GAPs for the CUT n Extreme GAPs Separation Open Questions Complexity Results for the Gap Inequalities Laura Galli Konstantinos Kaparis Adam N. Letchford 16 th Combinatorial Optimization Workshop, Aussois, France Complexity Results for the Gap Inequalities Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUT n Extreme GAPs Separation Open Questions Motivation In 1996, Laurent & Poljak introduced a very general class of valid inequalities for the cut polytope , called gap inequalities . Gap inequalities were largely ignored thereafter and Complexity Results for the Gap Inequalities Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUT n Extreme GAPs Separation Open Questions Motivation In 1996, Laurent & Poljak introduced a very general class of valid inequalities for the cut polytope , called gap inequalities . Gap inequalities were largely ignored thereafter and • the relevant complexity results are very limited, • while there is not known separation algorithm for them. Complexity Results for the Gap Inequalities Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUT n Extreme GAPs Separation Open Questions Outline 1 Gap Inequalities for the Cut Polytope 2 On Extreme Gap Inequalities 3 On the Complexity of Separation 4 Open Questions Complexity Results for the Gap Inequalities Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUT n Extreme GAPs Separation Open Questions The Cut Polytope Given a graph G = ( V , E ) the edge set δ ( S ) = { ( i , j ) ∈ E : i ∈ S , j ∈ ( V \ S ) , S ⊆ V } , defines a cut of G . A vector x ∈ { 0 , 1 } ( n 2 ) is the incidence vector of a cut in K n iff it satisfies the following set of triangle inequalities : ≤ 2 (1 ≤ i < j < k ≤ n ) x ij + x ik + x jk x ij − x ik − x jk ≤ 0 (1 ≤ i < j ≤ n ; k � = i , j ) The cut polytope (CUT n ), is the convex hull of such vectors. CUT n has been studied intensively in the literature and many classes of valid inequalities are known (see Deza & Laurent, 1997). Complexity Results for the Gap Inequalities Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUT n Extreme GAPs Separation Open Questions Gap Inequalities: Definition (Laurent & Poljak, 1997) Gap inequalities for CUT n , take the following form: σ ( b ) 2 − γ ( b ) 2 � � ( ∀ b ∈ Z n ) . � b i b j x ij ≤ / 4 1 ≤ i < j ≤ n Here, σ ( b ) denotes � i ∈ V b i , and � | z T b | : z ∈ {± 1 } n � γ ( b ) := min is the so-called gap of b . Every gap inequality defines a proper face of CUT n . Complexity Results for the Gap Inequalities Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUT n Extreme GAPs Separation Open Questions Gap Inequalities: Definition (Laurent & Poljak, 1997) Gap inequalities for CUT n , take the following form: σ ( b ) 2 − γ ( b ) 2 � � ( ∀ b ∈ Z n ) . � b i b j x ij ≤ / 4 1 ≤ i < j ≤ n Here, σ ( b ) denotes � i ∈ V b i , and � | z T b | : z ∈ {± 1 } n � γ ( b ) := min is the so-called gap of b . Every gap inequality defines a proper face of CUT n . Conjecture All facet defining gap inequalities have γ ( b ) = 1 . Complexity Results for the Gap Inequalities Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUT n Extreme GAPs Separation Open Questions Example 1 2 ① ① ❅ � � ❅ � ❅ � ❅ � ❅ ① ① 3 4 Complexity Results for the Gap Inequalities Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUT n Extreme GAPs Separation Open Questions Example ✬ ✩ ✬ ✩ S V \ S ❤❤❤❤❤❤❤❤❤❤❤❤❤ ✭ ✭✭✭✭✭✭✭✭✭✭✭✭✭ 1 t t 3 t ❤ 4 2 t ✫ ✪ ✫ ✪ Complexity Results for the Gap Inequalities Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUT n Extreme GAPs Separation Open Questions Example σ ( b ) = P i =4 i =1 b i ✬ ✩ ✬ ✩ S V \ S ❤❤❤❤❤❤❤❤❤❤❤❤❤ b 1 b 4 ✭ ✭✭✭✭✭✭✭✭✭✭✭✭✭ b 1 1 t t 3 b 3 t b 1 b 3 ❤ 4 b 2 2 t b 4 ✫ ✪ ✫ ✪ Complexity Results for the Gap Inequalities Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUT n Extreme GAPs Separation Open Questions Example σ ( b ) = P i =4 i =1 b i P i ∈ S b i = σ ( b ) / 2 P i ∈ V \ S b i = σ ( b ) / 2 ✬ ✩ ✬ ✩ S V \ S ❤❤❤❤❤❤❤❤❤❤❤❤❤ b 1 b 4 ✭ ✭✭✭✭✭✭✭✭✭✭✭✭✭ b 1 1 t t 3 b 3 t b 1 b 3 ❤ 4 b 2 2 t b 4 ✫ ✪ ✫ ✪ 1 ≤ i < j ≤ n b i b j x ij ≤ σ ( b ) 2 / 4 P Complexity Results for the Gap Inequalities Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUT n Extreme GAPs Separation Open Questions Example σ ( b ) = P i =4 i =1 b i = 4 P i ∈ S b i = 2 P i ∈ V \ S b i = 2 ✬ ✩ ✬ ✩ S V \ S ❤❤❤❤❤❤❤❤❤❤❤❤❤ ✭ ✭✭✭✭✭✭✭✭✭✭✭✭✭ b 1 = 1 1 t t 3 b 3 = 1 t ❤ 4 b 2 = 1 2 t b 4 = 1 ✫ ✪ ✫ ✪ P 1 ≤ i < j ≤ 4 x ij ≤ 4 Complexity Results for the Gap Inequalities Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUT n Extreme GAPs Separation Open Questions Example | z T b | : z ∈ {± 1 } n ¯ ˘ γ ( b ) := min σ ( b ) = P i =3 i =1 b i = 3 P i ∈ S b i = ( σ ( b ) + γ ( b )) / 2 P i ∈ V \ S b i = ( σ ( b ) − γ ( b )) / 2 ✬ ✩ ✬ ✩ S V \ S ✭ ✭✭✭✭✭✭✭✭✭✭✭✭✭ b 1 = 1 1 t t 3 b 3 = 1 b 2 = 1 2 t ✫ ✪ ✫ ✪ σ ( b ) 2 − γ ( b ) 2 ´ ` P 1 ≤ i < j ≤ n b i b j x ij ≤ / 4 Complexity Results for the Gap Inequalities Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUT n Extreme GAPs Separation Open Questions Example | z T b | : z ∈ {± 1 } n ¯ ˘ γ ( b ) := min σ ( b ) = P i =3 i =1 b i = 3 P i ∈ S b i = ( σ ( b ) + γ ( b )) / 2 P i ∈ V \ S b i = ( σ ( b ) − γ ( b )) / 2 ✬ ✩ ✬ ✩ S V \ S ✭ ✭✭✭✭✭✭✭✭✭✭✭✭✭ b 1 = 1 1 t t 3 b 3 = 1 b 2 = 1 2 t ✫ ✪ ✫ ✪ P 1 ≤ i < j ≤ 3 x ij ≤ 2 Complexity Results for the Gap Inequalities Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUT n Extreme GAPs Separation Open Questions Gap Inequalities: Dominated or Implied Inequalities Gap inequalities are remarkably general... Complexity Results for the Gap Inequalities Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUT n Extreme GAPs Separation Open Questions Gap Inequalities: Dominated or Implied Inequalities Gap inequalities are remarkably general... σ ( b ) 2 − γ ( b ) 2 � � ∀ b ∈ Z n b i b j x ij ≤ � / 4 1 ≤ i < j ≤ n gap Complexity Results for the Gap Inequalities Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUT n Extreme GAPs Separation Open Questions Gap Inequalities: Dominated or Implied Inequalities Gap inequalities are remarkably general... σ ( b ) 2 − γ ( b ) 2 � � ∀ b ∈ Z n � b i b j x ij ≤ / 4 1 ≤ i < j ≤ n X b i b j x ij ≤ ⌊ σ ( b ) 2 / 4 ⌋ and σ ( b ) odd ∀ b ∈ Z n 1 ≤ i < j ≤ n ✲ gap rounded psd Complexity Results for the Gap Inequalities Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUT n Extreme GAPs Separation Open Questions Gap Inequalities: Dominated or Implied Inequalities Gap inequalities are remarkably general... σ ( b ) 2 − γ ( b ) 2 � � ∀ b ∈ Z n � b i b j x ij ≤ / 4 1 ≤ i < j ≤ n X ∀ b ∈ R n b i b j x ij ≤ σ ( b ) 2 / 4 1 ≤ i < j ≤ n ✲ ✲ psd gap rounded psd Complexity Results for the Gap Inequalities Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUT n Extreme GAPs Separation Open Questions Gap Inequalities: Dominated or Implied Inequalities Gap inequalities are remarkably general... σ ( b ) 2 − γ ( b ) 2 � � ∀ b ∈ Z n � b i b j x ij ≤ / 4 1 ≤ i < j ≤ n X b ∈ Z n b i b j x ij ≤ σ ( b ) 2 / 4 and γ ( b ) = 0 ∀ 1 ≤ i < j ≤ n ✲ ✲ psd ✲ gap-0 gap rounded psd Complexity Results for the Gap Inequalities Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUT n Extreme GAPs Separation Open Questions Gap Inequalities: Dominated or Implied Inequalities Gap inequalities are remarkably general... σ ( b ) 2 − γ ( b ) 2 � � ∀ b ∈ Z n � b i b j x ij ≤ / 4 1 ≤ i < j ≤ n X b ∈ Z n b i b j x ij ≤ 0 and σ ( b ) = 0 ∀ 1 ≤ i < j ≤ n ✲ ✲ psd ✲ gap-0 ✲ negative-type gap rounded psd Complexity Results for the Gap Inequalities Laura Galli, Konstantinos Kaparis, Adam N. Letchford
GAPs for the CUT n Extreme GAPs Separation Open Questions Gap Inequalities: Dominated or Implied Inequalities Gap inequalities are remarkably general... σ ( b ) 2 − γ ( b ) 2 � � ∀ b ∈ Z n � b i b j x ij ≤ / 4 1 ≤ i < j ≤ n σ ( b ) 2 − γ ( b ) 2 ´ b ∈ Z n X ` b i b j x ij ≤ / 4 and γ ( b ) = 1 ∀ 1 ≤ i < j ≤ n ✲ ✲ psd ✲ gap-0 ✲ negative-type gap rounded psd ❆ ❆ ❆ ❆ ❆ ❆ ❯ gap-1 Complexity Results for the Gap Inequalities Laura Galli, Konstantinos Kaparis, Adam N. Letchford
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