Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem - PowerPoint PPT Presentation

Background Sublinear Upper Bounds Lower Bounds Final remarks Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut Robert Hildebrand* Rico Zenklusen Institute for Operations Research, ETH Z¨ urich & IBM T.J. Watson Research Center January 7, 2016 Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks Presentation Outline 1 Background 2 Sublinear Upper Bounds 3 Lower Bounds 4 Final remarks Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks Helly (1914) Let C be a collection of compact convex sets in R d . If ( d +1) � � C i � = ∅ ⇒ C � = ∅ . i =1 C ∈C Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks Helly (1914) Doignon (1973) Let C be a collection of compact convex sets in R n . If 2 n C i ∩ Z n � = ∅ C ∩ Z n � = ∅ . � � ⇒ i =1 C ∈C Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks Helly (1914) Doignon (1973) Averkov-Weismantel (2012) Let C be a collection of compact convex sets in R d + n . If ( d +1)2 n C i ∩ ( R d × Z n ) � = ∅ C ∩ ( R d × Z n ) � = ∅ . � � ⇒ i =1 C ∈C Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks • I. Aliev, R. Bassett, J. A. De Loera, and Q. Louveaux, A quantitative Doignon-Bell-Scarf theorem, (2014) • G. Averkov, On maximal S-free sets and the Helly number for S-convex sets, (2013) • G. Averkov and R. Weismantel, Transversal numbers over subsets of linear spaces, (2012) • G. Kalai and R. Meshulam, A topological colorful Helly theorem, (2005) • G. Kalai and R. Meshulam, Leray numbers of projections and a topological Helly-type theorem,(2008) • C. Knauer, H. R. Tiwary, and D. Werner, On the computational complexity of ham-sandwich cuts, Helly sets, and related problems, (2011) • L. Montejano and D. Oliveros, Colourful transversal theorems, (2008) • Montejano and D. Oliveros, Tolerance in Helly-type theorems, (2011) • L. Montejano, A New Topological Helly Theorem and Some Transversal Results, (2014) • P. Sober´ on, Quantitative (p, q) theorems in combinatorial geometry, (2015) • P. Sober´ on, Helly-type theorems for the diameter, (2015) • K. Swanepoel, Helly-type theorems for homothets of planar convex curves,(2003) • R. Wenger, Helly-type theorems and geometric transversals, (2004) Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks Discrete Quantitative Theorem Definition c ( n , k ) is the least integer such that for any A ∈ R m × n , and b ∈ R m , if { x ∈ R n : Ax ≤ b } has exactly k integer solutions, then there exits a subset S of the rows of A with | S | ≤ c ( n , k ) such that { x ∈ R n : A S x ≤ b S } has exactly k integer solutions. Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks Discrete Quantitative Theorem Definition c ( n , k ) is the least integer such that for any A ∈ R m × n , and b ∈ R m , if { x ∈ R n : Ax ≤ b } has exactly k integer solutions, then there exits a subset S of the rows of A with | S | ≤ c ( n , k ) such that { x ∈ R n : A S x ≤ b S } has exactly k integer solutions. Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks Discrete Quantitative Theorem Definition c ( n , k ) is the least integer such that for any A ∈ R m × n , and b ∈ R m , if { x ∈ R n : Ax ≤ b } has exactly k integer solutions, then there exits a subset S of the rows of A with | S | ≤ c ( n , k ) such that { x ∈ R n : A S x ≤ b S } has exactly k integer solutions. Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks Discrete Quantitative Theorem Definition c ( n , k ) is the least integer such that for any A ∈ R m × n , and b ∈ R m , if { x ∈ R n : Ax ≤ b } has exactly k integer solutions, then there exits a subset S of the rows of A with | S | ≤ c ( n , k ) such that { x ∈ R n : A S x ≤ b S } has exactly k integer solutions. Corollary (ABDL ’14) Let C be a finite collection of compact convex sets in R n . If � � c ( n , k ) � � � � � � � � � C i ∩ Z n � C ∩ Z n ≥ k + 1 ⇒ � ≥ k + 1 . � � � � � � � � i =1 � C ∈C � � Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks Bounds on c ( n , k ) Upper bounds c ( n , k ) ≤ ( k + 2) n 1 Bell (1977): c ( n , k ) ≤ ⌈ 2( k + 1) / 3 ⌉ 2 n − 2 ⌈ 2( k + 1) / 3 ⌉ + 2 2 ABDL (2014): 3 This talk: c ( n , k ) = o ( k )2 n Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks Bounds on c ( n , k ) Upper bounds c ( n , k ) ≤ ( k + 2) n 1 Bell (1977): c ( n , k ) � 2 3 k 2 n 2 ABDL (2014): 3 This talk: c ( n , k ) = o ( k )2 n Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks Bounds on c ( n , k ) Upper bounds c ( n , k ) ≤ ( k + 2) n 1 Bell (1977): c ( n , k ) � 2 3 k 2 n 2 ABDL (2014): 3 This talk: c ( n , k ) = o ( k )2 n 1 3 ) 4 Bell (1977): c (2 , k ) = O ( k Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks Bounds on c ( n , k ) Upper bounds c ( n , k ) ≤ ( k + 2) n 1 Bell (1977): c ( n , k ) � 2 3 k 2 n 2 ABDL (2014): 3 This talk: c ( n , k ) = o ( k )2 n 1 3 ) 4 Bell (1977): c (2 , k ) = O ( k Lower bounds 1 3 ) 1 ??? c (2 , k ) = Ω( k n − 1 n +1 ) 2 This talk: c ( n , k ) = Ω n ( k Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks Presentation Outline 1 Background 2 Sublinear Upper Bounds 3 Lower Bounds 4 Final remarks Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks Bell’s expanding polyhedron lemma ⇒ Lemma: There exists a polytope with c ( n , k ) facets, exactly one integer point on each facet found on its relative interior, and containing exactly k integer points not on the boundary. Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks Bell’s expanding polyhedron lemma ⇒ Lemma: There exists a polytope with c ( n , k ) facets, exactly one integer point on each facet found on its relative interior, and containing exactly k integer points not on the boundary. Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks Convex hull of lattice points Definition ℓ ( n , k ) is the least integer such that for any S ⊆ Z n with | S | ≥ ℓ ( n , k ) and S in convex position, | (conv( S ) ∩ Z n ) \ S | ≥ k , Lemma c ( n , k ) < ℓ ( n , k + 1) Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks Convex hull of lattice points Definition ℓ ( n , k ) is the least integer such that for any S ⊆ Z n with | S | ≥ ℓ ( n , k ) and S in convex position, | (conv( S ) ∩ Z n ) \ S | ≥ k , Lemma c ( n , k ) < ℓ ( n , k + 1) Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen

Background Sublinear Upper Bounds Lower Bounds Final remarks Convex hull of lattice points Definition ℓ ( n , k ) is the least integer such that for any S ⊆ Z n with | S | ≥ ℓ ( n , k ) and S in convex position, | (conv( S ) ∩ Z n ) \ S | ≥ k , Theorem: [ABDL ’14] ℓ ( n , k ) ≤ ⌈ 2 k / 3 ⌉ 2 n − 2 ⌈ 2 k / 3 ⌉ + 2 Sublinear bounds for a quantitative Doignon-Bell-Scarf Theorem Stephen R. Chestnut, Robert Hildebrand*, Rico Zenklusen