Linear Programming LP

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Linear Programming (LP)

• Vector Form
• Maximize cx
• Subject to Ax ? b
• c (c1, c2, , cn)
• x b
• A

• Summation Form
• Maximize ?cixi
• Subject to
• ?a1ixi ? b1
• ?a2ixi ? b2
• .
• .
• ?amixi ? bm

x1 . . xn
b1 . . bn
a11 a1n an1 amn
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Example LP

• n 2 m 4
• x1 x2 ? max
• x1 ? 0 ? (-1)x1 0x2 ? 0
• x2 ? 0 ? 0x1 (-1)x2 ? 0
• x1 ? 5 ? (-1)x1 0x2 ? 5
• x2 ? 6 ? 0x1 1x2 ? 6
• c (1, 1) b
• A

• Optimal solution is the unique point of
  intersection of the objective with the hyperplane
  feasible polytope.
• x2
• optimal solution x1 5 x2 6
• objective
• x1 x2 11
• x1

0 0 5 6
Feasible solution region
-1 0 0 -1 1 0 0 1
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Integer Linear Programming (ILP)

• Vector Form
• Maximize cx
• Subject to Ax ? b
• and x ? 0,1
• c (c1, c2, , cn)
• x b
• A

• Summation Form
• Maximize ?cixi
• Subject to
• ?a1ixi ? b1
• ?a2ixi ? b2
• .
• .
• ?amixi ? bm
• and x ? 0,1

x1 . . xn
b1 . . bn
a11 a1n an1 amn
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ILP for MIS

• Maximum Independent Set (MIS)
• - Find the maximum subset of nodes in graph G
  (V, E) which are pairwise non-adjacent
• ILP
• - For any v ? V make a variable xv ? 0, 1
• xv 0 ? v ? MIS which means 0 is not
  chosen
• xv 1 ? v ? MIS which means 1 is chosen
• - Maximize ?v?V xv
• Subject to ? e (u, v) ? V, xu xv
  ? 1

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Example ILP of MIS

• Max x1 x2 x3 x4 x5 x6
• subject to x1 x6 ? 1 x1 x2 ? 1 x2
  x3 ? 1 x3 x6 ? 1 x3 x5 ? 1 x6 x5
  ? 1 x3 x4 ? 1 x4 x5 ? 1 and x1, x2,,
  x6 ? 0,1

• Graph

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ILP for MaxClique

• ILP
• - ?xi ? max
• - Subject to
• xi xj ? 1 ? (i, j) ? E

• MaxClique
• - Given G (V, E)
• - Find
• X ? V ? x, x ? X
• (x, x) ? E
• X ? max

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ILP for Matching

• Matching
• - Given G (V, E)
• - Find X ? E ? e, e ? X
• e and e dont share endpoint.
• X ? max

• ILP
• - for any e ? E xe ?0, 1
• 0 is not in matching
• 1 is in matching
• - ?e?E xe ? max
• - Subject to
• ?e incident to V xe ? 1 ?v ? V
• e2 xe1 xe2 xe3 ? 1
• only one edge from
• e1 e3 matching can be
• incident to v

v
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LP relaxation (LPR) vs. ILPLP relaxation (LPR)
for MAX independent set problem (MISP) gives
larger value than the maximum size of
independent set.

• MISP
• ?xi ? max, i ? V
• xi xj ? 1, for each edge (xi,xj)? E
• xi ?0, 1

• LPR
• ? xi ? max, i ? V
• xi xj ? 1, for each edge (xi,xj)? E
• 0 ? xi ? 1

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Example 1 of ILP vs. LPR

• ?xi ? max
• x1 x6 ? 1
• x1 x2 ? 1
• x5 x6 ? 1
• x5 x2 ? 1
• x5 x4 ? 1
• x4 x3 ? 1
• x4 x2 ? 1
• x2 x3 ? 1

• ILP
• x1 x3 x5 1
• ?xi 3
• LPR
• xi ½
• ?xi 3

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1
5
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MISP Integrality Gap

• x1 x2 x3 ? max
• x1 x2 ? 1
• x1 x3 ? 1
• x2 x3 ? 1
• 0 ? x1 ? 1
• 0 ? x2 ? 1
• 0 ? x3 ? 1
• LPR (?) ? 3/2
• Implies LPR (?) 3/2
• So x1 x2 x3 ½ ? LPR (?) ? 3/2
• ILP (?) 1
• Integrality Gap (IG) LPR / ILP 3/2

• What is the integrality gap for (MISP)
• For a complete graph
• ILP (Kn) 1
• LPR (Kn) n/2
• Integrality Gap (IG) LPR / ILP
• Integrality gap may be as large as n/2

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LPR vs. ILP LP relaxation (LPR) for Minimum
Vertex Cover problem (MVCP) gives smaller value
than the minimum size of vertex cover

• LPR
• ? xi ? min , i ? V
• xi xj ? 1, for each e (xi,xj)? E
• 0 ? xi ? 1

• MVCP
• ?xi ? min, i ? V
• xi xj ? 1, for each e (xi,xj)? E
• xi ?0, 1

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MVCP Integrality Gap

• x1 x2 x3 ? min
• x1 x2 ? 1
• x1 x3 ? 1
• x2 x3 ? 1
• 0 ? x1 ? 1
• 0 ? x2 ? 1
• 0 ? x3 ? 1
• LPR (?) ? 3/2
• Implies LPR (?) 3/2
• So x1 x2 x3 ½ ? LPR (?) ? 3/2
• ILP (?) 2
• Integrality Gap (IG) ILP / LPR 4/3

• What is the integrality gap for (MVCP)
• For a complete graph
• ILP (Kn) n - 1
• LPR (Kn) n/2
• Integrality Gap (IG) ILP / LPR
• Integrality gap may be as large as 2 (2 / n)
• For more information
• http//www-unix.mcs.anl.gov/otc/Guide/faq/linear-p
  rogramming-faq.html

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