DOCSLIB.ORG

Polyhedral Theory, Flows an D Ne Twork S CO@W Berlin

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

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 .
Recommended publications
  • 7 LATTICE POINTS and LATTICE POLYTOPES Alexander Barvinok

    7 LATTICE POINTS and LATTICE POLYTOPES Alexander Barvinok

    7 LATTICE POINTS AND LATTICE POLYTOPES Alexander Barvinok INTRODUCTION Lattice polytopes arise naturally in algebraic geometry, analysis, combinatorics, computer science, number theory, optimization, probability and representation the- ory. They possess a rich structure arising from the interaction of algebraic, convex, analytic, and combinatorial properties. In this chapter, we concentrate on the the- ory of lattice polytopes and only sketch their numerous applications. We briefly discuss their role in optimization and polyhedral combinatorics (Section 7.1). In Section 7.2 we discuss the decision problem, the problem of finding whether a given polytope contains a lattice point. In Section 7.3 we address the counting problem, the problem of counting all lattice points in a given polytope. The asymptotic problem (Section 7.4) explores the behavior of the number of lattice points in a varying polytope (for example, if a dilation is applied to the polytope). Finally, in Section 7.5 we discuss problems with quantifiers. These problems are natural generalizations of the decision and counting problems. Whenever appropriate we address algorithmic issues. For general references in the area of computational complexity/algorithms see [AB09]. We summarize the computational complexity status of our problems in Table 7.0.1. TABLE 7.0.1 Computational complexity of basic problems. PROBLEM NAME BOUNDED DIMENSION UNBOUNDED DIMENSION Decision problem polynomial NP-hard Counting problem polynomial #P-hard Asymptotic problem polynomial #P-hard∗ Problems with quantifiers unknown; polynomial for ∀∃ ∗∗ NP-hard ∗ in bounded codimension, reduces polynomially to volume computation ∗∗ with no quantifier alternation, polynomial time 7.1 INTEGRAL POLYTOPES IN POLYHEDRAL COMBINATORICS We describe some combinatorial and computational properties of integral polytopes.
  • Planar Hypohamiltonian Graphs on 40 Vertices

    Planar Hypohamiltonian Graphs on 40 Vertices

    Planar Hypohamiltonian Graphs on 40 Vertices Mohammadreza Jooyandeh, Brendan D. McKay Research School of Computer Science, Australian National University, ACT 2601, Australia Patric R. J. Osterg˚ard,¨ Ville H. Pettersson Department of Communications and Networking, Aalto University School of Electrical Engineering, P.O. Box 13000, 00076 Aalto, Finland Carol T. Zamfirescu Department of Applied Mathematics, Computer Science and Statistics, Ghent University, Krijgslaan 281 - S9, 9000 Ghent, Belgium Abstract A graph is hypohamiltonian if it is not Hamiltonian, but the deletion of any single vertex gives a Hamiltonian graph. Until now, the smallest known planar hypohamiltonian graph had 42 vertices, a result due to Araya and Wiener. That result is here improved upon by 25 planar hypohamiltonian graphs of order 40, which are found through computer-aided generation of certain families of planar graphs with girth 4 and a fixed number of 4-faces. It is further shown that planar hypohamiltonian graphs exist for all orders greater than or equal to 42. If Hamiltonian cycles are replaced by Hamilto- nian paths throughout the definition of hypohamiltonian graphs, we get the definition of hypotraceable graphs. It is shown that there is a planar hypo- traceable graph of order 154 and of all orders greater than or equal to 156. We also show that the smallest planar hypohamiltonian graph of girth 5 has 45 vertices. Email addresses: [email protected] (Mohammadreza Jooyandeh), [email protected] (Brendan D. McKay), [email protected] (Patric R. J. Osterg˚ard),¨ [email protected] (Ville H. Pettersson), [email protected] (Carol T. Zamfirescu) URL: http://www.jooyandeh.com (Mohammadreza Jooyandeh), http://cs.anu.edu.au/~bdm (Brendan D.
  • Every Graph Occurs As an Induced Subgraph of Some Hypohamiltonian Graph

    Every Graph Occurs As an Induced Subgraph of Some Hypohamiltonian Graph

    View metadata, citation and similar papers at core.ac.uk brought to you by CORE Received: 27 October 2016 Revised: 31 October 2017 Accepted: 20 November 2017 provided by Ghent University Academic Bibliography DOI: 10.1002/jgt.22228 ARTICLE Every graph occurs as an induced subgraph of some hypohamiltonian graph Carol T. Zamfirescu1 Tudor I. Zamfirescu2,3,4 1 Department of Applied Mathematics, Com- Abstract puter Science and Statistics, Ghent University, Krijgslaan 281 - S9, 9000 Ghent, Belgium We prove the titular statement. This settles a problem 2Fachbereich Mathematik, Universität of Chvátal from 1973 and encompasses earlier results Dortmund, 44221 Dortmund, Germany of Thomassen, who showed it for 3, and Collier and 3 “Simion Stoilow” Institute of Mathematics, Schmeichel, who proved it for bipartite graphs. We also Roumanian Academy, Bucharest, Roumania show that for every outerplanar graph there exists a pla- 4College of Mathematics and Information Science, Hebei Normal University, 050024 nar hypohamiltonian graph containing it as an induced Shijiazhuang, P.R. China subgraph. Correspondence Carol T. Zamfirescu, Department of KEYWORDS Applied Mathematics, Computer Sci- hypohamiltonian, induced subgraph ence and Statistics, Ghent University, Krijgslaan 281 - S9, 9000 Ghent, Belgium. MSC 2010. Email: czamfi[email protected] 05C10, 05C45, 05C60 Funding information Fonds Wetenschappelijk Onderzoek 1 INTRODUCTION Consider a non-hamiltonian graph . We call hypohamiltonian if for every vertex in , the graph − is hamiltonian. In similar spirit, is said to be almost hypohamiltonian if there exists a vertex in , which we will call exceptional, such that − is non-hamiltonian, and for any vertex ≠ in , the graph − is hamiltonian. For an overview of results on hypohamiltonicity till 1993, see the survey by Holton and Sheehan [7].
  • On Stable Cycles and Cycle Double Covers of Graphs with Large Circumference

    On Stable Cycles and Cycle Double Covers of Graphs with Large Circumference

    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector Discrete Mathematics 312 (2012) 2540–2544 Contents lists available at SciVerse ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc On stable cycles and cycle double covers of graphs with large circumference Jonas Hägglund ∗, Klas Markström Department of Mathematics and Mathematical Statistics, Umeå University, SE-901 87 Umeå, Sweden article info a b s t r a c t Article history: A cycle C in a graph is called stable if there exists no other cycle D in the same graph such Received 11 October 2010 that V .C/ ⊆ V .D/. In this paper, we study stable cycles in snarks and we show that if a Accepted 16 August 2011 cubic graph G has a cycle of length at least jV .G/j − 9 then it has a cycle double cover. We Available online 23 September 2011 also give a construction for an infinite snark family with stable cycles of constant length and answer a question by Kochol by giving examples of cyclically 5-edge connected snarks Keywords: with stable cycles. Stable cycle ' 2011 Elsevier B.V. All rights reserved. Snark Cycle double cover Semiextension 1. Introduction In this paper, a cycle is a connected 2-regular subgraph. A cycle double cover (usually abbreviated CDC) is a multiset of cycles covering the edges of a graph such that each edge lies in exactly two cycles. The following is a famous open conjecture in graph theory. Conjecture 1.1 (CDCC).
  • Improved Bounds for Hypohamiltonian Graphs∗

    Improved Bounds for Hypohamiltonian Graphs∗

    Also available at http://amc-journal.eu ISSN 1855-3966 (printed edn.), ISSN 1855-3974 (electronic edn.) ARS MATHEMATICA CONTEMPORANEA 13 (2017) 235–257 Improved bounds for hypohamiltonian graphs∗ Jan Goedgebeur , Carol T. Zamfirescu Department of Applied Mathematics, Computer Science & Statistics, Ghent University, Krijgslaan 281-S9, 9000 Ghent, Belgium In loving memory of Ella. Received 23 February 2016, accepted 16 January 2017, published online 6 March 2017 Abstract A graph G is hypohamiltonian if G is non-hamiltonian and G − v is hamiltonian for every v 2 V (G). In the following, every graph is assumed to be hypohamiltonian. Aldred, Wormald, and McKay gave a list of all graphs of order at most 17. In this article, we present an algorithm to generate all graphs of a given order and apply it to prove that there exist exactly 14 graphs of order 18 and 34 graphs of order 19. We also extend their results in the cubic case. Furthermore, we show that (i) the smallest graph of girth 6 has order 25, (ii) the smallest planar graph has order at least 23, (iii) the smallest cubic planar graph has order at least 54, and (iv) the smallest cubic planar graph of girth 5 with non-trivial automorphism group has order 78. Keywords: Hamiltonian, hypohamiltonian, planar, girth, cubic graph, exhaustive generation. Math. Subj. Class.: 05C10, 05C38, 05C45, 05C85 1 Introduction Throughout this paper all graphs are undirected, finite, connected, and neither contain loops nor multiple edges, unless explicitly stated otherwise. A graph is hamiltonian if it contains a cycle visiting every vertex of the graph.
  • Polyhedral Combinatorics, Complexity & Algorithms

    Polyhedral Combinatorics, Complexity & Algorithms

    POLYHEDRAL COMBINATORICS, COMPLEXITY & ALGORITHMS FOR k-CLUBS IN GRAPHS By FOAD MAHDAVI PAJOUH Bachelor of Science in Industrial Engineering Sharif University of Technology Tehran, Iran 2004 Master of Science in Industrial Engineering Tarbiat Modares University Tehran, Iran 2006 Submitted to the Faculty of the Graduate College of Oklahoma State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY July, 2012 COPYRIGHT c By FOAD MAHDAVI PAJOUH July, 2012 POLYHEDRAL COMBINATORICS, COMPLEXITY & ALGORITHMS FOR k-CLUBS IN GRAPHS Dissertation Approved: Dr. Balabhaskar Balasundaram Dissertation Advisor Dr. Ricki Ingalls Dr. Manjunath Kamath Dr. Ramesh Sharda Dr. Sheryl Tucker Dean of the Graduate College iii Dedicated to my parents iv ACKNOWLEDGMENTS I would like to wholeheartedly express my sincere gratitude to my advisor, Dr. Balabhaskar Balasundaram, for his continuous support of my Ph.D. study, for his patience, enthusiasm, motivation, and valuable knowledge. It has been a great honor for me to be his first Ph.D. student and I could not have imagined having a better advisor for my Ph.D. research and role model for my future academic career. He has not only been a great mentor during the course of my Ph.D. and professional development but also a real friend and colleague. I hope that I can in turn pass on the research values and the dreams that he has given to me, to my future students. I wish to express my warm and sincere thanks to my wonderful committee mem- bers, Dr. Ricki Ingalls, Dr. Manjunath Kamath and Dr. Ramesh Sharda for their support, guidance and helpful suggestions throughout my Ph.D.
  • 3. Linear Programming and Polyhedral Combinatorics Summary of What Was Seen in the Introductory Lectures on Linear Programming and Polyhedral Combinatorics

    3. Linear Programming and Polyhedral Combinatorics Summary of What Was Seen in the Introductory Lectures on Linear Programming and Polyhedral Combinatorics

    Massachusetts Institute of Technology Handout 6 18.433: Combinatorial Optimization February 20th, 2009 Michel X. Goemans 3. Linear Programming and Polyhedral Combinatorics Summary of what was seen in the introductory lectures on linear programming and polyhedral combinatorics. Definition 3.1 A halfspace in Rn is a set of the form fx 2 Rn : aT x ≤ bg for some vector a 2 Rn and b 2 R. Definition 3.2 A polyhedron is the intersection of finitely many halfspaces: P = fx 2 Rn : Ax ≤ bg. Definition 3.3 A polytope is a bounded polyhedron. n n−1 Definition 3.4 If P is a polyhedron in R , the projection Pk ⊆ R of P is defined as fy = (x1; x2; ··· ; xk−1; xk+1; ··· ; xn): x 2 P for some xkg. This is a special case of a projection onto a linear space (here, we consider only coordinate projection). By repeatedly projecting, we can eliminate any subset of coordinates. We claim that Pk is also a polyhedron and this can be proved by giving an explicit description of Pk in terms of linear inequalities. For this purpose, one uses Fourier-Motzkin elimination. Let P = fx : Ax ≤ bg and let • S+ = fi : aik > 0g, • S− = fi : aik < 0g, • S0 = fi : aik = 0g. T Clearly, any element in Pk must satisfy the inequality ai x ≤ bi for all i 2 S0 (these inequal- ities do not involve xk). Similarly, we can take a linear combination of an inequality in S+ and one in S− to eliminate the coefficient of xk. This shows that the inequalities: ! ! X X aik aljxj − alk aijxj ≤ aikbl − alkbi (1) j j for i 2 S+ and l 2 S− are satisfied by all elements of Pk.
  • Linear Programming and Polyhedral Combinatorics Summary of What Was Seen in the Introductory Lectures on Linear Programming and Polyhedral Combinatorics

    Linear Programming and Polyhedral Combinatorics Summary of What Was Seen in the Introductory Lectures on Linear Programming and Polyhedral Combinatorics

    Massachusetts Institute of Technology Handout 8 18.433: Combinatorial Optimization March 6th, 2007 Michel X. Goemans Linear Programming and Polyhedral Combinatorics Summary of what was seen in the introductory lectures on linear programming and polyhedral combinatorics. Definition 1 A halfspace in Rn is a set of the form fx 2 Rn : aT x ≤ bg for some vector a 2 Rn and b 2 R. Definition 2 A polyhedron is the intersection of finitely many halfspaces: P = fx 2 Rn : Ax ≤ bg. Definition 3 A polytope is a bounded polyhedron. n Definition 4 If P is a polyhedron in R , the projection Pk of P is defined as fy = (x1; x2; · · · ; xk−1; xk+1; · · · ; xn) : x 2 P for some xkg. We claim that Pk is also a polyhedron and this can be proved by giving an explicit description of Pk in terms of linear inequalities. For this purpose, one uses Fourier-Motzkin elimination. Let P = fx : Ax ≤ bg and let • S+ = fi : aik > 0g, • S− = fi : aik < 0g, • S0 = fi : aik = 0g. T Clearly, any element in Pk must satisfy the inequality ai x ≤ bi for all i 2 S0 (these inequal- ities do not involve xk). Similarly, we can take a linear combination of an inequality in S+ and one in S− to eliminate the coefficient of xk. This shows that the inequalities: aik aljxj − alk akjxj ≤ aikbl − alkbi (1) j ! j ! X X for i 2 S+ and l 2 S− are satisfied by all elements of Pk. Conversely, for any vector (x1; x2; · · · ; xk−1; xk+1; · · · ; xn) satisfying (1) for all i 2 S+ and l 2 S− and also T ai x ≤ bi for all i 2 S0 (2) we can find a value of xk such that the resulting x belongs to P (by looking at the bounds on xk that each constraint imposes, and showing that the largest lower bound is smaller than the smallest upper bound).
  • ADDENDUM the Following Remarks Were Added in Proof (November 1966). Page 67. an Easy Modification of Exercise 4.8.25 Establishes

    ADDENDUM the Following Remarks Were Added in Proof (November 1966). Page 67. an Easy Modification of Exercise 4.8.25 Establishes

    ADDENDUM The following remarks were added in proof (November 1966). Page 67. An easy modification of exercise 4.8.25 establishes the follow­ ing result of Wagner [I]: Every simplicial k-"complex" with at most 2 k ~ vertices has a representation in R + 1 such that all the "simplices" are geometric (rectilinear) simplices. Page 93. J. H. Conway (private communication) has established the validity of the conjecture mentioned in the second footnote. Page 126. For d = 2, the theorem of Derry [2] given in exercise 7.3.4 was found earlier by Bilinski [I]. Page 183. M. A. Perles (private communication) recently obtained an affirmative solution to Klee's problem mentioned at the end of section 10.1. Page 204. Regarding the question whether a(~) = 3 implies b(~) ~ 4, it should be noted that if one starts from a topological cell complex ~ with a(~) = 3 it is possible that ~ is not a complex (in our sense) at all (see exercise 11.1.7). On the other hand, G. Wegner pointed out (in a private communication to the author) that the 2-complex ~ discussed in the proof of theorem 11.1 .7 indeed satisfies b(~) = 4. Page 216. Halin's [1] result (theorem 11.3.3) has recently been genera­ lized by H. A. lung to all complete d-partite graphs. (Halin's result deals with the graph of the d-octahedron, i.e. the d-partite graph in which each class of nodes contains precisely two nodes.) The existence of the numbers n(k) follows from a recent result of Mader [1] ; Mader's result shows that n(k) ~ k.2(~) .
  • Combinatorial Optimization, Packing and Covering

    Combinatorial Optimization, Packing and Covering

    Combinatorial Optimization: Packing and Covering G´erard Cornu´ejols Carnegie Mellon University July 2000 1 Preface The integer programming models known as set packing and set cov- ering have a wide range of applications, such as pattern recognition, plant location and airline crew scheduling. Sometimes, due to the spe- cial structure of the constraint matrix, the natural linear programming relaxation yields an optimal solution that is integer, thus solving the problem. Sometimes, both the linear programming relaxation and its dual have integer optimal solutions. Under which conditions do such integrality properties hold? This question is of both theoretical and practical interest. Min-max theorems, polyhedral combinatorics and graph theory all come together in this rich area of discrete mathemat- ics. In addition to min-max and polyhedral results, some of the deepest results in this area come in two flavors: “excluded minor” results and “decomposition” results. In these notes, we present several of these beautiful results. Three chapters cover min-max and polyhedral re- sults. The next four cover excluded minor results. In the last three, we present decomposition results. We hope that these notes will encourage research on the many intriguing open questions that still remain. In particular, we state 18 conjectures. For each of these conjectures, we offer $5000 as an incentive for the first correct solution or refutation before December 2020. 2 Contents 1Clutters 7 1.1 MFMC Property and Idealness . 9 1.2 Blocker . 13 1.3 Examples .......................... 15 1.3.1 st-Cuts and st-Paths . 15 1.3.2 Two-Commodity Flows . 17 1.3.3 r-Cuts and r-Arborescences .
  • Polyhedral Combinatorics : an Annotated Bibliography

    Polyhedral Combinatorics : an Annotated Bibliography

    Polyhedral combinatorics : an annotated bibliography Citation for published version (APA): Aardal, K. I., & Weismantel, R. (1996). Polyhedral combinatorics : an annotated bibliography. (Universiteit Utrecht. UU-CS, Department of Computer Science; Vol. 9627). Utrecht University. Document status and date: Published: 01/01/1996 Document Version: Publisher’s PDF, also known as Version of Record (includes final page, issue and volume numbers) Please check the document version of this publication: • A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the official published version of record. People interested in the research are advised to contact the author for the final version of the publication, or visit the DOI to the publisher's website. • The final author version and the galley proof are versions of the publication after peer review. • The final published version features the final layout of the paper including the volume, issue and page numbers. Link to publication General rights Copyright and moral rights for the publications made accessible in the public portal are retained by the authors and/or other copyright owners and it is a condition of accessing publications that users recognise and abide by the legal requirements associated with these rights. • Users may download and print one copy of any publication from the public portal for the purpose of private study or research. • You may not further distribute the material or use it for any profit-making activity or commercial gain • You may freely distribute the URL identifying the publication in the public portal.
  • 36 COMPUTATIONAL CONVEXITY Peter Gritzmann and Victor Klee

    36 COMPUTATIONAL CONVEXITY Peter Gritzmann and Victor Klee

    36 COMPUTATIONAL CONVEXITY Peter Gritzmann and Victor Klee INTRODUCTION The subject of Computational Convexity draws its methods from discrete mathe- matics and convex geometry, and many of its problems from operations research, computer science, data analysis, physics, material science, and other applied ar- eas. In essence, it is the study of the computational and algorithmic aspects of high-dimensional convex sets (especially polytopes), with a view to applying the knowledge gained to convex bodies that arise in other mathematical disciplines or in the mathematical modeling of problems from outside mathematics. The name Computational Convexity is of more recent origin, having first ap- peared in print in 1989. However, results that retrospectively belong to this area go back a long way. In particular, many of the basic ideas of Linear Programming have an essentially geometric character and fit very well into the conception of Compu- tational Convexity. The same is true of the subject of Polyhedral Combinatorics and of the Algorithmic Theory of Polytopes and Convex Bodies. The emphasis in Computational Convexity is on problems whose underlying structure is the convex geometry of normed vector spaces of finite but generally not restricted dimension, rather than of fixed dimension. This leads to closer con- nections with the optimization problems that arise in a wide variety of disciplines. Further, in the study of Computational Convexity, the underlying model of com- putation is mainly the binary (Turing machine) model that is common in studies of computational complexity. This requirement is imposed by prospective appli- cations, particularly in mathematical programming. For the study of algorithmic aspects of convex bodies that are not polytopes, the binary model is often aug- mented by additional devices called \oracles." Some cases of interest involve other models of computation, but the present discussion focuses on aspects of computa- tional convexity for which binary models seem most natural.