that this inequality can X

xi 

X

j 2C

be generalised.

Proposition 2 For any cover C and any   0 X

j 2C

daj exj 

X

j 2C

daj euj ? de

(6)

is valid for P.

Note that in the special case  = a1 (6) is precisely (5). When are these constraints facet-de ning for PC = P \ fx j xj = 0; 8j 2= C g? Under some conditions each of the constraints can be facet-de ning, however in general they are not. Below we de ne a stricter notion of minimality of a cover which guarantees that (4) is facet-de ning. Under these conditions, (5) and (4) are identical.

De nition 2 A strongly minimally dependant (SMD) set is a dependent set C P such that uj aj + (ui ? 1)ai  b for all i 2 C . j 2C nfig

Proposition 3 If C is strongly minimally dependant then (4) is facet-de ning for PC .

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We now de ne a less strict condition than strong minimal dependence under which we determine a facet-de ning constraint for PC .

De nition 3 A generalised strongly minimally dependant (GSMD) set is a P minimally dependent set C such that uj aj + (ui ? 1)ai  b for all j 2C nfig i 2 C n fmg, where m = arg minj2C faj g. Theorem 2 If C is GSMD then X

j 2C nfmg

xj + xm 

X

j 2C nfmg

uj + u m ? j

(7) k

is both valid for P and facet-de ning for PC , where = am .

GSMD is a weaker condition than SMD. If C is SMD then C is GSMD with

= 1, and (7) becomes precisely (4). Thus all of the facets obtained from (4) under the SMD condition are also obtained from (7) under the GSMD condition, however (7) can be facet-de ning on occasions when (4), (5) and (6) cannot. When a cover C is GSMD we can use (7) in step 3 of Algorithm 1, and be sure that the result is facet-de ning. However an arbitrary cover may not be GSMD. The following proposition shows that given a cover, it is always possible to nd a GSMD cover from among the cover variables and thus obtain a starting point for Algorithm 1 that is usually better than xi  i . Let xi be de ned for all i 2 C by

8 > > > uj ; > > > < xij = > ui ? 1; > > > > > : 0;

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j 2 C n fig j=i j 2N nC

Proposition 4 Let C be any minimally dependent cover for P that is not SMD. Then for each m 2 C with xm 2= P , the subcover Sm = fi 2 C j xi 2 P g [ fmg is GSMD for the polytope Pm = P \ fx j xj = uj , j 2 C nSmg. Corollary 1 Given a cover C and a GSMD subcover Sm X

j 2Sm nfmg

xj + xm 

X

j 2Sm nfmg

uj + u m ? j

(8) k

is valid for Pm and facet-de ning for Pm \ PC , where = am .

We can use the facet (8) of Pm \ PC as a starting point for the lifting procedure in Algorithm 1. Then we only need to lift in n ? jSm j variables. In the best case Sm = C , and we only lift n ? jC j variables; in the worst case Sm contains a single variable, m, and we start with xm  m .

4 Extended Covers Similar to the de nition for binary covers, we de ne an extended cover

E (C ) = C [ f1; :::; j1 ? 1g where j1 = arg maxj2C faj g. De ne j2 = arg maxj2C nfj1 g faj g.

Proposition 5 If C is GSMD then X

j 2E (C )nfmg

xj + xm 

X

j 2C nfmg

uj + um ?

(9)

is valid for P , and facet-de ning for PE(C ) if Condition 1, de ned below, holds. P aj uj ? 2aj1 + a1  b and uj1 Condition 1 Either j 2C P aj uj ? aj1 ? aj2 + a1  b and uj1 = 1. j 2C

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 2, or

Corollary 2 If C is GSMD, E (C ) = N and Condition 1 holds then the constraint (9) is facet-de ning for P .

In the event that E (C ) 6= N , C is not GSMD or Condition 1 does not hold, then we can still nd a facet of a polytope de ned over a subset of the variables. De ne xp for each p 2 E (C ) n C by

8 > > > ui ; > > > > > > > < ui ? 2; p xi = > > > 1; > > > > > > > : 0;

i 2 C n fj1 g i = j1 i=p otherwise:

Proposition 6 If Sm is a GSMD subcover, uj1 2, and D = fp 2 E(Sm )nSm : xp 2 P g then X

j 2(Sm [D)nfmg

xj + xm 

X

j 2Sm nfmg

uj + u m ?

(10)

is valid for P and facet-de ning for the space

P \ fx j xj = uj ; j 2 C n Sm ; xj = 0; j 2 N n (C [ D)g: If uj1 = 1 a similar result holds. We can also nd coecients for some variables in N n E (C ) without having to lift. If

P aj uj ? aj1 + ap  b for some p 2 N n E (C ) then yp de ned by j 2C 8 > > > ui ; i 2 C n fj1 g > > > > > > > < ui ? 1; i = j1 p yi = > > > 1; i=p > > > > > > > : 0; otherwise

8

is feasible and satis es the constraint (10) at equality, i.e. a lifting coecient of zero for the variable xp is maximal. If Condition 2, de ned below, holds then the maximal lifting coecient for xp is zero for all p 2 N n E (C ).

Condition 2

P aj uj ? aj1 + ap  b, where j 2C

p = arg maxj2N nE(C ) faj g:

Proposition 7 If G = fp 2 N nE (C ) j yp 2 P g, Sm is a GSMD subcover, uj1  2, and D = fp 2 E (Sm ) n Sm j xp 2 P g then (10) is both valid and facetde ning for the space P \fx j xj = uj ; j 2 C n Sm ; xj = 0; j 2 N n (C [ D [ G)g.

5 Summary If a cover C is GSMD and both Condition 1 and Condition 2 hold then X

j 2E (C )nfmg

xj + xm 

X

j 2C nfmg

uj + um ?

is facet-de ning for P . Note that this includes the special cases where C is GSMD and a) C = N (as both Condition 1 and Condition 2 hold vacuously), b) E(C ) = N and Condition 1 (as Condition 2 holds vacuously), or c) C = E(C ) and Condition 2 (as Condition 1 holds vacuously). If none of these conditions hold, but C is a minimal cover then X

j 2(Sm [D)nfmg

xj + xm 

X

j 2Sm nfmg

uj + u m ?

is both valid and facet-de ning for the space

P \ fx j xj = uj ; j 2 C n Sm ; xj = 0; j 2 N n (C [ D [ G)g: 9

We can nd a facet for P by repeatedly applying Theorem 1 to lift in the additional n ? jSm j ? jDj ? jGj variables.

References [1] Sebastian Ceria, Cecile Cordier, Hugues Marchand, and Laurence A. Wolsey. Cutting planes for integer programs with general integer variables. Technical report, CORE, 1995. [2] Zonghao Gu, George L. Nemhauser, and Martin W. P. Savelsbergh. Lifted cover inequalities for 0-1 integer programs: Computation. Technical Report COC-94-09, Georgia Tech., 1995. [3] Zonghao Gu, George L. Nemhauser, and Martin W. P. Savelsbergh. Lifted cover inequalities for 0-1 integer programs: Complexity. Technical Report COC-94-10, Georgia Tech., 1995. [4] Zonghao Gu, George L. Nemhauser, and Martin W. P. Savelsbergh. Lifted cover inequalities for 0-1 integer programs: Fast alogrithms. Technical Report COC-95-xx, Georgia Tech., 1995. [5] Laurence A. Wolsey. Facets and strong valid inequalities for integer programs. Operations Research, 24:367{372, 1976.

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