} Facets of the Set-Covering Polytope with coefficients in {0,1,2,3 Anureet Saxena

Transcript } Facets of the Set-Covering Polytope with coefficients in {0,1,2,3 Anureet Saxena

Facets of the Set-Covering Polytope with coefficients in {0,1,2,3 } Anureet Saxena ACO PhD Student, Tepper School of Business, Carnegie Mellon University.

Credits

• Prof. Egon Balas • Prof. Francois Margot • Prof. Stephan Hedetniemi and Prof. Alice Mc. Rae • Dr. Kent Anderson Anureet Saxena, TSoB 1

Taylor Expansion of 0/1 Polytopes • Calculus • Chvatal-Gomory Closure • Disjunctive Closure • Split Disjunction Closure • Facets with small coefficients f(x) = [ f(0) f’(0) f’’(0) f’’’(0) . . . ] Anureet Saxena, TSoB 2

Taylor Expansion of 0/1 Polytopes • Calculus • Chvatal-Gomory Closure • Disjunctive Closure • Split Disjunction Closure • Facets with small coefficients P I = [ P lp P 1 P 2 P 3 . . . ] P k = CG-rank

inequalities Anureet Saxena, TSoB 3

of the

constrained variables 0/1 Anureet Saxena, TSoB 4

Taylor Expansion of 0/1 Polytopes • Calculus • Chvatal-Gomory Closure • Disjunctive Closure • Split Disjunction Closure • Facets with small coefficients P I P 1 = [ P lp = P 1 P 2

split closure

P 3 of P lp . . . ] P 2 =

split closure

of P 1 … Anureet Saxena, TSoB 5

Taylor Expansion of 0/1 Polytopes • Calculus • Chvatal-Gomory Closure • Disjunctive Closure • Split Disjunction Closure • Facets with small coefficients P I = [ P lp P 1 P 2 1. 8 j2 N, |  j | · k.

2. |  | · k P 3 . . . ] P k = Polytope defined by valid inequalities  x ¸  such that Anureet Saxena, TSoB 6

Taylor Expansion of 0/1 Polytopes • Chvatal-Gomory Closure – Optimizing over elementary closure is NP-Complete (Eisenbrand ‘99) – Computational Experiments (Fischetti and Lodi ’05) – Separation Problem (Brady Hunsaker ’04) • Disjunctive Closure – Optimization in polynomial time.

– Computational Experiments (Pierre Bonami ‘04) • Split Disjunction Closure – Cook et al ’90, Cornuejols et al ’04 – Optimizing over elementary closure is NP-Complete (Caprara et al ‘03) • Facets with small coefficients – Set-Covering Polytope – Balas and Ng (‘89), Cornuejols and Sassano (‘89), Sanchez et al (‘98), Bienstock and Zuckermann (’04) Anureet Saxena, TSoB 7

Set-Covering Polytope

Attributes

Family of Ineq Minimality Facetness Gen. Proc FSSP Lifting Proc Gen All Ineq via Lft Simultaneous Lft Critical Matrices

Balas

et al ‘89 {0,1,2}

Sanchez

et al ‘98 {0,1,2,3}

Saxena

‘04 {0,1,2,3} Anureet Saxena, TSoB 8

Set-Covering Polytope

Attributes

Family of Ineq Minimality Facetness Gen. Proc FSSP Lifting Proc Gen All Ineq via Lft Simultaneous Lft Critical Matrices

Balas

et al ‘89 {0,1,2}

Sanchez

et al ‘98 {0,1,2,3}

Saxena

‘04 {0,1,2,3} Anureet Saxena, TSoB 9

Notations

• A :- 0/1 matrix • M :- row index set of A • N :- column index set of A • P A = conv { x2 {0,1} N | Ax ¸ 1} (Set-Covering Polytope) • For Jµ N, M(J) is defined to be those set of rows which are not covered by columns in J, M(J) = { i2 M |  j2 J A ij =0 } Anureet Saxena, TSoB 10

Notations

• If  2 {0,1,…,k} N , then for t2 J  t {0,1,…,k} = { j2 N |  j =t } eg:  = (3,2,3,1,0,2) then J  0 J  1 J  2 J  3 = { 5 } = { 4 } = { 2, 6 } = { 1, 3 } • • I A k – set of valid inequalities with coeff. in {0,1,…,k} M A k – set of minimally valid inequalities with coeff. in {0,1,…,k} Anureet Saxena, TSoB 11

Cover Hypergraphs

Def: Let  2 I A k. The such that

Cover-hypergraph

of  x ¸ k is a hypergraph H V(H) = Nn(J  0 [ J  k ) and S µ Nn(J  0 [ J  k ) is an edge of H iff S[ J  0  is a cover of A.

(S) =  j2 S  j = k.

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Cover Hypergraphs

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Cover Hypergraphs

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Minimality

Theorem

(Saxena’04): Let  inequality if and only if 2 I A k . Then  x ¸ k is a minimal valid 1.

The cover hypergraph of 8 j2 J  k  , 8 i2 M(J  0 ) : A ij =1 x ¸ k has no isolated vertex.

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Facetness

Theorem

(Balas and Ng ’89): Let  2 I A 2 . Then  x ¸2 defines a facet of P A iff  2 M A 2 .

Every component of the cover hypergraph of  x ¸ 2 contains 3.

an odd cycle.

For every j2 J  0 , 9 x j 2 P A , such that x j j =0 and  x j = 2.

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Facetness

Theorem

(Balas and Ng ’89): Let  2 I A 2 . Then  x ¸2 defines a facet of P A iff  2 M A 2 .

Every component of the cover hypergraph of  x ¸ 2

contains

odd cycle

For every j2 J  0 , 9 x j 2 P A , such that x j j =0 and  x j = 2.

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Facetness

Lemma: Let G be a connected graph, then the edge-vertex incidence matrix of G is of full column rank iff G contains an odd cycle.

Proof: Anureet Saxena, TSoB 18

Kernel Hypergraphs

Defn: A hypergraph H=(V,E) is a

kernel hypergraph

if the vertex edge incidence matrix of H is a square non-singular matrix with at least two ones in every row.

Eg: Anureet Saxena, TSoB 19

Strong Connectivity of Hypergraphs Defn: H=(V,E) and S µ V. A sequence of vertices {v 1 ,v 2 ,…,v q } in VnS (where q=| VnS |) is said to be a

S-connected sequence

if 8 t2 {1,2,…,q}, 9 e t 2 E s.t. 1.

v t 2 e t e t µ S[ {v 1 ,v 2 ,…,v t } H is said to be

strongly connected

w.r.t S if there exists a S connected sequence.

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Facetness

Theorem

(Saxena’04): Let  iff 1.

2 I A k . Then  x ¸ k defines a facet of P  2 M A k .

Every component (say H i ) of the cover hypergraph of  has a kernel subhypergraph , say K i , such that H i x ¸ k is strongly connected w.r.t V(K i ).

For every j2 J  0 , 9 x j 2 P A , such that x j j =0 and  x j = k.

A Anureet Saxena, TSoB 21

Fixed Support Separation Problem Given S µ N, let F S k = {  2 M A k |  j  0 if and only if j2 S} Theorem: (Balas and Ng ‘89) 1.

| F S 2 If F S 2 | · 1   , then the inequality in F S 2 is a

CG-rank 1

inequality .

The separation problem over F S 2 can be solved in linear time . Anureet Saxena, TSoB 22

Fixed Support Separation Problem Given S µ N, let F S k = {  2 M A k |  j  0 if and only if j2 S} Theorem: (Saxena ‘04) 1.

| F S 3 | can have exponential number (in |S|) of inequalities.

Inequalities in F S 3 typically are higher rank inequalities.

The separation problem over F S 3 is

NP-Complete .

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Fixed Support Separation Problem Given S µ N, let F S k = {  2 M A k |  j  0 if and only if j2 S} Theorem: (Saxena ‘04) There exists a ( poly-time constructible) hypergraph H=(V,E), and a subset J µ V, such that there is a 1.

precise one-one correspondence between the following:  2 F S 3 .

Independent dominating sets

of H contained in J.

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Fixed Support Separation Problem

Theorem

: (Balas and Ng’89) Let x be a fractional solution to { x | Ax ¸ 1, 0 · x · 1} and let If  2 M A 2 I = { j2 N | x j =1 }.

such that  x < 2, then 1.

I Å J  2 I Å J  1 =  .

=  .

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Fixed Support Separation Problem

Theorem

: (Saxena’04) Let x be a fractional solution to { x | Ax ¸ 1, 0 · x · 1} and let Furthermore, suppose @ that  x < 3, then I = { j2 N | x j =1 }.

 2 M A 2 such that  x < 2. If  2 M A 3 such 1.

I Å J  3 I Å J  2 I Å J  1 =  .

= =   .

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Research Diagram

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Lifting Theory

Let N 1 µ N, B=A N1 and  2 M B .

F  = {  2 M A |  j =  j , 8 j2 N 1 }.

Questions 1.

Give a procedure to generate inequalities in F  .

Identify inequalities which can be generated by that procedure.

Modify the procedure to generate those inequalities which cannot be generated by the incumbent procedure –

simultaneous lifting , relaxation-lifting

. Anureet Saxena, TSoB 28

Determining Lifting Coefficient

• • • Given: N  1 µ N, B=A 2 M B 3.

N1, k 2 Nn N 1.

To Determine: The minimum value of  such that the inequality,  j2 N1  j x j +  x k ¸ 3, is a minimally valid inequality of the polytope P A Å {x | x j = 0, 8 j  N 1 [ { k } }.

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Determining Lifting Coefficient

Lifting Procedure:

(c3): A ik =1, 8 i2 M( J  0 (c2): 9 j2 J  1 , A ij + A ik ) ¸ 1, 8 i2 M( J  0 ) (c1a): 9 j2 J  2 , A ij + A ik (c1b): 9 j,h2 J  2 , A ij + A ¸ 1, 8 i2 M( J ih + A ik  0 ) ¸ 1, 8 i2 M( J  0 ) Set  = 3 if (c3) holds 2 if (c2) holds but (c3) does not 1 if (c1a) or (c1b) holds but (c3) and (c2) do not 0 otherwise.

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Determining Lifting Coefficient

x 1 + x 2 + x 3 + 2x 4 ¸ 3 minimally valid for N 1 = {1,2,3,4} Lifting Sequence: ( 5, 6, 7, 8, 9, 10 ) Anureet Saxena, TSoB 31

Lifting x

5 x 1 + x 2 + x 3  + 2x 4 = 1 +  x 5 ¸ 3 Anureet Saxena, TSoB 32

Lifting x

5 x 1 + x 2 + x 3  + 2x 4 = 1 +  x 5 ¸ 3 Anureet Saxena, TSoB 33

Lifting x

6 x 1 + x 2 + x 3 + 2x 4  = 2 + x 5 +  x 6 ¸ 3 Anureet Saxena, TSoB 34

Lifting x

7 x 1 + x 5 + x 2 + 2x + x 3 6 +  + 2x 4 x 7 ¸ 3  = 2 Anureet Saxena, TSoB 35

Lifting x

8 x 1 + x 2 + 2x 6 + x 3 + 2x 7  + 2x 4 + = 1  x 8 + x 5 ¸ 3 Anureet Saxena, TSoB 36

Lifting x

9 x 1 + 2x 6 + x 2 + x + 2x 7 3 + 2x 4 + x 8 +  + x 5 x 9 ¸ 3  =1 Anureet Saxena, TSoB 37

Lifting x

10 + 2x 6 x 1 + x 2 + 2x 7 + x + x 8  3 + 2x 4 + x = 1 9 + x 5 +  x 10 ¸ 3 Anureet Saxena, TSoB 38

Fundamental Theorem of Lifting

Let N 1 µ N, B=A N1 and F   2 M B 3 , = {  2 M A 3 |  j =  j , 8 j2 N 1 }.

Theorem

(Saxena’04):  lifting the inequality  x ¸ 3, (  2 F  ) be obtained by sequentially x ¸ 3, if and only if

the cover hypergraph of



x ¸ 3 is strongly connected w.r.t the cover hypergraph of



x ¸ 3.

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Lifting Theory

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Lifting Theory

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Lifting Theory

Given  2 M A 3 , the support of any inequality which gives rise to  x ¸ 3

via

lifting is bounded below by the number of components in the cover-hypergraph of  x ¸ 3.

Is it possible to

modify

the lifting procedure such that any minimally valid inequality  x ¸ 3 can be obtained by lifting an inequality which has a sufficiently small support?

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Lifting Theory

Given  2 M A 3 , the support of any inequality which gives rise to  x ¸ 3

via

lifting is bounded below by the number of components in the cover-hypergraph of  x ¸ 3.

Is it possible to

modify

the lifting procedure such that any minimally valid inequality  x ¸ 3 can be obtained by lifting an inequality which has a sufficiently small support?

A: Cover Hypergraph Enrichment

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Cover Hypergraph Enrichment

Inequality: x 1 + x 2 + x 3 x 5 + x 6 + x 7 + 2x 4 + + 2x 8 ¸ 3 Cover Hypergraph: Anureet Saxena, TSoB 44

Cover Hypergraph Enrichment

Inequality: x 1 + x 2 + x 3 x 5 + x 6 + x 7 + 2x 4 + + 2x 8 ¸ 3 Cover Hypergraph: Anureet Saxena, TSoB 45

Cover Hypergraph Enrichment

Inequality: x 1 + x 2 + x 3 x 5 + x 6 + x 7 + 2x 4 + + 2x 8 ¸ 3 Cover Hypergraph: Anureet Saxena, TSoB 46

Cover Hypergraph Enrichment

Defn: Let  2 M A 3 . The process of successively dropping rows of the matrix A such that  x ¸ 3 remains a valid inequality for the resulting matrices is termed cover-hypergraph enrichment .

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Cover Hypergraph Enrichment

Defn: Let  2 M A 3 . The process of successively dropping rows of the matrix A such that  x ¸ 3 remains a valid inequality for the resulting matrices is termed cover-hypergraph enrichment .

Defn: A 0/1 matrix 1.

is said to 3-critical w.r.t  (  2 {0,1,2,3} N ), if the following conditions hold true:  2 M A 3   I B 3 where B is full-column proper submatrix of A (i.e B=A S for some S ( M) Anureet Saxena, TSoB 48

Cover Hypergraph Enrichment

Theorem: (Saxena ’04) If A is a 3-critical w.r.t  , then there exist h,j,k 2 J  1 such that (H is the cover-hypergraph of  x ¸ 3) 1.

{h,j,k} is a 3-edge of H.

H is strongly connected w.r.t {h,j,k}.

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Cover Hypergraph Enrichment

Theorem: (Saxena ’04) If A is a 3-critical w.r.t  , then there exist h,j,k 2 J  1 such that (H is the cover-hypergraph of  x ¸ 3) 1.

{h,j,k} is a 3-edge of H.

H is strongly connected w.r.t {h,j,k}.

Corollary: Let  x ¸ 3 be facet defining inequality of P A , then there exists columns h,j,k,p 2 J  1 such that  x ¸ 3 can be obtained by lifting the inequality x h + x j + x k +  x p ¸ 3 (  2 {1,2}) using the

modified

lifting procedure.

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Research Diagram

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3-Critical Matrices

Observation:

Every facet



x ¸ 3 of P A is associated with a 3-critical submatrix of A.

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3-Critical Matrices

Observation:

Every facet



x ¸ 3 of P A is associated with a 3-critical submatrix of A.

Implication:

3-critical submatrices of A can be used as source of strong valid inequalities of P A .

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Lifting from 3-Critical Matrices

Given: B is 3-critical w.r.t Problem: To lift   and  x ¸ 3 defines facet of P B .

x ¸ 3 to obtain facet of P A .

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Lifting from 3-Critical Matrices

  x + = 0  x k ¸ 3 is a minimally valid inequality.

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Lifting from 3-Critical Matrices

  x + = 3  x k ¸ 3 is a minimally valid inequality.

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Lifting from 3-Critical Matrices

 x +  j2 J3 3x j +  x k ¸ 3 is a minimally valid inequality.

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Lifting from 3-Critical Matrices

 x +  j2 J3   0 3x j +  x k ¸ 3 is a minimally valid inequality.

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Lifting from 3-Critical Matrices

  x +  j2 J3 = 1 or 2 3x j +  x k ¸ 3 is a minimally valid inequality.

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Lifting from 3-Critical Matrices

  x +  j2 J3 = 2 3x j +  x k ¸ 3 is a minimally valid inequality.

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Lifting from 3-Critical Matrices

 x +  j2 J3 3x inequality.

j +  j2 J2 2x j +  h x h +  k x k ¸ 3 is a minimally valid  h +  k ¸ 3 Anureet Saxena, TSoB 61

Lifting from 3-Critical Matrices

 x +  j2 J3 3x inequality. j +  j2 J2 2x j +  h x h +  k x k ¸ 3 is a minimally valid (2  h ) + (2  k ) · 1 Anureet Saxena, TSoB 62

Lifting from 3-Critical Matrices

• • Summary Sequence Independent Lifting Coefficients for columns in J 3 , J 2 , J 1 .   J: remaining columns 8 j2 J,  j 2 { 1, 2 }, i.e (2  j ) 2 { 0 , 1 } 8 h,k 2 J, such that A ih (2 + A ik  h ) + (2 ¸ 1 8 i2 M, then  k ) · 1. Define a graph G on vertex-set J such that h,k2J are adjacent iff A ih + A ik ¸ 1 8 i2 M. Anureet Saxena, TSoB 63

Lifting from 3-Critical Matrices

Theorem: (Saxena’04) There is a precise one-one correspondence between 1.

Maximal Independent sets of G Set of inequalities which can be obtained by sequentially lifting  x ¸ 3.

In particular, if S µ J is a maximal independent set of G, the corresponding inequality is:  x +  j2 J3 3x j +  j2 J2 2x j +  j2 J (2 – y j ) x j ¸ 3, where y j = 1 if j2 S and y j =0 if j 2 Jn S.

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Undominated Extreme Points

Defn: x is an undominated extreme point of P, if 1.

x is an extreme point of P.

P Å { y | y¸ x} = { x }.

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Lifting from 3-Critical Matrices

Theorem: (Saxena’04) There is a precise one-one correspondence between 1.

Undominated extreme points of the vertex-edge relaxation of the stable set polytope of G.

Set of inequalities which can be obtained by sequentially or simultaneously lifting  x ¸ 3.

In particular, if y is an undominated extreme point of G, the corresponding inequality is:  x +  j2 J3 3x j +  j2 J2 2x j +  j2 J (2 – y j ) x j ¸ 3.

Anureet Saxena, TSoB 66

Research Diagram

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Research Diagram

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What Next?

Kernel Hypergraphs

• Complete Characterization?

• To what extent do they generalize the properties of odd cycles to hypergraphs?

• Do NP-Complete problems on general hypergraphs have polynomial time algorithms if H is not strongly connected w.r.t a kernel hypergraph ?

– Existence of spanning tree?

– 3-Dimensional Matching ?

– ….

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What Next?

Strong Connectivity

• Do other known conditions for facetness have a similar generalization?

– Generalization of sufficient condition for facetness of Sassano.

– Applications to facets of the form   is an arbitrary integer.

x ¸  , where  j 2 {1, 2} and – Analysis of set-covering polytope • Analysis of set-covering polytope of circulant matrices.

• Analysis of the dominating set polytope of graphs.

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What Next?

Lifting Theory

• Generalizing the

Fundamental theorem of Lifting

• Understanding the interpretation of simultaneous lifting in terms of cover hypergraphs and suitable generalization of strong connectivity.

• Is it true that every facet defining inequality,  x ¸ 4, can be derived by lifting an inequality with support of just 5 elements?

• Is it true that every facet defining inequality with coefficients in {0,1,…, k} can be obtained by lifting an inequality with a support which is a polynomial of k?

• Simultaneous lifting in cases where the range of lifting coefficients is exactly 1 – Cover inequalities (Knapsack Polytope, Balas and Zemel ’80) – Odd anti-hole inequalities (Set-packing polytope) Anureet Saxena, TSoB 71

thank you

Anureet Saxena, TSoB 72