SIA.M J. Ol'TlM.Vol. 21, .\o. 3. pp. (il7 032

© 2UU Socicly ful Itutu^lliul utui AppUtrtl Muttu'liuiticü

A PROBABILISTIC COMPARISON OF SPLIT AND TYPE 1TRIANGLE CUTS FOR TWO-ROW MIXED-INTEGER PROGRAMS*

QIE HE*, SHABBIR AHMED', AND GEORGE L, NEMHAUSER'

Abstract . We provide a probabilistic coiiiparisou of split and tyiK' 1 triangle cuts for uiixed-iutegerprograms with two rows and two integer vaiiables in terms of cut coefficients and volume cutoff. Under aspecific probabilistic model of the problem paiametere, we show that for the above measure, the probabilitythat a split cut is better than a type 1 triangle cut is higher than the probability that a type 1 triangle cut isbetter than a split cut.

Key words, mixed integer programs, split cuts, triangle cuts, probabilistic conipari8i>n

AMS subject classifîcations. 90011, 90C57

DOI. 10,1137/100797254

1. Introduction. This paper is concerned with valid inequalities for a two-rowmixed-integer program (MIP) with two integer variables of the form

(1,1)

where/ € Q^ \ Z^andW e Q^ \ {0}forallj.LetXdenotethesetofsolutionsto(l.l).Ithasbeen shown (e.g., Andersen et al. (1|) that any valid inequality for conv( A') that cuts off theinfeasible point (a;, y) = (/, 0) is an intersection cut (Balas [2]), corresponding to a convexset I e R with int(I) n Z^ = 0 (i.e., integer-free) and / € int(L). Such a cut is of theform

(L2)

where

(1.3)

K is given by

0 r e rec.cone(L),1 /I > 0 , / + Ar G boundary(Z,).

Furthermore, minimal inequalities of the form (1.2) can be derived from niiiximalinteger-free sets in R^ with nonempty interior. Such sets are of one of the following types(Lovász 112]):

• a split S: c < axi -t- bx2 < c + 1, where a, b,c € Z and gcd(a, 6) = 1;• a triangle with an integer point in the relative interior of each of the edges;

these can be further classified into one of the following three types (Dey andWolsey [10]):

•Received by the editoi-s .June L 2010; accepted for publication (in revi,sed form) March 7, 2011; publishetielectronically .July 19, 2011, This work was supported by NSF giant CMMI-0758234 awarded to the Ge<.)rgiaInstitute of Technology,

http://www,siain.org/journals/siopt/21-3/79725.htnil'H, Milton Stewart School of Incliistrial & Systems Engineering, Georgia Institute of Technology, Atlanta,

GA 30332-0205 (qlie3<agatech,edu, sahiiied@isye.gatecli,edu, geoige.nemhaiiser^yisye.gatecli.edu).

617

618 QIE HE. SHABBIR AHMED. AND GEORGE L. NEMHAUSER

- a type 1 triangle Tj : a triangle with integer vertices and exactly one integerpoint in the relative interior of each edge;

- a type 2 triangle T^: a triangle with more than one integer ¡xjint on one edgeand exactly one integer point in the relative interior of each of the othertwo edges;

- a type 3 triangle Ty. a triangle with exactly one integer point in the relativeinterior of each edge and nonintegral vertices;

• a quadrilateral Q with exactly one integer point in the relative interior of eachedge such that the four integer ])oints foriii a parallelogram of area one.

Inequalities of the foriu (1.2) corresponding to the above sets are called split, (tyjje 1,2, or 3)triangle, and quadrilateral cuts, resp(>ctively. From the niaxiniality of the above integer-freesets, it follows that any uoiitriyial facet of coiiv(yY) is either a split, triangle, or quadrilateralcut 111, [61.

Split cuts are the classical Goniory-niixecl-iateger (GMI) or mixed-integer roundingcuts |13|. Recently there has been a great deal of actiyity in comparing triangle and( Hiadrilateral cuts to split cuts for two-row MIPs. Ba^u et aJ. [4| compared the rank-1 closure! (the coiiyex set obtained by adding in a single round all possible cuts fromthe fauiily) corresponding to the three cut classes. They showed that the triangle closure(c onsidering all three types of triangle cuts) and the quadrilateral closure are containedill the split closure, suggesting that triangle and (¡uadrilateral cuts are in some sensestronger than split cuts. Dey and Louyeaux |9| showed that type 2 and type 3 trianglecuts and qtiadrilateral cuts haye finite split ranks (i.e., such a cut can be constructed yiaa finite sequence of split cuts), while only type 1 triangle cuts can haye infinite split rank.However, empirical studies demonstrating the success of triangle and quadrilateral cutsill comparison to split (or GMI) cuts have been limited. Espinoza ¡11] reported somesuccess with intersection cuts generated from some classes of integer-free trianglesand quatlrilaterals. Basu et al. |3| considered strengthened yersions of a class of type2 triangle cuts and showed that combining these cuts with GMI cuts gives somewhatbetter performance than GMI cuts alone. Dey et al. 18| presented computational resultson raiicloiiily generated uiultikiiapsack instances and showed that a subclass of type 2triangle cuts can close more gap than GMI cuts.

We present a probabilistic comparison of type 1 triangle cuts and split cuts. Spe-cifically we address the question. What is the likelihood that a split cut will dominatewith respect to cut coefficients or cut off more yolunie from the linear programmingrcîlaxatioii than a type 1 triangle cut for an arbitrary instance of the two-row MIP(1.1), giyeu a specific probability distribution of the problem parameters? Our analysisreveals that, for the given distribution of the instances, such likelihood is high. The ana-lysis also suggests some guidelines on when type 1 triangle cuts are likely to be moreeffectiye than split cuts and yice versa.

2. Setup, la this section, we discuss the distributional model for instances of thetwo-row MIP (1.1) and the two metrics used in our probabilistic comparison of type 1triangle and split cuts.

Without loss of generality (by translating x by [/J and scaling yj by ||r'||2), we canassume that 0 < / , < 1 for i — 1.2 and ||r^||2 = 1 for all j in (1.1). Then r[ = cos 6j andr-l, = sin ffj. where Ö, is the angle between r and the positive Xj-axis.

The input model. We consider instances of (1.1) where / is a realization of a randomvector f that is uniformly distributed with support U ••= (0, l)'^, i.e., the open unit squareill the ])laiK'. and Oj is a realization of a raiuloni variable Oj that is uniformly distributedoyer [U, 2;?) for all j . (When / is on the boundary of c\{U), the coefficients for some

COMPARISON OF SPLIT AND TRIANGLE CUTS

Si ^ ,

(0,1)

(0,0)

X2

52

Xl

(1.0)

T2(0,1)

(0,0)

T l

619

T4

(1.0)

T3

(a) Two simple splits. (b) Four type 1 triangles.

FIG. 2.1. The integer-free bodie» selected fur comparison.

split and type 1 triangle cuts can be +00, causing technical issues in their comparison.)Moreover, f. 0/ öj are independent random variables.

Under this probabilistic input model, the cut corresponding to the integer-free bodyL is of the form

(2.1)

where the cut coefficient i/fi(f. 6j) of variable j/j is a random variable depending on f and0j and is given by (1.3). Our analysis compares the random ctit (2.1) when the set L is asplit or a type 1 triangle. To guarantee that / S int(L) with probability 1, we consideronly integer-free splits and type 1 triangles that contain U. This ensures that the in-equality (2.1) corresponding to L cuts off the infeasible point (/. 0) for every realization/ of f. There are only two splits containing U (the valid inequality corresponds to theGMI cut for each row of system (1.1)), and there are only four type 1 triangles containingÍ/, with one of the four vertices of /7 as its right-angle vertex (see Figtire 2.1).

There are various criteria for comparing cuts. We choose two criteria suitable forcomparing two individual cuts rather than cut families. The first one is based on cutdominance.

DEFINITION 2.1. Suppose Ci : X JLj ajTjj > 1 and C2: Z^j=i b^y^ > 1 are two distinctvalid inequalities for system (1.1); then Ci dominates C2 ifüj < bj forj = 1 k withat least one of the inequalities being strict. We use C] >-o C^ to denote that C) dom-inates C2.

If Cl )~p C2, then C2 is implied by Cj. The second criteria is based on the volumecut off by the cuts from the linear relaxation.

DEFINITION 2.2. Suppose Ci : Y.'j^i a^y^ > 1 and C^ : X]j=i bjy^ > 1 are two distinctvalid inequalities for system {I.I). LetXip be the linear relaxation of {\.\). Then Ci>~vC2 if Cl cuts off more volume than C2 from X^p; i.e.,

> vol

620 QIE HE. SHABBIR AHMED. AND GEORGE L. NEMHAUSER

We probabilistically compare split and type 1 triangle cuts with respect to these twometrics.

3. Conditional probabilities with respect to / . We first analyze the condi-tional probabilities of split cuts dominating and cutting off more volume than trianglecuts with respect to the fractional point / . This analysis helps with computing the totalprobabilities in .sec:tion 4, and it also provides some insight into values of / for which type1 triangle cuts are likely to be Vjetter than split caits and vice versa.

3.1. Cut coefficient comparison. Without loss of generality, we select one splitfrom tlie two splits and one type 1 triangle from the four type 1 triangles in Figure 2.1.The analysis easily extends to the other splits and type 1 triangles by synmietry. Thechosen split 5] and type 1 triangle Tj are shown in Figure 3.1. The split ^i is defined byAD and BC, and the type 1 triangle Tj is defined by AAEF. Suppose that Cg, is thesplit cut for Si and Cr, is the triangle cut for Tj. and recall that \l/^^{t,Oj) andi/';'i(f.öy) are the corresponding (random) cut coefficients for variable yj. We usePv[\¡f¡'^{{.Oj) < \l/g^{f.0j)\f] to denote the conditional probability of the eventiA,.(f.öy)<iAs,(f.ö>) whenf = / .

LEMMA 3.1. For each j = l k, Pr[iA. . (f.Öy) < V^s,(f.Öy)|/] = a ( / ) , PrlV^^-(f,ffj) = ilfr,{f.0j)\f] = ß{f). and Pr[iA,, (f.ö,-) ' fr,{f-0j)\f] = Y{f). where

arceos arceos«(/) =

•2nß{f) =

2n

arceos ,... ['*'.'• ' ., _.,. + arceos

Yif) =

Proof. Since Oj (j = 1 k) are independent and identically distributed (i.i.d.),we need only prove the result for some j . P or simplicity, we supress the index j here andprove it for some ray r = (™^ 0-

The triangle

F

D

A

VcKM

B

. The split S\

R

\N

E

u. 3.1. CumiiutmyPi-[fs{i.O)

COMPARISON OF SPLIT AND TRIANGLE CUTS G21

As shown In Figure 3.1, Í7 is the unit square with vertices A,D, C, and D, and 0 isthe fractional point /. Let OR be the ray defined by / -I- kr. Let OM be the lino i>arallelto the Xj-axis that intersects 5 and Ti at M and N, respectively. Then 0 is the anglebetween OM and OR in the counterclockwise direction. Let the symbol /. denote anangle less than n. Since the probability density function of Ö is j^l{0€ [0,2.;r)). V)vthe law of total probability.

(3.1) i: 2nde.

where I{A) is the indicator function of event A.By (1.3), i,sM-O)=^, where / +As,(™J^) e boundary(5), and xlfr,{f-O) =

j ^ , where / + ^ ,( 1 ) € boundary(r,). Therefore, V.v,(/.Ö) - V^r,(/.ö) if the ray

/ + ' (s"ne) ^i** ^^^ boundary of T, first, and ^T,{f-(^) < i^s^f-G) if the rayf + ki'^g) hits the boundary of S^ first. When d&[Q.zMOC) or 0G{2n — ZMOB.27T), OR is contained in the cone bounded by OB and OC and hitsthe boundary of S first, so Vr, (/- <9) < V s, (/- 0). Similarly, when d e {^MOC, /.MOE)or 9 £{2ii - /.M0A,2n - ^MOB), V^s.l/.ö) < Vr,(/.ö); when d £[/.M0E,2n-^M0Ä\ or G is equal to zMOC or 27r - /.MOB, Vs' (/-ö) = V/' (/.<9)- Therefore,by (3.1),

= ^¡fr,{ï•e)\f] =

In ABOC, \0B\ =the law of cosines.

fl \0C\ = T ^ , and \DC\ = 1. By

cos ^BOC =2\ÓB\\ÓC\

Therefore,

Similarly, Z/IOF = 2jr^(/), and zAOB -|- /COE = 27iy{f). Therefore,

Ü

Lemma 3.1 provides the probabilities that a single coefficient of the split cut Cg^ issmaller than, equal to, and larger than that ofthe triangle cut CT¡ as a function of/. Tocompare the other split and type 1 triangles in Figure 2.1, we need only change /] to1 - /i or /2 to 1 - ¡2 in «(/), ß{f), and yif) by symmetry. The following theorem givesthe conditional probability that the split cut Cg^ dominates the triangle cut C^, withrespect to / and the number of continuous variables k.

• J • ' •

G22 QIE HE. SHABBIR AHMED. AND GEORGE L. NEMHAUSER

THEOREM 3.2. i ' •

PT[CS, y O Cr, I/] = ißif) + vif)]' - [ßif)]'.

Proof.

where the second equality follows from the assumption that 6j {j = I A:)are i.i.d. D

Given integer-free bodies L| and L>, let

The following corollary follows from Theorem 3.2.CORDLLAHY 3.3 .

= {/G Î/: and ,. 5,) =

By symmetry, after appropriately translating / , we can similarly describe the regionsTi) and Rp{ Tj. 5,) for i = 1 . 2 and j = 1.2.3.4 corresponding to any of the two

splits and four type 1 triangles in Figure 2.1. Figure 3.2(a)-(b) shows the regionsi~ij=i RDÍ^I. TJ) and nj^i /?D(52. TJ), respectively, shaded in black. The white regionsin these figuri-s indicate UJ^, Ro{ Tj, Si ) and u'^i R[){ Tj. S2), respectively. Since the un-ion of the two black regions covers the unit square, there is no / for which a type 1 trianglecut Cr satisfies that Pr[C7. )~D Cs,\f] > PT[CS, >-D Crlf] {i = 1.2). It follows from thediscussion above that if we are allowed to add only one cut, when/ ëflj^i R[){Si. Tj),wewould select ^ i , and when / €UJ^j Rp{Tj, Si), we would select 59.

3.2. Volume comparison. In this section, we compare cuts based on the volumecut off from the linear relaxation of system (1.1). First we describe how the volume cutoffis computed. , ,

01 0.2 0.3 0,4 0,5 0.6 0.7 0,8 0,9 1

(a) The region nj Tj). (b) The region n^^,

Flo. 3.2. The ityiuii.

COMPARISON OF SPLIT AND TRIANGLE CUTS 623

Suppose that C: X *Li ijVj > li with a > 0 for all j , is a valid inequality for system(1.1). Consider the linear relaxation of (1.1)

(3.2) X =

Let Xip be the set of feasible solutions of system (3.2) and

Let vol(5c) denote the volume of the polyhedron 5^, which is also the volume cut offfrom S by the valid inequality C. The following lemma gives the volume of Sc-

LEMMA 3.4.

-1-00 if 3j such that aj = 0,otherwise.(3.3)

where a is a constant depending on the rays r^ r*.Proof. When ßj = 0 for some j , Sc is an unbounded polyhedron, and

vo\{Sc) = 4-00. When Oj > 0 for all j , SQ is a /c-dimensional polytope containing(/,0). Let

Proj¡,(5c) = {2/ G K ' : 3a; e R2 such that {x, y) e Sc}

be the projection of Sc onto the y space. Projj,(5(7) is /c-dimensional simplex with (J,^ e^ ... , ^ e*" as its (A: -I-1) vertices, where e^ is the jth unit vector. Therefore,

vol(Proj,(Sc)) = - - • • • - = — i .n! «1 «j. n! nî=i «j

Each point in 5^ is just an affine transformation of a point in the simplex Proju(5c), soYo\{Sc) and vol(Proj^(5(7)) differ only by a factor a depending on the rays r' r \Thusvol(5c.)=-rrñ—. •

By Lemma 3.4, it suffices to compute the product of cut coefficients when we com-pare cuts based on the volume cut off from the linear relaxation.

Now consider the split Si and type 1 triangle Tj as in section 3.1. As before, theanalysis easily extends to another pair of split and type 1 triangle bodies by symmetry.Note that for fixed / € (0,1)^, \¡/'j'^ if'^j) > 0 with probability 1. Moreover, since Oj iscontinuously distributed, Pr[3j s.t. i/s,{f,Oj) = 0] = Pr[3j s.t. Oj = f or ^] = 0.

THEOREM 3.5.

hProof. From Definition 2.2, Lemma 3.4, and the fact that \lfs^{f.Oj) > 0 and(/iöy) > 0 with probabihty 1, we have that

624 QIE HE, SHABBIR AHMED, AND GEORGE L, NEMHAUSER

yy Cr,\f] = Pr[vol(5c.,_) >

Next we analyze the asymptotic behavior of the probability Pr[(7,v, y\- C-y \f\ asthe number of continuous variables k increases. Before presenting further results, we givetwo technical lemmas.

LEMM,\ 3.6.

/ " In cos xdx = — arid / ( In cos x)'^dx < cx).

Proof. See Appendix A. D ^

To simplify the notation, let Xj{f) = In .'i .'g, for every j = I k. Note that

for a fixed / e (0. l)'^, the random variable Xj{f) is uniquely determined by dj. The as-

suniptioii that Oj, for j = 1 k, are i.i.d, implies that Xj(f), for j = 1 k, are alsoi,i.(l. Let ßf = E[Xj{f)] and aj = Yt\r[Xj(f)] for any j = 1 k.

LEMMA 3.7.

\fj.f\ < 00 and cr'j < c«. '•'"

Proof. See Appendix B. DNow we ])resent the asymptotic result on the probability that a split cut cuts off

more voluiiic than a type 1 triangle cut as the number of continuous variables increases.THEDHEM 3.8.

f l ifßf<0,{ 1/2 ^f^Xf = O.[Q ifHf>Q.

limil-—KX

Proof From Theorem 3.5. we know Pr[Cs^ •^y Cr,\f] = Pr[J2j^i Xj{f) < 0].

Since Xj(f) (j = 1 k) are i.i.d., we can apply the Weak Law of Large Numbers

and the Central Limit Theorem. Let Xi^{f) = ' \, '—. Since \\i¡\ is finite (Lemma 3.7),

by the Weak Law of Large Numbers, linij.^,^ Px\\Xk(f) - /Xy| < f] = 1 for any f > O.We

consider three cases: yi¡ • Ü, /i^ > Ü, and fij = Q.

(1) /X| < 0. Choose e = - ^ . Then

0] > Pv[X,{f) - ß, < €\ > Pv[\X,if) -

Thus, hill iiif,^^ Pr[E'=i Xjif) <0]> lim inf,^^ PAl^kif) - M/l < e] =

/ ) - M/l ' . e] = 1. Since lim sup ,^^ Pr[EÍ=i Xj(f) < 0] <

(2) ßj> 0. Choose e = ^. Then

GOMPARISON OF SPLIT AND TRIANGLE CUTS

= Pr[X,{f) < 0] < Pi[X,if) -

< Pr[|X,(/) - M/l > e].

625

-e]

Thus, lim supt._^^ PrEi=i ^jif) < 0] < um sulim,_^ Pr[|Â:,(/) - IXf\ >e] = 0. Since lim

E Î . , , ( / ) ] ;(3) /i^ = 0. From Lemma 3.7, aj is finite. By the Central Limit Theorem, !/T'

conyerges to the standard normal random yariable A/'(0.1) in distribution.Thus,

lim Pr \y Xj{f) < ol = lim Pr

Define Rv[Si.Ty) = {/G U\\x¡<Q} and ßv,(Ti,5i) = {/€ C / : M | > 0 } . Then,Ry{Si. Ti) indicates the region where the split cut Cs^ cuts off more yolume thanthe type 1 triangle cut Cj-^ with probability close to 1 when A: is large, andRY{TI,SI) indicates the region where the type 1 triangle cut Cj; cuts off more yolumethan the split cut C5, with probability close to 1 when A: is large. Even though 6j has asimple distribution, it is difficult to analytically compute ¡Xj. Howeyer, we can estimate¡Xj by Monte Carlo simulation for a giyen value of / , and we can identify the regionsRy{Si,Ti) and Ry{Ti,Si). The black and white regions in Figure 3.3 indicateRy{Si. Ti) and Ry{Ti, Si), respectiyely. These haye been identified as follows. Firstwe randomly generate 10"'' fractional points / in U; then for each / , we independently

generate 1000 dj uniformly from [0,27r) and check if the sample mean of In . \ , A is less

or greater than zero to identify if the corresponding / is in Ry{Si, Tj) or Ry{Ti. Si).The area of the black region is approximately 0.9. Unless /i is close to 1, the split cut Cs

Kiü. 3.3. The shape ofRv{St, T,) and Iiy(T¡,S¡).

626 QIE HE. SHABBIR AHMED, AND GEORGE L. NEMHAUSER

cuts off more volume than the type 1 triangle cut C'¡-^ with probability close to 1 when itis large, and therefore Cg^ is preferred.

4. Total probabilities. In this section, we use the conditional probabilities fromthe previous section to compute coefficient dominance and volume cutoff probabilitiesfor split and type 1 triangle cuts when / is random. As l)efore, we focus on the split cutCjj^ and the type 1 triangle cut Cj^. and note that the analysis and conclusions extendto another pair of split and type 1 triangle bodies by symmetry. The total probabilityanalysis provides some insight on how these cuts are likely to perform when no informa-tion about the instance is available.

4.1. Cut coefficient comparison. By the law of total probability,

(f.Ö,.) < iAr,(f.Ö;) Vj]

PrlV,-, {Î.Oj) < Vy. {ï,ej)\f\Yd^{f),

where <!>(/) is the cumulative distribution function of f. and the last equality followsfrt)ni the fact that 9j iue i.i.d. for J = 1 k. Recall that the conditional probabilityPr[V'.S|(f.öy) • V r, (f'^y)!/] is given in Lemma 3.1. The following theorem describes theperformance of the split cut Cs^ and type 1 triangle cut Cr^ when there is only onecontinuous variable.

THEOREM 4.1. Ifk=l, then

yo CrJ «0.426 > 0.25 =

Proof Note that ADOC. ZAOB, and zCOF are shown in Figure 3.1. Then

Similarly,

zAOD+zCOF—

The proof then follows from Lemma 4.2. DLKMM/V 4.2.

arid

C ^ 0.176.

Proof. See Appendix C D 'Now we consider the case k > \.THEOREM 4.3. For any k, Pr[Cs, ^D CT,\ > Pr[C'y, ^ß C5,

COMPARISON OF SPLIT AND TRIANGLE GUTS

Proof.

Px[Cs

627

f

The second equality follows from symmetry since / is uniformly distributed in(0.1)'-. D

Theorem 4.3 states that a single split cut is more likely to dominate a single type 1triangle cut under our probabilistic model no matter how many continuous variablesthere axe in system (1.1). We also use Monte Carlo simulation to estimate the magnitudeof the probabilities that one cut dominates another. The result is shown in Figure 4.1.

From Figure 4.1, although Pv[Cs, ^D CyJ > Pr[Cv, >[) Cs,] for all fc, both prol)-abilities are very small when fc > 5, indicating that it is unlikely that one cut totallydominates another when there are many continuous variables.

4.2. Volume comparison. In this section we estimate Pr[C5| y y Cr¡] with re-spect to the number of continuous variables fc. Recall that Pr[C5| y y Cr,] =

Prin'-i ,''''I,ai < 11- We use Monte Carlo simulation to estimate the above probabil-

ities as follows. For each fc S {1 1000}, we randomly generate N = lO'' samples offi,f2,Oi,...,0i. according to our probabilistic input model. The probability

' ' • - ' I ] is then estimated by the proportion of the A samples with

4 5 6 7Tlio nmiilKU' of VAytr, k

10

FIG. 4.1. -^ >~p CgJ with respect to tfie rmrriber of rays k.

628 QIE HE. SHABBIR AHMED. AND GEORGE L. NEMHAUSER

— 0.65

20 25 30

vfu'iahlis k

Vu:. 4.2. Estimated Pr[n)=i jf^jfe

200 400 600 800 1000Nuiiiboi' of continuous vuriiiblos k

" " ' • ' ' •to k.

- ''^' estimated probabilities with respect to k are shown in Figure 4.2.The estimated probabihty that Cg^ cuts off more volume from the linear relaxation thanCx, increases as the number of continuous variables increases, converging to approxi-mately 0.9. To explahi this, note that ;, -^.j,,

Since PrfC^i >-i' Cx,\f] is bounded, l\y interchanging limit and integral and applyingTheorem 3.8, we have , .

= Pr[f 6

where I{A) is tfie indicator function of event A and R\-{Si. T^) is defined in section 3.2.Figure 4.2 presents Pr[C5| "^-y Cx,] with respect to the number of continuous variablesk (in two different scales). Recall that, as observed in Figure 3.3, the area of RY{SÍ. T¡)

is approximately 0.9, which coincides with the observation in Figure 4.2. We can con-chide Cg, is more likely to c;ut off more volume than Cx, when k is not too small, givenany instance of (1.1) with parameters distributed according to our probabilistic inputmodel.

5. Conclusions. In this paper, we propose a probabilistic model to compare splitc uts and type 1 triangle cuts. The analysis can be extended to other classes of facetdefining intersection cuts where the corresponding integer-free body contains the unitscjuare, such as type 2 triangles and quadrilaterals containing U. In particular, for thecomparison of volume cutoff, results similar to those in Theorem 3.5, Lemma 3.7, andTheorem 3.8 can be derived, since the type 2 triangles and quadrilaterals are allbounded, and the corresponding cut coefficients are strictly greater than zero. Althoiigliit might be difficult to compute the associated probabilities analytically, we can stillestimate the probability numerically and obtain regions of / where one cut dominates

COMPARISON OF SPLIT AND TRIANGLE CUTS 629

another or cuts off more volume. The analysis for type 3 triangles is much less obvioussince such a triangle does not contain U. Another interesting question is how to extendour probabilistic analysis on cut comparisons to the model with explicit bounds on the yvariables. In this model, the region cut off from the LP relaxation by an individual cut isnot always a simplex, and therefore the volume comparison Incomes more complicated.It would also be interesting to study how to extend our analysis on volume comparisonto multiple rounds of cuts. Finally we note that, recently, after the first submission ofthecurrent paper, two groups, Del Pia, Wagner, and Weismantel [7| and Basu, Cornuójols,and Molinaro [5], have also conducted probabilistic analyses of the strength of variousfamilies of two-row cuts, using different probabilistic models and comparisoncriteria.

Appendix A. Proof of Lemma 3.6. By substitution of variables,

^ln COS xdx = fl\n sin xdx. Then,

í ' In sin xdx = In I 2 sin - cos - ) dx0 7() V 2 2;

= / " In 2dx-\- I In sin -dx+ I In cos —dxJa Jn 2 ./o 2

+ 2 / ^ In sin ydy + 2 ' \n cos zdzJo Ja

I In sin ydy + 2 / hi sin ydyJa ' ./i

/ In sin ydy.Ja

nln22

;r I n 2

2

Ä I n 2

+ 2

+ 2

Therefore, / g in sin xdx = — " 'j* ,

By substitution of variables, yo(ln cos x)'^dx = fl(ln sin x)^dx. Since 0 < sin x <

X for 0 < I < 1 , then 0 < (In sin x)"^ < (In x)^. Moreover, / ( I n x)^dx = x(\n x)^—

2xlnx + 2x+c/, where d is a constant. Thus, / |(ln x)'^di = f (lnf)^ - n-In f +

;r < 00. Therefore, / |(ln sin x)^dx is finite. D

Appendix B. Proof of Lemma 3.7. To simplify the notation, let

= E[ln - E[ln

By (1.3), ifT,if'Oj) is bounded and strictly positive for fixed / G (0,1)^. Thus,

In ifr,{f'Oj) is bounded and E[ln \lfr,{f-Oj)] is finite. By (1.3), i/.s,{f.Oj) = •¿-, where

Z + ' i (s"neO ^^^^ ^^^ boundary of the split S^. Thus, fi+^s^ cos Oj = 1 when

OJ e [0,f) and Oj £(^,2JT), and / i 4-As, cos Oj = 0 when Oj e (f , ^ ) . Therefore.

^sM,0j)="-^ when Oj e [O,f) and Oj £ ( f ,2;r), and V^s,(/,Ö,) = _ Í 2 ^ when

Oj G (f , ^ ) . The probability density function of öy is ^IiOj e [0.2;r)). Therefore,

630 QIE HE. SHABBIR AHMED. AND GEORGE L. NEMHAUSER

bj]h\ w s { f . 0 i ) \ = I h i \l/^ i f . 0 : ) — f i é ' ,

1 r fi eos Oj rf - cos Oj r2K COS 0j ]

" 2^ [./„ " 1 - / , ^ " .A '" / i "^ ^ i f " 1 - /i '^^'l

= -— / " In cos 0,d0. - / " lii(l - f,)d0, + / • l n ( - cos 0Ad0.!

- / " In /irföj + / hl cos öji/öj - / lii(l - fi)d&j

= ¿ [4 ^ hi cos ö^fiö, - TT 111 /, (1 - / , ) ] .

By Lemma 3.6, ß\n cos 6» (/öj = - ^ . Therefore, E[\n \l/s,{f.6j)] is finite andßf < OO.

It remains only to yerify that Oj is finite. Since CT? = E[{Xj{f))'^] — ¡xj, we need toverify that E{{Xj(f))'^] is finite. By expanding E[{Xj(f))'%

= E[(h. V.s',{/.ö;))-] - 2E[ln V'i'.i/.Ö.Ohi ^r,(/.Ö,-)] + E[(lu

Since we have shown that In ^T,U<OJ) '^^ bounded and E[ln ^s,{f'Oj)] is finite forfixed /, the last two terms in the above equation are finite. For the first termE[{\\\ ^j/^^{f.Oj))-], substitute the formula for In \l/<j^{f.0j) and expand it as anintegration.

/•f/ COsé^A-í 1 , , y f / - C O S ( 9 . \ 2 1 f2n / COS 6», V-' 1

,/o V l - / i / -TT i f V / i / 27r - . / A V 1 - / 1 / 2;r

= - ! - 4 / * ( l n c o s ( 9 j ) 2 í t ó l j - 4 1 n / i ( l - / i ) / " I n cos 6i,rfö,27r L ./() ./()

By Lemma 3.6, / |(ln cos Oj)'^d0j and y|ln cos OjdOj are both finite. Thus,

Therefore, Var(Xj) is finite. D

Appendix C. Proof of Lemma 4.2. Indeed, since O is uniformly distributedover U,

In A COD. \ÓC\ = v/(r^^77?TTr^^7¡F, \'ÖD\ = ^f\ + (l - f-^f, and |C7i)| = 1. Bythe law of cosines.

COS zCOD =

COMPARISON OF SPLIT AND TRIANGLE CUTS

• -f- \ÖC\^ - lÔDp

631

2\0D\\0C\

Therefore, zCOD = arceos ^^^¡;^;-:^^;Xu_,^,y ^'^"

zBOC = arceos

By substitution of variables.

-'Z.COD

fí=l-g¡.f2=9¡

nr-nr--nr-nr-nr

arceosdf.dfi

arceos

2n

arceos

2n

arceos

2n

Similarly, we can show

Je .le

Therefore,

nr 2n

2n ' ' •"•

i-< zBOC + /.COD + zDOA + zAOB

Thus,

r\-c2n

= 0.25.

632 . QIE HE, SHABBIR AHMED. AND GEORGE L. NEMHAUSER

Now we compute f^-^^^

arceos= lim / / ^MÉÏIEimÎ2Em,if^,if., , 0.176.

111 the final step, we used the MATLAB function "dblquad" with e = 10"** for the nu-merical calculation. D

Acknowledgments . The authors would like to thank an anonymous associate edi-tor aiul three auoiiymous referees for their thoughtful comments.

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