Facets of the SetCovering Polytope with coefficients in 0,1,2,3

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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.

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Credits

• Prof. Egon Balas
• Prof. Francois Margot
• Prof. Stephan Hedetniemi and Prof. Alice Mc. Rae
• Dr. Kent Anderson

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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) . . .

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Taylor Expansion of 0/1 Polytopes

• Calculus
• Chvatal-Gomory Closure
• Disjunctive Closure
• Split Disjunction Closure
• Facets with small coefficients

• PI Plp P1 P2 P3 . . .
• Pk CG-rank k inequalities

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Taylor Expansion of 0/1 Polytopes

• Calculus
• Chvatal-Gomory Closure
• Disjunctive Closure
• Split Disjunction Closure
• Facets with small coefficients

• PI Plp P1 P2 P3 . . .
• Pk inequalities which can be obtained by
  imposing integrality on k of the n 0/1
  constrained variables

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Taylor Expansion of 0/1 Polytopes

• Calculus
• Chvatal-Gomory Closure
• Disjunctive Closure
• Split Disjunction Closure
• Facets with small coefficients

• PI Plp P1 P2 P3 . . .
• P1 split closure of Plp
• P2 split closure of P1

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Taylor Expansion of 0/1 Polytopes

• Calculus
• Chvatal-Gomory Closure
• Disjunctive Closure
• Split Disjunction Closure
• Facets with small coefficients

• PI Plp P1 P2 P3 . . .
• Pk Polytope defined by valid inequalities ? x
  ? such that
• 8 j2 N, ?j k.
• ? k

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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)

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Set-Covering Polytope
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Set-Covering Polytope
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Notations

• A - 0/1 matrix
• M - row index set of A
• N - column index set of A
• PA conv x2 0,1N 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 JAij0

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Notations

• If ? 2 0,1,,kN, then for t2 0,1,,k
• J?t j2 N ?jt
• eg ? (3,2,3,1,0,2) then
• J?0 5
• J?1 4
• J?2 2, 6
• J?3 1, 3
• IAk set of valid inequalities with coeff. in
  0,1,,k
• MAk set of minimally valid inequalities with
  coeff. in 0,1,,k

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Cover Hypergraphs

• Def Let ? 2 IAk. The Cover-hypergraph of ?x k
  is a hypergraph H such that
• V(H) Nn(J?0 J?k)
• and Sµ 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 (Saxena04) Let ? 2 IAk. Then ? x k is
  a minimal valid inequality if and only if
• The cover hypergraph of ? x k has no isolated
  vertex.
• 8 j2 J?k , 8 i2 M(J?0) Aij1

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Facetness

• Theorem (Balas and Ng 89) Let ? 2 IA2. Then ?
  x2 defines a facet of PA iff
• ? 2 MA2.
• Every component of the cover hypergraph of ? x
  2 contains an odd cycle.
• For every j2 J?0, 9 xj2 PA, such that xjj0 and ?
  xj 2.

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Facetness

• Theorem (Balas and Ng 89) Let ? 2 IA2. Then ?
  x2 defines a facet of PA iff
• ? 2 MA2.
• Every component of the cover hypergraph of ? x
  2 contains an odd cycle.
• For every j2 J?0, 9 xj2 PA, such that xjj0 and ?
  xj 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

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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

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Strong Connectivity of Hypergraphs

• Defn H(V,E) and Sµ V. A sequence of vertices
  v1,v2,,vq in VnS
• (where q VnS ) is said to be a S-connected
  sequence if
• 8 t2 1,2,,q, 9 et2 E s.t.
• v t 2 et
• et µ S v1,v2,,vt
• H is said to be strongly connected w.r.t S if
  there exists a S-
• connected sequence.

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Facetness

• Theorem (Saxena04) Let ? 2 IAk. Then ? x k
  defines a facet of PA iff
• ? 2 MAk.
• Every component (say Hi) of the cover hypergraph
  of ? x k has a kernel subhypergraph, say Ki,
  such that Hi is strongly connected w.r.t V(Ki).
• For every j2 J?0, 9 xj2 PA, such that xjj0 and ?
  xj k.

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Fixed Support Separation Problem

• Given Sµ N, let
• F Sk ? 2 MAk ?j? 0 if and only if j2 S
• Theorem (Balas and Ng 89)
• F S2 1
• If FS2 ? ?, then the inequality in F S2 is a
  CG-rank 1 inequality .
• The separation problem over FS2 can be solved in
  linear time.

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Fixed Support Separation Problem

• Given Sµ N, let
• F Sk ? 2 MAk ?j? 0 if and only if j2 S
• Theorem (Saxena 04)
• F S3 can have exponential number (in S) of
  inequalities.
• Inequalities in FS3 typically are higher rank
  inequalities.
• The separation problem over FS3 is NP-Complete.
  

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Fixed Support Separation Problem

• Given Sµ N, let
• F Sk ? 2 MAk ?j? 0 if and only if j2 S
• Theorem (Saxena 04) There exists a (poly-time
  constructible) hypergraph H(V,E), and a subset
  Jµ V, such that there is a precise one-one
  correspondence between the following
• ? 2 FS3.
• Independent dominating sets of H contained in J.

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Fixed Support Separation Problem

• Theorem (Balas and Ng89)
• Let x be a fractional solution to x Ax 1,
  0 x 1 and let
• I j2 N xj1 .
• If ? 2 MA2 such that ? x lt 2, then
• I Å J?2 ?.
• I Å J?1 ?.

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Fixed Support Separation Problem

• Theorem (Saxena04)
• Let x be a fractional solution to x Ax 1,
  0 x 1 and let
• I j2 N xj1 .
• Furthermore, suppose _at_ ? 2 MA2 such that ? x lt 2.
  If ? 2 MA3 such that ? x lt 3, then
• 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 N1µ N, BAN1 and ? 2 MB.
• F? ? 2 MA ?j ?j, 8 j2 N1 .
• Questions
• 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.

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Determining Lifting Coefficient

• Given
• N1µ N, BAN1,
• ? 2 MB3.
• k 2 Nn N1.
• To Determine The minimum value of ? such that
  the inequality,
• ?j2 N1 ?j xj ? xk 3,
• is a minimally valid inequality of the polytope
• PA Å x xj 0, 8 j? N1 k .

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Determining Lifting Coefficient

• Lifting Procedure
• (c3) Aik1, 8 i2 M( J?0 )
• (c2) 9 j2 J?1, Aij Aik 1, 8 i2 M( J?0 )
• (c1a) 9 j2 J?2, Aij Aik 1, 8 i2 M( J?0 )
• (c1b) 9 j,h2 J?2, Aij Aih Aik 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

• x1 x2 x3 2x4 3
• minimally valid for N1 1,2,3,4
• Lifting Sequence
• ( 5, 6, 7, 8, 9, 10 )

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Lifting x5

• x1 x2 x3 2x4 ? x5 3
• ? 1

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Lifting x5

• x1 x2 x3 2x4 ? x5 3
• ? 1

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Lifting x6

• x1 x2 x3 2x4 x5 ? x6 3
• ? 2

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Lifting x7

• x1 x2 x3 2x4
• x5 2x6 ? x7 3
• ? 2

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Lifting x8

• x1 x2 x3 2x4 x5
• 2x6 2x7 ? x8 3
• ? 1

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Lifting x9

• x1 x2 x3 2x4 x5
• 2x6 2x7 x8 ? x9 3
• ? 1

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Lifting x10

• x1 x2 x3 2x4 x5
• 2x6 2x7 x8 x9 ? x10 3
• ? 1

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Fundamental Theorem of Lifting

• Let N1µ N, BAN1 and ? 2 MB3,
• F? ? 2 MA3 ?j ?j, 8 j2 N1 .
• Theorem (Saxena04) ? x 3, (? 2 F?) be
  obtained by sequentially lifting the inequality ?
  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 MA3, 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.
• Q 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 MA3, 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.
• Q 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
• x1 x2 x3 2x4
• x5 x6 x7 2x8 3
• Cover Hypergraph

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Cover Hypergraph Enrichment

• Inequality
• x1 x2 x3 2x4
• x5 x6 x7 2x8 3
• Cover Hypergraph

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Cover Hypergraph Enrichment

• Inequality
• x1 x2 x3 2x4
• x5 x6 x7 2x8 3
• Cover Hypergraph

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Cover Hypergraph Enrichment

• Defn Let ? 2 MA3. 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 MA3. 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 A is said to 3-critical w.r.t
  ? (? 2 0,1,2,3N ), if the following conditions
  hold true
• ? 2 MA3
• ? ? IB3 where B is full-column proper submatrix
  of A (i.e BAS for some S ( M)

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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)
• 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)
• 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 PA, then there exists columns
  h,j,k,p 2 J?1 such that ? x 3 can be obtained by
  lifting the inequality
• xh xj xk ? xp 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 PA is associated with a
  3-critical submatrix of A.

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3-Critical Matrices

• Observation
• Every facet ? x 3 of PA 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 PA.

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Lifting from 3-Critical Matrices

• Given B is 3-critical w.r.t ? and ? x 3 defines
  facet of PB.
• Problem To lift ? x 3 to obtain facet of PA.

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Lifting from 3-Critical Matrices

• ? x ? xk 3 is a minimally valid inequality.
• ? 0

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Lifting from 3-Critical Matrices

• ? x ? xk 3 is a minimally valid inequality.
• ? 3

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Lifting from 3-Critical Matrices

• ? x ?j2 J3 3xj ? xk 3 is a minimally valid
  inequality.

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Lifting from 3-Critical Matrices

• ? x ?j2 J3 3xj ? xk 3 is a minimally valid
  inequality.
• ? ? 0

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Lifting from 3-Critical Matrices

• ? x ?j2 J3 3xj ? xk 3 is a minimally valid
  inequality.
• ? 1 or 2

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Lifting from 3-Critical Matrices

• ? x ?j2 J3 3xj ? xk 3 is a minimally valid
  inequality.
• ? 2

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Lifting from 3-Critical Matrices

• ? x ?j2 J3 3xj ?j2 J2 2xj ?h xh ?k xk 3
  is a minimally valid inequality.
• ?h ?k 3

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Lifting from 3-Critical Matrices

• ? x ?j2 J3 3xj ?j2 J2 2xj ?h xh ?k xk 3
  is a minimally valid inequality.
• (2-?h) (2-?k) 1

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Lifting from 3-Critical Matrices

• Summary
• Sequence Independent Lifting Coefficients for
  columns in
• J3, J2, J1.
• J remaining columns
• 8 j2 J, ?j 2 1, 2 , i.e (2 - ?j ) 2 0 , 1
  
• 8 h,k 2 J, such that Aih Aik 1 8 i2 M, then
• (2 - ?h) (2 - ?k) 1.
• Define a graph G on vertex-set J such that h,k2J
  are adjacent iff
• Aih Aik 1 8 i2 M.

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Lifting from 3-Critical Matrices

• Theorem (Saxena04) There is a precise one-one
  correspondence between
• 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 3xj ?j2 J2 2xj ?j2 J (2 yj)
  xj 3,
• where yj 1 if j2 S and yj0 if j 2 Jn S.

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Undominated Extreme Points

• Defn x is an undominated extreme point of P, if
• x is an extreme point of P.
• P Å y y x x .

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Lifting from 3-Critical Matrices

• Theorem (Saxena04) There is a precise one-one
  correspondence between
• 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 3xj ?j2 J2 2xj ?j2 J (2 yj)
  xj 3.

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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 ? x ?, where
  ?j 2 1, 2 and ? is an arbitrary integer.
• 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)

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• thank you