Basics of polyhedral theory, flows an d ne twor ks CO@W Berlin Mar tin Gr ötschel 28.09.2015 Martin Grötschel . Institut für Mathematik, Technische Universität Berlin (TUB) . DFG-Forschungszentrum “Mathematik für Schlüsseltechnologien” (MATHEON) . Konrad-Zuse-Zentrum für Informationstechnik Berlin (ZIB) [email protected] http://www.zib.de/groetschel 2 Graph Theory: Super Quick CO@W . Graph G=(V,E), nodes, edges e=ij . Digraph D=(V,A), nodes arcs a=(u,v) Concepts . Chain, walk, path, cycle, circuit . clique, stable set, matching . coloring, clique cover, clique partitioning, edge coloring . … . Optimization problems associated with these . Polynomial time solvability, NP-hardness I assume that this is known Martin Grötschel 3 Special „simple“ combinatorial optimization problems CO@W Finding a . minimum spanning tree in a graph . shtthortest pa thith in a ditddirected graph . maximum matching in a graph . minimum capacity cut separating two given nodes of a gpgraph or dig gpraph . cost-minimal flow through a network with capacities and costs on all edges . … These problems are solvable in polynomial time. Martin Grötschel 4 Special „hard“ combinatorial optimization problems CO@W . travelling salesman problem (the prototype problem) . location und routing . set-packing, partitioning, -covering . max-cut . linear ordering . scheduling (with a few exceptions) . node and edge colouring . … These problems are NP-hard (in the sense of complexity theory). Martin Grötschel 5 Contents CO@W 1. Linear programs 2. Polyhedra 3. Algorithms for ppyolyhedra - Fourier-Motzkin elimination - some Web resources 4. Semi-algebraic geometry 5. Faces of polyhedra 6. Flows, networks, min-max results 7. The travelling salesman ppypolytope Martin Grötschel 6 Contents CO@W 1. Linear programs 2. Polyhedra 3. Algorithms for ppyolyhedra - Fourier-Motzkin elimination - some Web resources 4. Semi-algebraic geometry 5. Faces of polyhedra 6. Flows, networks, min-max results 7. The travelling salesman ppypolytope Martin Grötschel 7 Linear Programming CO@W max cxT max cx11 cx 2 2... cxnn subject to Ax b ax ax... ax b 11 1 12 2 1nn 1 x 0 ax21 1 ax 22 2... ax 2nn b 2 . linear program in standard form axmm11 ax 2 2 ... ax mnnm b xx12, ,..., xn 0 Martin Grötschel 8 Linear Programming CO@W max cxT linear max cxT program Axb Ax b in standard Axb x 0 form x 0 linear max cxT program max cxTT cx Axb in AxAxIsb “polyhedralpolyhedral form” xxs,, 0 ()x xx Martin Grötschel 9 Contents CO@W 1. Linear programs 2. Polyhedra 3. Algorithms for ppyolyhedra - Fourier-Motzkin elimination - some Web resources 4. Semi-algebraic geometry 5. Faces of polyhedra 6. Flows, networks, min-max results 7. The travellinggpyp salesman polytope Martin Grötschel 10 A Polytope in the Plane CO@W Martin Grötschel 11 A Polytope in 3 -dimensional space CO@W Martin Grötschel 12 „beautiful“ polyehedra CO@W •a tetrahedron, •a cube , •an octahedron, •a dodecahedron, •an icosahedron, •a cuboctahedron, •an icosidodecahedron, and •a rhombitruncated cuboctahedron. Martin Grötschel 13 “Real” polyhedra CO@W Martin Grötschel 14 Rhombicuboctahedron CO@W Herrnhuter Stern Germany’s most popular Christmas star Martin Grötschel 15 Crystallographic classifications CO@W http://de. wikipedia.org/wiki/Kristallmorphologie Martin Grötschel 16 Polyhedra-Poster http://www.peda .com/posters/Welcome . html CO@W PtPoster which di spl ays all convex polyhedra with regular polygonal faces Martin Grötschel 17 http://www.eg-models.de/ CO@W Martin Grötschel 18 http://www.ac-noumea.nc/maths/amc/polyhedr/index_.htm CO@W Martin Grötschel 19 CO@W Martin Grötschel 20 Kepler’s solar system CO@W Martin Grötschel 21 Polyhedra have fascinated people during all periods of our history CO@W . book illustrations . magic objects . pieces of art . objjyyects of symmetry . models of the universe Martin Grötschel 22 Definitions CO@W Linear programming lives (for our purposes) in the n-dimensional real (in patiepractice: rational) vetoector spac e. convex polyhedral cone: conic combination (i. e ., nonnegative linear combination or conical hull) of finitely many points K = cone(E), E a finite set in n. polytope: convex hull of finitely many points: P = conv(V), V a finite set in n. polyhedron: intersection of finitely many halfspaces Px{|}Rn Axb Martin Grötschel 23 Important theorems of polyhedral theory (LP -view) CO@W When is a polyhedron nonempty? Martin Grötschel 24 Important theorems of polyhedral theory (LP -view) CO@W When is a polyhedron nonempty? The Farkas-Lemma (1908): A polyhedron defined by an inequality system Ax b is empty, if and only if there is a vector y such that yy0,,,TTTTA 0 , y b 0 Theorem of the alternative Martin Grötschel 25 Important theorems of polyhedral theory (LP -view) CO@W Which forms of representation do polyhedra have? Martin Grötschel 26 Important theorems of polyhedral theory (LP -view) CO@W Which forms of representation do polyhedra have? Minkowski (1896), Weyl (1935), Steinitz (1916) Motzkin (1936) Theorem: For a subset P of R n the following are equivalent: (1) P is a polyhedron. (2) P is the intersection of finitely many halfspaces , i. e., there exist a matrix A und ein vector b with Px{|}Rn Axb . (exterior representation) (3) P is the sum of a convex polytope and a finitely generated (polyhedral) cone , i .e ., there exist finite sets V and E with P conv(V)+cone (E). (interior representation ) Martin Grötschel 27 Representations of polyhedra CO@W Carathéodory‘s Theorem (1911), 1873 Berlin – 1950 München Let xP conv(V)+cone(E) , there exist s vv00,...,ssi V, ,...,R , 1 i0 and est1 ,...,e E, s+1 ,..., t R with t n such that s t x iive+ i i i1i1is1=s+1 Martin Grötschel 28 Representations of polyhedra CO@W (1) - x2 <= 0 The -representation (2) -x1 -x2<2 <=-1 ()(exterior representation) (3) - x1 + x2 <= 3 (4) + x1 <= 3 (5)+x1+2x2<=9(5) + x1 + 2x2 <= 9 Ax b (4) (1) Martin Grötschel 29 Representations of polyhedra CO@W The -representation (interior representation) P conv(V)+cone(E). P V E Martin Grötschel 30 Example: the Tetrahedron CO@W 01 00 yconv 0,0,1,0 0001 yyy1231 0 y 0 0 1 1 y 0 0 2 1 1 0 0 0 0 Martin 0 y 0 Grötschel 0 3 31 Example: the cross polytope CO@W 2n points n P conv eii, e | i 1,..., n R Martin Grötschel 32 Example: the cross polytope CO@W 2n points n P conv eii, e | i 1,..., n R 2n inequalities n T n Px R |1 ax a 1,1 Martin Grötschel 33 Example: the cross polytope CO@W 2n points n P conv eii, e | i 1,..., n R The “power” of |.|. n n Px R |1 xi i1 2n inequalities n T n Px R |1 ax a 1,1 Martin Grötschel 34 All 3-dimensional 0/1-polytopes CO@W Martin Grötschel 35 Contents CO@W 1. Linear programs 2. Polyhedra 3. Alggpyorithms for polyhedra - Fourier-Motzkin elimination - some Web resources 4. Semi-algebraic geometry 5. Faces of polyhedra 6. Flows, networks, min-max results Martin Grötschel 36 Polyedra in linear programming CO@W . The solution sets of linear programs are polyhedra. If a polyhedron P conv(V)+cone(E) is given explicitly via finite sets V und E, linear programming is trivial. In linear programming, polyhedra are always given in -representation. Each solution method has its „standard form“. Martin Grötschel 37 Fourier-Motzkin Elimination CO@W . Fourier, 1847 . Motzkin, 1938 . Method: successive projection of a polyhedron in n- dimensional space into a vector space of dimension n-1 by elimination of one variable. Projection on y: (0,y) Martin Projection on x: (x,0) Grötschel 38 AFourierA Fourier-Motzkin step CO@W 0 â1 . 1 a1 + . 1 . -1 al + . 0 ân -1 am 0 b1 0 b1 . copy . 0 bk 0 bk Martin Grötschel 39 Fourier-Motzkin elimination proves the Farkas Lemma CO@W When is a polyhedron nonempty? The Farkas-Lemma (1908): A polyhedron defined by an inequality system Ax b is empty, if and only if there is a vector y such that yy0,,,TTTTA 0 , y b 0 Martin Grötschel 40 Fourier-Motzkin Elimination: an example CO@W min/max + x1 + 3x2 (1) - x2 <= 0 (2) - x1 - x2 <=-1 (3) - x1 + x2 <= 3 (4)+x1(4) + x1 <= 3 (5) + x1 + 2x2 <= 9 (4) (1) Martin Grötschel 41 Fourier-Motzkin Elimination: an example CO@W (1) - x2 <= 0 (2) - x1 - x2 <=-8 (3) - x1 + x2 <= 3 (4)+x1(4) + x1 <= 3 (5) + x1 + 2x2 <= 9 (4) (1) Martin Grötschel 42 Fourier-Motzkin Elimination: an example, call of PORTA CO@W DIM = 3 INEQUALITIES_SECTION (1) - x2 <= 0 (1) - x2 <= 0 (2) - x1 - x2 <=-8 (2) - x1 - x2 <=-8 (3) - x1 + x2 <= 3 (3) - x1 + x2 <= 3 (4) + x 1 <= 3 (4) + x 1 <= 3 (5) + x1 + 2x2 <= 9 (5) + x1 + 2x2 <= 9 ELIMINATION_ORDER 1 0 Martin Grötschel 43 Fourier-Motzkin Elimination: an example, call of PORTA CO@W DIM = 3 DIM = 3 INEQUALITIES_SECTION INEQUALITIES_SECTION (1) (1) -x202 <= 0 (1) - x2 <= 0 (2,4) (2) - x2 <= -5 (2) - x1 - x2 <=-8 (2,5) (3) + x2 <= 1 (3) - x1 + x2 <= 3 (3,4) (4) + x 2 <= 6 (4) + x 1 <= 3 (3,5) (5) + x2 <= 4 (5) + x1 + 2x2 <= 9 ELIMINATION_ORDER 1 0 Martin Grötschel 44 Fourier-Motzkin Elimination: an example, call of PORTA CO@W DIM = 3 DIM = 3 INEQUALITIES_SECTION INEQUALITIES_SECTION (1) (1) -x202 <= 0 (2,3) 0 <= -4 (2,4) (2) - x2 <= -5 (2,5) (3) + x2 <= 1 (3,4) (4) + x 2 <= 6 (3,5) (5) + x2 <= 4 ELIMINATION_ORDER 0 1 Martin Grötschel 45 Fourier-Motzkin elimination proves the Farkas Lemma CO@W When is a polyhedron nonempty? The Farkas-Lemma (1908): A polyhedron defined by an inequality system Ax b is empty, if and only if there is a vector y such that yy0,,,TTTTA 0 , y b 0 Martin Grötschel 46 Which LP solvers are used in practice? CO@W .