New Facets of the Linear Ordering Polytope 1

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8, 14] derives facet-de ning inequalities, which are violated by current ... The rotation method for the linear ordering polytope generalizes facets induced.
New Facets of the Linear Ordering Polytope 1 Bolotashvili G., Kovalev M.

2

and Girlich E.

3

Abstract The linear ordering problem has many applications and was studied by many authors

(see [5, 9, 12] for survey). One approach to solve this problem, the so-called cutting plane method [8, 14] derives facet-de ning inequalities, which are violated by current nonfeasible solution and adds them to the system of inequalities of current linear programming problem. We present a method (rotation method) for generating new facets of polyhedra by using already known ones. The rotation method for the linear ordering polytope generalizes facets induced by subgraphs called m-fences, Mobius ladders and Zm -facets introduced by Reinelt [12], (m; k)fences introduced by Bolotashvily [1] and t-reinforced m-fences introduced by Leung and Lee [10]. We introduce ten collections of inequalities representing facets of the linear ordering polytope. Among them there are three that coincide with earlier known ones: m-wheel-facets introduced by Reinelt [12], augmented m-fences introduced by McLennan [11] and augmented t-reinforced m-fences introduced by Leung and Lee[10].

Key words. linear ordering polytope, facets, linear ordering, ranking. AMS subject classi cation. primary 52B12, secondary 90B10

Partially supported by Fundamental Research Foundation of Republic Belarus and DAAD Faculty of Applied Mathematics and Informatics, State University of Belarus, 220080, Minsk, Belarus Email:[email protected] Fax: 00375-172-207914 3 Faculty of Mathematics, Otto-von-Guericke-University, PSF 4120, 39106 Magdeburg, Germany Email: [email protected] Fax:049-391-67 11171 1

2

1

1 Introduction A linear ordering of an n-element set N is a bijection  : f1; 2; : : : ; ng ! N . The linear ordering polytope Pn is a convex hull of n! points in Rn ?n . Each of these points one-to-one corresponds to some  = ((1); :::; (n)) by the following rule: xij = 1, if ?1 (i) < ?1 (j ) and xij = 0, if ?1 (i) > ?1 (j ); i 6= j . 2

Let G = (N; A) be a complete directed graph (digraph) with node set N and arc set A = N  N (without loops). A directed subgraph (N; T ) is a spanning tournament, if for every pair of distinct nodes u; v 2 N exactly one of arcs (u; v) and (v; u) is in T . Given a linear ordering  of the nodes N of a digraph, the arc set f(u; v) : ?1 (u) < ?1 (v)g forms an acyclic spanning tournament on N ; conversely, an acyclic spanning tournament (N; T ) induces a unique ordering of N . Thus the linear ordering polytope Pn is the convex hull of the incidence vectors of the acyclic spanning tournaments on N . The following system of inequalities and equations

xij  1 xij + xjk + xki  2 xij + xji = 1

i 6= j; i; j 2 N; i 6= j; i 6= k; j 6= k; i; j; k 2 N; i 6= j; i; j 2 N:

(1) (2) (3)

de nes the relaxation polytope Bn of the polytope Pn . Every point x is a vertex of Bn . In addition to these integer vertices, the polytope Bn for n  6 has noninteger vertices (note that in earlier publications it was believed that Pn = Bn (cf. [4])). The simplest classes of noninteger vertices are [1]: 1) x0is iq = x0jsjq = 21 ; x0isjq = 0; x0jq is = 1; s 6= q; 1  s; q  m; x0isjs = x0js is = 12 ; 1  s  m; 1 00 2) x00 is iq = xjsjq = 2 ; s 6= q; 1  s; q  m; 00 x00 is jq = 0; xjq is = 1; (jq ; is ) 2 B; 1 00 x00 is jq = xjq is = 2 ; (jq ; is ) 62 B; where I = (i1 ; : : : ; im ); J = (j1 ; : : : ; jm ) are arbitrary disjoint subsets of N , 3  m  n2 , k is such that m + 1  0 (mod k), and

B=

m k[ ? [

1

s=1 q=1

f(js q ; is); (js?q ; is )g; jm p = jp; j ?p = jm?p ; p = 1; : : : ; m: +

+

2

1

+1

It is proved in [1, 2, 12] that the inequality m X

s=1

x is js ?

m X m X s=1 q=1 q6=s

xisjq  1

(4)

cuts o the noninteger vertex x0 and de nes a facet of Pn , whereas the inequality m X

s=1

x is js ?

m kX ? X

1

s=1 q=1

(xis js q + xis js?q )  m k+ 1 ? 1 +

(5)

cuts o the noninteger vertex x00 and de nes a facet of Pn [2]. Every facet-de ning inequality for Pn can be represented in normal form [12, 13], i.e. in a form with nonnegative coecients. Indeed, facet-de ning inequality (4) can be rewritten as (since xij + xji = 1) m m X m X X x(F ) = xis js + xjq is  m2 ? m + 1 (4') s=1

where F =

s=1 q=1 q6=s

Sm (f(i ; j )g Sf Sm f(j ; i ) : q 6= sgg).

s=1

s s

q=1

q s

In this form it has been constructed in [12]. Inequality (5) with nonnegative coecients has the following form:

x(F 0 ) =

S Sm

m X

s=1

xis js +

m kX ? X

1

s=1 q=1

(xjs q is + xjs?q is )  m k+ 1 ? 1 + 2m(k ? 1) +

(5')

where F 0 = B f (is ; js )g. s=1 If an inequality ax  a0 is in normal form, then it induces a weighted digraph corresponding to the non-zero coecients of the inequality and with w(i; j ) = aij , where w(i; j ) is the weight of arc (i; j ) in the digraph. Conversely, a weighted digraph can be understood to induce an inequality by associating a coecient of w(i; j ) with every arc in the digraph, 0 for all other arcs, and appropriately de ned righthand side. Digraphs induced by facet-de ning inequalities (4') and (5') are called m-fences and (m; k)-fences, respectively. Figure 1a illustrates an m-fence for m = 4, Figure 1b illustrates an (m; k)-fence for m = 5; k = 2. Evidently, if m is odd, then an m-fence is a special case of an (m; k)-fence when m + 1 = 2k. Throughout the gures in the paper, arcs shown without numerical labels should be interpreted as having weight equal to 1. Using substitution xij = 1 ? xji (it follows from equations (3)) we can replace every facetde ning inequality ax  b by the equivalent inequality b  ax. For example, inequality (4) can be rewritten as m m X m X X m ? 1  xjs is + xiq js ; (4") s=1

s=1 q=1

s6=q

3

i1iH iP YP

i iH ii ii 1  *  *  Y   P  H H  I@ HPHP?P@IPHH?@I ? @ @??HHHP@?P?PHHH@?@? @@?HH@@PPP?HH@ ?   @ ?? HH@ ?? PPPHH@ ?    ? ?   P ji i j ji ji 1

2

3

4

2

3

4

a) 4-fence

: ii ii i i  X  X  I@ ? @IXXXX? @I ? @I @  @@?? @?? @??XXXX@?X@? @ ? @ X@XXX??@ ? ? @ @  ? @j?i? @j?i?XXX@X ji? @j?i? ji? yXXX i2i i1iX 1

2

3

4

5

3

4

5

b) (5,2)-fence Figure 1: and inequality (5) can be rewritten as

m ? m k+ 1 + 1 

m X s=1

xjsis +

m kX ? X

1

s=1 q=1

(xis js q + xis js?q ):

(5")

+

Lemma 1 (Trivial lifting lemma [12, 13]) Facet-de ning inequalities of Pn de ne also facets for

Pn , n1 > n. 1

2 The rotation method We present now a new rotation method for generating new facets of the linear ordering polytope. The idea of the method was introduced in [3]. Let P be a polyhedron in Rn with the set of vertices vert P and be an ane mapping of Rn into itself. If vert P = vert (P ) then is called a rotation mapping of P . Evidently, the rotation mappings realize one-to-one mappings of the facet set and corresponding facetde ning inequalities onto themselves. Hence, having a facet-de ning inequality ax  a0 for the polyhedron P , we have a facet-de ning inequality a (x)  a0 for the same polyhedron P .

Remark 1 If a facet F is de ned as convfx ; : : : ; xs g, then (F ) = convf (x ); : : : ; (xs )g. 1

1

4

A trivial rotation mapping for linear ordering polytope Pn is arc reversal mapping which de ned as xij = xji. This mapping transforms every facet ax  a0 into the facet bx  a0 , where bij = aji for all 1  i; j  n [12, 13]. The trivial rotation mapping does not generate new facets for the most known facets of the polytope Pn , because in many cases the mapping generates another member of the same facet family, i.e. the corresponding digraphs are isomorphic (for example, an m-fence maps to an m-fence, and so on). For a given vertex r 2 N , we introduce a mapping r : Rn ?n ! Rn ?n , de ned as 2

2

xrj = xjr ; xjr = xrj ; j 6= r; j 2 N; xij = xij + xjr ? xir ; i 6= j; i = 6 r; j 6= r; i; j 2 N:

(6) (7)

An equivalent version of (7) using (3) is

xij = xij + xjr + xri ? 1; i 6= j; i 6= r; j 6= r; i; j 2 N: Observe that this mapping is not a rotation in the strict sense of the word; thus the term "rotation mapping" is not restricted to mappings that are rotational in the conventional sense, but can include re ections as well.

Remark 2 The rotation

r of the linear ordering

ordering r + 1; : : : ; n; r; 1; : : : ; r ? 1.

Theorem 1 The mapping r 2 N.

1; 2; : : : ; r ? 1; r; r + 1; : : : ; n maps to the

r is a rotation mapping of the linear ordering polytope for every

Proof. To prove that r is a rotation mapping for Pn it is sucient to show that r transforms the relaxation polytope Bn into itself and all its integer vertices into integer vertices. Indeed, since r is a nonsingular ane mapping there exists the inverse mapping r?1 . The mapping ?1 r is de ned by the following equalities

xrj = xjr ; xjr = xrj ; xij = xij + xjr ? xir i 6= j; i 6= r; j 6= r; i; j; r 2 N: (8) Therefore, the polytope r (Bn ) is de ned by conditions 1  xij = xij + xjr ? xir i 6= j; i 6= r; j 6= r; i; j; r 2 N; 2  xij + xjr + xri = xij + xjr ? xir + xrj + xir = xij + 1; 1 = xij + xji = xij + xji; which combined with the equalities xij + xji = 1 in coordinates xij are identical to (1)-(3), that is, r (Bn ) = Bn . Equalities (6)-(7) and (8) imply that x is an integer point of Bn if and only if x is an integer point of r (Bn ), that is, vert( r (Bn )) \ Z n(n?1) = vert(Bn ) \ Z n(n?1) , or vert( r (Pn )) = vert(Pn ).

Theorem 1 and the de nition of rotation mapping r directly imply the following technique for obtaining new facets of the linear ordering polytope from known ones. 5

Theorem 2 If the inequality P P aij xij  a de nes a facet for Pn, then the inequality 1 0 n BX n X B@ aij (xij + xjr ? xir ) + air xri + arixir CCA  a 0

i=1

0

j =1

i6=r j 6=r

de nes a facet for Pn .

We distinguish two cases of rotation mapping r : the rst, when the variables xir and xrj are PP a x  a and the second when not. In the rst involved in the facet-de ning inequality ij ij 0 case we'll speak about facet routing, and in the second case, about facet lifting. Let a facet-de ning inequality ax  b be in normal form, G = (N (U ); U ) be a subdigraph of a complete digraph Gn = (N; A) and arc set U = f(i; j ) 2 A : aij > 0g. It is clear that the case r 62 N (U ) corresponds to facet lifting, and r 2 N (U ) to facet routing. The lifting transforms the digraph G = (N (U ); U ) into digraph Gr = (N (U ) [ r; Ur ), by adding the two arcs (r; i) and (j; r) for every arc (i; j ) 2 U . Notice that an arc (r; k) or (k; r) might be added multiple times; the aggregate multiplicity corresponds to the coecient of the new arc in the lifted inequality. The routing transforms a digraph G = (N (U ); U ) into digraph G0 = (N (U ); U 0 ), by adding the two arcs (r; i) and (j; r) for every arc (i; j ) 2 U : i; j 6= r and by changing the orientation of the arcs (r; j ) and (i; r) 2 U .

Remark 3 If indeg(s) = outdeg(s) for all s 2 N (U ) then lifting does not give new facets. The routing transforms the digraph G = (N (U ); U ) into digraph Gr = (N (U ); U 0 ), by adding two arcs (r; i) and (j; r) for every arc (i; j ) 2 U : i; j 6= r and by changing the orientation of arcs (r; j ) 2 U and (i; r) 2 U . Let ax  a0 be a facet-de ning inequality; then for any i and j , a j ( i (x))  a0 is equivalent either to ax  a0 or to a k (x)  a0 , for some k 2 N . It does not allow to apply the rotation method repeatedly. We call a valid inequality ax  a0 for Pn regular, if

X

i2N

ais =

X

i2N

asi; for all s 2 N:

If aij 2 f0; 1g then condition (9) is equivalent to indeg(s) = outdeg(s) for all s 2 N (U ). For r 2 N (U ) let

(

0 ; if i = r or j = r; aij ; otherwise ! X X 1 ar0 = a0 ? 2 air + ari : i2N i2N

arij

=

The next lemma is a generalization of the McLennan transitivity proposition [11]. 6

(9)

Lemma 2 (Routing lemma) If ax  a is a regular inequality for Pn and r 2 N (U ) then ax  a represents a facet of Pn i ar x  ar represents a facet of Pn . 0

0

0

To prove the lemma it suces to note that

routing r : ax  a0 ) ar x  ar0 and lifting r : ar x  ar0 ) ax  a0 :

Remark 4 If a digraph G = (N (U ); U ) induces a facet F and if for all s 2 N (U ), indeg(s) = outdeg(s), then deletion any vertex s gives a digraph, which itself induces a facet.

3 Reducing forms of new facet-de ning inequalities The rotation r of a facet the following coecients:

P a x  a yields the facet-de ning inequality P a x  a with ij ij ij ij 0

0

aij = aij ; i 6= r; j 6= r; X ajs; asr = j;s)2U

(

ars =

X

s;j )2U

(

a0 = a0 +

asj ;

X

aij

(lifting);

aij

(routing):

i;j )2U

(

a0 = a0 +

X

i;j )2U

(

i;j 6=r

The equality xij + xji = 1 allows one to reduce opposite directed arcs in the digraph G, and it corresponds to the following coecients correction:

a0ij a0sr a0rs a00

= = = =

aij = aij ; i 6= r; j 6= r; maxfasr ? ars ; 0g; maxfars ? asr ; 0g; X a0 ? min(asr ; ars )

a00 = a0 ?

s2N (U )

X

s2N (U )nfrg

(lifting);

min(asr ; ars ) (routing):

(10) (11) (12) (13) (14)

If a facet-de ning inequality has the following form

x(U ) =

X

i;j )2U

(

7

xij  a0

(15)

(all above described facets have this form), then the rotation r of (15) is:

X

i;j )2U 0

xij +

(

P

X

(maxf0; indeg(s) ? outdeg(s)gxsr +

s2N (U 0 )

maxf0; ?indeg(s) + outdeg(s)gxrs )  a00 ;

minfindeg(s); outdeg(s)g; U 0 = f(i; j ) 2 U : i; j 6= rg and where a00 = a0 + jU 0 j ? s2N (U 0 ) indeg(s)(outdeg(s)) is the number of incoming (outcoming) arcs for a node s. Remember, that ars = 1; asr = 0, if (r; s) 2 U; ars = 1; ars = 0, if (s; r) 2 U .

4 Rotations of m-fences The proof of the next theorem (and of all other similar theorems in this paper) directly follows from Theorem 2 and equations (10)-(14).

Theorem 3 The following inequalities de ne facets of the linear ordering polytope: a) the routing of m-fences for all r 2 fi ; : : : ; im ; j ; : : : ; jm g: 0 1 m m X BB X C xjq is C @xis js + (m ? 2)(xis r + xrjs ) + A  2(m ? 1) ; 1

1

2

s=1

q=1

is 6=r or js 6=r

q6=s;jq 6=r

b) the lifting of m-fences for all r 62 fi1 ; : : : ; im ; j1 ; : : : ; jm g:

0 1 m B@xisjs + (m ? 2)(xis r + xrjs ) + X xjq is CCA  2m ? 3m + 1:

m B X s=1

2

q=1

q6=s

The facets which are depicted in Figure 2a, 2b cut o noninteger vertices x = r (x0 ); r (x00 ), respectively. Note that the facet in Figure 2a for m = 3 is isomorphic to the Mobius ladder in Figure 4.

5 Rotations of t-reinforced m-fences A generalization of m-fences was presented in [6, 10, 11]. If (I [ J; F ) is an m-fence, then for any nonnegative integer t  m ? 2 the inequality m m X m X X 1) t xisjs + xjs iq  m(m ? 1) + t(t + 2 s=1 s=1 q =1

q6=s

represents a facet of Pn and called a t-reinforced m-fence [10]. Application of the rotation method to t-reinforced m-fences yields new classes of facets. 8

i1iH Y

i2i

m?2

m?2

p p p *imi m ? 2 H  @I@?? I@ HH?? @ ~~jit @ ?HH@ ? m?2 j i i t ( : (  ( ? ? @ @ @((H(?H(H(H(@((((m ? 2 ?   ?(((( @j?i?p p p H@jm?i ji? Y Y m?2 1

2

m?2

a) (r = jt ) m?2

m?2

p p p *imi  @I  ? I@HHH?? @ @ ?HH@@?? ?@ HH?@ ?  @@ ??? HH@H@ ?  ? ?  i jiY p p p jmi j Y m?2 i1iH Y

1

i2i

m?2 m?2

j~~ri

2

m?2

b)

Figure 2: Routing a) and lifting b) of m-fence

Theorem 4 The following inequalities de ne facets of the linear ordering polytope: a) the routing of t-reinforced m-fences, for any t = 1; 2; : : : ; m ? 2: m m X X 1) ; (txis js + (m ? t ? 1)(xis r + xrjs ) + xjq is  2(m ? 1) + tm + t(t + 2 2

s=1

q=1

is 6=r or js 6=r

q6=s;jq 6=r

b)the lifting of t-reinforced m-fences, for any t = 1; 2; : : : ; m ? 2:

0 1 m B@txisjs + (m ? 1 ? t)(xis r + xrjs ) + X xjq is CCA  (2m ? 2 ? t)m + t(t + 1) :

m B X s=1

q=1

q6=s

2

The last class of facets coincides with augmented t-reinforced m-fences introduced by McLennan [11] and Leung and Lee [10].

6 Rotations of (m,k)-fences An example of the routing of an (m; k)-fence is presented in Figure 3. 9

Theorem 5 The following inequalities de ne facets of the linear ordering polytope: a) the routing of (m; k)-fences for any r 2 fi ; : : : ; im ; j ; : : : ; jm g: 1

1

0 m B kX ? BB X (xjs BBxisjs + (2k ? 3) (xis r + xrjs ) + s @ q is 6 r js q 6 r 1

1 CC C q is + xjs?q is )C CA

+

=1 =

=1 + =

js 6=r

js?q 6=r

 m k+ 1 ? 1 + 2(m ? 1)(k ? 1) + m(2k ? 3);

b) the lifting of (m; k)-fences for any r 62 fi1 ; : : : ; im ; j1 ; : : : ; jm g:

0 m X @

s=1

xis js + (2k ? 3)(xis r + xrjs ) +

kX ?1 q=1

1 (xjs q is + xjs?q is )A +

 m k+ 1 ? 1 + 2m(k ? 1) + m(2k ? 3): i1iX yX

ii ii ii ii :    X  X   @I I @ @I X @ ??XXXX?X?XXX@ @??? ?@@ ?? XX@X@X ? @@ ? X??XXXX @ ? @?i?   ? ? ? R @  j  i i   X*ji jY j j ji 1

2

3

4

5

2

(= r)

3

4

5

Figure 3: Routing of (5,2)-fence for r = j3 .

7 Rotations of Mobius ladders The following class of facet-de ning inequalities was constructed in [9] and [12]: X x(M ) = xij  jM j ? m 2+ 1 ; (i;j )2M

(16)

where M is set of arcs of the Mobius ladder, which, in turn, is an ordered sequence of odd number of di erent dicycles C1 ; : : : ; Cm of length 3 or 4 (Figure 4), satisfying the following condition: every two adjacent dicycles have a common arc, and if any two of these dicycles Ci and Cj , Ci < Cj have a common node then this node belongs to either all dicycles Ci; : : : ; Cj , or to all dicycles Cj ; : : : ; Cm ; C1 ; : : : ; Ci (exact de nition see in [12]). Note that, for an (m; 2)-fence, 10

1jH

 56j  

6 HHH

HHj 3j  6HHH  HHHjR  2jH 6j   HH HHHj  4j Figure 4: Mobius ladder, m = 5. inequality (5') has the form m X

s=1

(xis js + xjs is + xjs? is )  m 2+ 1 + 2m ? 1 +1

1

(17)

and coincides with Mobius ladder with all dicycles of length 4 (4-dicycles). If m = 3 then inequality (17) coincides with inequalities (4') and (5'), i.e. 3-fences, (3,2)-fences and Mobius ladders with three 4-dicycles are isomorphic.

Theorem 6 The following inequalities de ne facets of the linear ordering polytope: a) the routing of Mobius ladders:

X

i;j )2M 0

X

(xij + xjr + xri ) +

(

i;r)2M

xri +

X r;i)2M

(

(

xir  jM j + jM 0 j ? m 2+ 1 ;

where M 0 = f(i; j ) 2 M : i; j 6= rg; b) the lifting of Mobius ladders:

X

i;j )2M

(

(xij + xjr + xri )  2jM j ? m 2+ 1 :

Let M consist of 4-dicycles only. Note that there exists a unique Mobius ladder (up to isomorphism) generated by m 4-dicycles. Observe that such a Mobius ladder is isomorphic to an (m; 2)-fence. The lifting of the Mobius ladder with every dicycle of length 4 (or, in other words, lifting of the (m; 2)-fence), is the following facet m ? X

s=1

xis js + xjs

+1

is + xjs?1 is + xris + xjs r

  7m ? 1 2

named in [12] as an m-wheel with center in r. The corresponding digraph is depicted in Figure 5, where is = 2s ? 1; js = 2s; s = 1; 2; 3; 4; 5 (cf. Figure 1b). 11

- 1i i i 10Y *7 I



4i



j ri? I

 8iy

W3i O j: 9i

R 2iY U ? * 5i 6i Figure 5: Lifting of a Mobius ladder.

8 Rotations of Zm-facets In [12] the facet-de ning digraph Zm = (N; A) is introduced as a generalization of Mobius ladders. Let M be the set of arcs of a Mobius ladder in which there exists only one 4-dicycle (j1 ; im ; jm ; i2 ) and all 3-dicycles (is ; js?1 ; h), (is ; js ; h), s = 2; : : : ; m have a common node h. Then A = M [ f(jm?1 ; i1 ); (j2 ; i1 ); (i1 ; jm ); (i1 ; j1 )g. See Figure 6a for the corresponding digraph Zm = (I [ J [ h), where m = 5, h = 11, is = 2s ? 1, js = 2s, s = 1; 2; : : : ; 5.

Theorem 7 The following inequalities de ne facets of the linear ordering polytope:

a) the routing of Zm -facets:

X

(i;j )2A0

(xij + xjr + xri ) +

X i;r)2A

xri +

(

X r;i)2A

xir  jAj + jA0 j ? m ? 2;

(

where A0 = f(i; j ) 2 A : i; j 6= rg; b) the lifting of Zm -facets:

X

(xij + xjr + xri )  2jAj ? m ? 2:

i;j )2A

(

It should be noted, that Zm is a facet of Pn only for m  5, n  (2m + 1). Example of routing 6 of Z5 is depicted in Figure 6b.

9 Acknowledgments The authors would like to thank the referees for helpful suggestions which have improved the presentation of this paper. 12

1i q 10i) PP 7 6So  2i

)P 7 16iSo q 2i 10i P 

 J PPPSSP I JJ

SPPPq 3i )  9iP iPPP J S1 J^11i  S  PP 1  iPPP S    W8i

6JP PPS 4i J J M

 J

 J^ 7i - 6i 5i

 J PPPSSP I JJ

SPPPq 3i )  9iP iPPP J S1  PP1J^11i Pi S  W8i

JPJ PPPSPS 4i J M

 J

 J^ 5i 7iY U * i 6

a) Z5

b) Routing of Z5 , r = 6 Figure 6:

References [1] G.G. Bolotashvili (1986), On the facets of the permutation polytope, Communic. Georgien Academy of Sciences, 121, N 2, pp. 281-284 (in Russian). [2] G.G.Bolotashvili (1987), A class of facets of the permutation polytope and a method for constructing facets of the permutation polytope, Preprint VINITI N3403-B87, N3405-B87 (in Russian). [3] G.G.Bolotashvili, M.M.Kovalev (1988), The partial order polytope, in: Proceedings of the VIII conference "Problems in theoretical cybernetics", Gorky (in Russian). [4] V. Bowman (1972), Permutation polyhedra, SIAM J.Appl.Math., v.22, N 3, pp.586-589. [5] P.Fishburn (1990), Binary probabilities induced by rankings, SIAM J.Disc.Math. V.3, N.4, pp.478-488. [6] P.Fishburn (1992), Induced binary probabilities and the linear ordering polytope: a status report, Math. Social Sci., v.23, pp.67-80. [7] E.Girlich, G.G.Bolotashvili, M.M.Kovalev (1995), The poset polytope, European Chapter on Combinatorial Optimization VIII, Poznan. [8] M.Grotschel, M.Junger, G.Reinelt (1984), A cutting plane algorithm for the linear ordering problem, Oper. Res., 32, pp.1195-1220 [9] M.Grotschel, M.Junger, G.Reinelt (1985), Facets of the linear ordering polytope, Math. Progr., N 33, pp.43-60. 13

[10] J.Leung, J.Lee (1994), More facets from fences for linear ordering and acyclic subgraph polytopes, Discrete Applied Math., v.50, pp.185-200. [11] A.McLennan (1990), Binary stochastic choice, in J.S. Chipman, D. McFadden, M.K. Richter, ed., Preferences, Uncertainity and Optimality, Westview Press, Boulder, pp.187-202. [12] G.Reinelt (1985), The linear ordering problem : Algorithms and Applications, Research and Exposition in Mathematics 8, Heldermann Verlag, Berlin. [13] G.Reinelt (1993), A note on small linear ordering polytope, Discrete Comput. Geom., v.10, N 1, pp.67-78. [14] V.A.Yemelichev, M.M.Kovalev, M.K.Kravtsov (1984), Polytopes, graphs, optimization, Cambridge University Press.

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