Algebraic-Geometric ideas in Discrete Optimization ...This book presents recent advances in the mathematical theory of discrete optimization, particularly those supported by methods

Algebraic-Geometric ideas in Discrete Optimization(Universidad de Sevilla, 2014)

Jesus A. De Loera, UC Davis

Lectures based on the book:

Algebraic & Geometric Ideas in the Theory of Discrete OptimizationBy J. De Loera, R. Hemmecke & M. Koppe

July 21, 2014

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Jesús A

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This book presents recent advances in the mathematical theory of discrete optimization, particularly those supported by methods from algebraic geometry, commutative algebra, convex and discrete geometry, generating functions, and other tools normally considered outside the standard curriculum in optimization.

Algebraic and Geometric Ideas in the Theory of Discrete Optimization • offersseveralresearchtechnologiesnotyetwellknownamongpractitioners

of discrete optimization,

• minimizesprerequisitesforlearningthesemethods,and

• providesatransitionfromlineardiscreteoptimizationtononlineardiscrete optimization.

This book can be used as a textbook for advanced undergraduates or beginning graduate students in mathematics, computer science, or operations research or as a tutorial for mathematicians, engineers, and scientists engaged in computation whowishtodelvemoredeeplyintohowandwhyalgorithmsdoordonotwork.

Jesús A. De Loera is a professor of mathematics and a member of the Graduate Groups in Computer Science and Applied Mathematics at University of California, Davis.HisresearchhasbeenrecognizedbyanAlexandervonHumboldtFellowship,theUCDavisChancellorFellowaward,andthe2010INFORMSComputingSocietyPrize. He is an associate editor of SIAM Journal on Discrete Mathematics and Discrete Optimization.

Raymond Hemmecke is a professor of combinatorial optimization at Technische Universität München. His research interests include algebraic statistics, computer

algebra, and bioinformatics.

Matthias Köppe is a professor of mathematics and a member of the Graduate Groups in Computer Science and Applied Mathematics at University of California, Davis. He is an associate editor of Mathematical Programming, Series A and Asia-Pacific Journal of Operational Research.

MO14

SocietyforIndustrial and Applied Mathematics

3600MarketStreet,6thFloorPhiladelphia,PA19104-2688USA

+1-215-382-9800•[email protected]•www.siam.org

MathematicalOptimizationSociety3600MarketStreet,6thFloor

Philadelphia,PA19104-2688USA+1-215-382-9800x319 Fax+1-215-386-7999

[email protected]•www.mathopt.org

ISBN 978-1-611972-43-6

MOS-SIAM Series on Optimization

AlgebrAic And geometric ideAs in the theory

of discrete optimizAtion

Jesús A. De LoeraRaymond HemmeckeMatthias Köppe

J.A.DeLoera,R.Hemmecke,M.Köppe

MO14_DeLoera-Koeppecover09-24-12.indd 1 10/12/2012 10:35:59 AM

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Menu for Seville Lectures

For Lecture ONE: Motivation and Overview

Non-Linear Polynomials and Discrete Optimization.

TAPAS of Non-Linear Discrete Optimization

Rest of the week: Dos raciones enterasFor Lectures TWO+THREE Augmentation Algorithms in Linear and IntegerOptimization

Graver Bases Methods

For Lecture THREE+FOUR Using Polynomials when one does not expect them!

Hilbert Nullstellensatz and Colorability problems.

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Historical Transitioning from linear

to non-linear constraints in models

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Now we have INTEGER or BINARY variables

max f (x)

subject to gi (x) ≤ 0, i = 1, 2, . . . , k ,

hj(x) = 0, j = 1, 2, . . . ,m,

x ∈ Rn1 × Zn2 .

(1)

Here the objective function f and the constraint functions gi , hj are assumed to bearbitrary real-valued functions.

KEY POINT: The study of these problems requires more ideas from Algebra,Geometry and Topology.

LET US GO BACK IN TIME....

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A Classical Example from the beginning of DiscreteOptimization

Initial work by Kantorovich (1939), T.C Koopmans (1941), von Neumann(1947).The Transportation problem: A company builds laptops in four factories,each with certain supply power. Four cities have laptop demands. There is acost ci,j for transporting a laptop from factory i to city j . What is the bestassignment of transport in order to minimize the cost?

ON FOUR CITIES

DEMANDS

220

215

93

64

108

286

71

127

SUPPLIES

BY FACTORIES

A silly way to solve this: run through all possibilities! Well how do I do this??Not so easy... If supply and demand are all ONE and if number of cities andfactories is n = 35, and a computer took 10−9 seconds to check onepossibility, it would take 200,000 years to solve!

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Modeling with LINEAR equations and inequalities

Let xi,j be a variable indicating number of laptops factory i provides to city j .xi,j can only take non-negative integer values, xi,j ≥ 0.

Then Since factory i produces ai laptops we have

n∑j=1

xi,j = ai , for all i = 1, . . . , n.

and since city j needs bj laptops

n∑i=1

xi,j = bj , for all j = 1, . . . , n.

Now we minimize∑

ci,jxi,j .

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Overview LINEAR Discrete Optimization (circa 1990)

Efficient computation with Convex Sets & Lattices ⇐⇒ Efficient Optimization

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Linear Discrete Optimization: The state of the art

Traditional Algorithms

Dual (polyhedral) techniques

max c>

x2

x1

x0max c>

x2

x0

x1

Cutting plane algorithms– based on polyhedral theory

Enumeration

max c> x0max c> x0max c> x0

Branch-and-bound

Adhoc methods

special structure(e.g. network,matroids, etc.)

Mathematical modelling – Strong initial IP formulation

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max c>

x2

x1

x0max c>

x2

x0

x1


Enumeration


Branch-and-bound

Adhoc methods



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max c>

x2

x1

x0

max c>

x2

x0

x1


Enumeration


Branch-and-bound

Adhoc methods



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max c>

x2

x1

x0

max c>

x2

x0

x1


Enumeration


Branch-and-bound

Adhoc methods



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max c>

x2

x1

x0

max c>

x2

x0

x1


Enumeration

max c> x0

max c> x0max c> x0

Branch-and-bound

Adhoc methods



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max c>

x2

x1

x0

max c>

x2

x0

x1


Enumeration

max c> x0

max c> x0

max c> x0

Branch-and-bound

Adhoc methods



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max c>

x2

x1

x0

max c>

x2

x0

x1


Enumeration

max c> x0max c> x0

max c> x0

Branch-and-bound

Adhoc methods



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max c>

x2

x1

x0

max c>

x2

x0

x1


Enumeration

max c> x0max c> x0

max c> x0

Branch-and-bound

Adhoc methods



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max c>

x2

x1

x0

max c>

x2

x0

x1


Enumeration

max c> x0max c> x0

max c> x0

Branch-and-bound

Adhoc methods



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MANY CHALLENGES!!

LIFE IS NON-LINEAR!!

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Example: Non-linear transportation polytopes

1 In the traditional transportation problem cost at an edge is a constant. Thuswe optimize a linear function.

2 but due to congestion or heavy traffic or heavy communication load thetransportation cost on an edge could be a non-linear function of the flow ateach edge.

3 For example cost at each edge is fij(xij) = cij |xij |aij for suitable constant aij .This results on a non-linear function

∑fij which is much harder to minimize.

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Reality is NON-LINEAR and worse!!

Non-linear Discrete Optimization

max/min f (x1, . . . , xd)

subject to gj(x1, . . . , xd) ≤ 0,

for j = 1 . . . s, and with

with xi integer

with f , gj Non-Linear

WHAT CAN BE DONE IN THISGENERAL CONTEXT??

Prove good theorems? Are thereefficient algorithms?

BAD NEWS: The problem isINCREDIBLY HARDTheorem It is UNDECIDABLEalready when f ,gi ’s arearbitrary polynomials (Jeroslow,1979).

EVEN WORSE Theorem: Itundecidable even with number ofvariables=10. (Matiyasevich andDavis 1982).

No theorem or algorithm performance canbe proved without ASSUMPTIONS

Let us see two nice theorems that wereproved in the last years

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max/min f (x1, . . . , xd)



with xi integer








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max/min f (x1, . . . , xd)



with xi integer








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max/min f (x1, . . . , xd)



with xi integer








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max/min f (x1, . . . , xd)



with xi integer








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ANOTHER GOOD REASON: Even baby problemsunsolvable with traditional techniques!

Market Share problem (Corneujols- Dawande, Williams)

minimizePm

i=1 |si | subject to the constraints

nXj=1

ai,jxj + si = di , i = 1, . . . , m

xj ∈ {0, 1}, j = 1, . . . , n, and all si ∈ integer

Nasty Knapsack problems (Aardal, Bixby et al)

Minimize or maximizeP10

i=1 xi , subject to xi ≥ 0 and

3719x1 + 20289x2 + 29067x3 + 60517x4 + 64354x5 + 65633x6 + 76969x7 +

102024x8 + 106036x9 + 119930x10 = 13385100

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MANY MORE CHALLENGES (not discussed here)

How to deal with Mixed Integer Non-Linear Programming?

How to deal with Multiobjective Programming?

How to deal with Uncertainty, Stochasticity? How to deal with error of data?Robustness?

How to deal with large-scale problems? Heuristics and Approximation?

We will see examples of new algebraic ideas to attack these problems

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TAPAS OF

ALGEBRAIC-GEOMETRIC

IDEAS FOR

DISCRETE OPTIMIZATION

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TAPAS 1 & 2Linear Optimization

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PART I: The Simplex Method

One of “10 top most-influential algorithms in the 20th century” by SIAM newsGeorge Dantzig, inventor of the simplex algorithm

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The simplex method

Lemma: Feasible region is a polyhedron. An optimal solution for an LP isamong the vertices of the polytope.

The simplex method walks or searches along the graph of the polytope, eachtime moving to a better and better cost!

Performance of the simplex method depends on the diameter of the graph ofthe polytope: largest distance between any pair of nodes.

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Bounding the Diameter

QUESTION: Is there a polynomial bound of the diameter in terms of the numberof facets and the dimension? The best general bounds are

Barnette-Larman: 2dim(P)−2

3(#facets(P)− dim(P) + 5/2).

Kalai-Kleitman: (#facets(P))log(dim(P))+1.

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The Hirsch Conjecture on a Polytopes

The story so far:

1957: Hirsch proposes that a d-dimensional polytope with n facets hasdiameter (maximal distance between any two vertices) no more than n − d .Known to be true in many instances, e.g. for polytopes with 0/1 vertices.

2010: Santos constructs a counterexample (43 dimensions and 86 facets)

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Could the diameter be still LINEAR ???

Experiments suggest that it is!!

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Diameters of Simplicial Complexes

We can use topological arguments! Idea goes back to the 1980’s.

Definition

The distance between two facets, F1,F2, is the length k of the shortestsimplicial path F1 = f0, f1, . . . , fk = F2.

The diameter of a simplicial complex is the maximum over all distancesbetween all pairs of vertices.

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Weak vertex-Decomposability

Definition

A d-dimensional complex ∆ is weakly vertex-decomposable if We recursively“peel off” the simplicial complex using a sequence of vertices so that finally wearrive to a simplex.

2

1

3

4

5 \2

1

3

4

5 \13

4

5

2-simplex!

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Why should we care??

Theorem (Billera,Provan, 1980)

If ∆ is a weakly vertex-decomposable complex, (n = f0(∆)):

diam(∆) ≤ 2f0(∆) = 2n.

BIG OLD QUESTION from 1980: All simplicial polyhedra weaklyvertex-decomposable?Theorem (2012, JDL and S. Klee) Not all simplicial polytopes are weaklyvertex decomposable!

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Interior-Point Methods

Narendra Karmarkar, inventor of interior point methods

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The Central Path of a Linear Program: Optimizers view

Linear Program: Maximizex∈Rn c · x s.t. A · x = b and x ≥ 0.

Replace by : Maximizex∈Rn fλ(x) s.t. A · x = b,

where λ ∈ R+ and fλ(x) := cT · x + λ∑n

i=1 log |xi |.

The maximum of the function fλ is attained by a unique point x∗(λ) in the theopen polytope {x ∈ (R>0)n : A · x = b}.

The central path is {x∗(λ) : λ > 0}.As λ→ 0 , the path leads from theanalytic center of the polytope, x∗(∞),to the optimal vertex, x∗(0).

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i=1 log |xi |.



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i=1 log |xi |.



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i=1 log |xi |.



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The Central Path of a Linear Program


Interior point methods = piecewise-linear approx. of this path

Key point: The convergence of Newton steps will depend on how “curvy” how“twisted” is the central path!!!

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CURVATURE: measuring how “twisted” is the central path

The total curvature of a curve C in Rm is the arc length of its image under theGauss map. γ : C → Sm−1.

Bounds on total curvature → bounds on # Newton steps.() July 21, 2014 29 / 55



Bounds on total curvature → bounds on # Newton steps.

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Bounds on total curvature → bounds on # Newton steps.() July 21, 2014 29 / 55

Central path is part of an algebraic curve!!!!

Consider the linear optimization problem in its primal and dual formulation:

(P) Maximize cT x subject to Ax = b and x ≥ 0;

(D) Minimize bT y subject to AT y − s = c and s ≥ 0.

The primal-dual central curve is the real curve that satisfies

Ax = b , AT y − s = c , and xi si = λ for i = 1, 2, . . . , n. (2)

We want to understand the PROJECTION of this curve into the x coordinates (ory variables).

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The Central Path is part of an Algebraic Curve

The central curve C is the Zariski closure of the central path.It contains the central paths of all polyhedra in the hyperplane arrangement{xi = 0}i=1,...,n ⊂ {A · x = b}.

−→Zariski closure

Bayer-Lagarias (1989) first studied the central path as an algebraic curve andsuggest the problem of identifying its defining equations.

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The Central Path is part of an Algebraic Curve

The central curve C is the Zariski closure of the central path.It contains the central paths of all polyhedra in the hyperplane arrangement{xi = 0}i=1,...,n ⊂ {A · x = b}.

−→Zariski closure

Bayer-Lagarias (1989) first studied the central path as an algebraic curve andsuggest the problem of identifying its defining equations.

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Questions and Contributions

Question 1: Find the defining equations for the central curve, what is the degree of the

curve?

Question 2: What is the degree and the maximum total curvature of the central pathgiven a matrix A?

Our contribution: We found explicit algebraic equations of the central curve, itsdegree, and refined bounds on its degree and total curvature.

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curve?



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curve?



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Example (continued!)

(n = 5, d = 2)

A =

(1 1 1 0 00 0 0 1 1

)c =

(1 2 0 4 0

)b =

(32

)Equations for C:

−2x2x3 + x1x3 + x1x2,4x2x4x5 − 4x1x4x5 + x1x2x5 − x1x2x4,4x3x4x5 − 4x1x4x5 − x1x3x5 + x1x3x4,4x3x4x5 − 4x2x4x5 − 2x2x3x5 + 2x2x3x4

x1 + x2 + x3 = 3

x4 + x5 = 2

h = (1, 2, 2) ⇒ deg(C) = 5, total curvature(C) ≤ 12π (≤ 16π)

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(n = 5, d = 2)

A =

(1 1 1 0 00 0 0 1 1

)c =

(1 2 0 4 0

)b =

(32

)Equations for C:


x1 + x2 + x3 = 3

x4 + x5 = 2


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(n = 5, d = 2)

A =

(1 1 1 0 00 0 0 1 1

)c =

(1 2 0 4 0

)b =

(32

)Equations for C:


x1 + x2 + x3 = 3

x4 + x5 = 2


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(n = 5, d = 2)

A =

(1 1 1 0 00 0 0 1 1

)c =

(1 2 0 4 0

)b =

(32

)Equations for C:


x1 + x2 + x3 = 3

x4 + x5 = 2


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OPEN QUESTIONS!!!

What is the exact total curvature of the central path?

Conjecture:( Deza, Terlaky, Zinchenko) The total curvature of the centralpath in a polyhedron is ≤ 2π(#number of facets).

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OPEN QUESTIONS!!!



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OPEN QUESTIONS!!!



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TAPAS 3Algebraic Combinatorial Optimization

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A Typical Combinatorial Feasibility Problem

Stable Set: Given a graph G and an integer k , does there exist a subset ofthe vertices of size k such that no two vertices in the subset are adjacent?

Recall, the stability number of a graph is the size of the largest stable set inthe graph, and is denoted by α(G ).

Turan Graph T (5, 3): no stable set of size bigger than 2.

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Stable Set as a System of Polynomial Equations (L. Lovasz 1989)

Given a graph G and an integer k :

one variable per vertex

For every vertex i = 1, . . . , n, let x2i − xi = 0

For every edge (i , j) ∈ E (G ), let xixj = 0

Finally, let (− k +

n∑i=1

xi

)= 0

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Turan Graph T (5, 3): =⇒ System of Polynomial Equations

Figure: Does T (5, 3) have a stable set of size 3?

x1x3 = 0, x1x4 = 0, x1x5 = 0, x2x3 = 0, x21 − x1 = 0, x2

2 − x2 = 0

x2x4 = 0, x2x5 = 0, x3x5 = 0, x4x5 = 0, x23 − x3 = 0, x2

4 − x4 = 0

x1 + x3 + x5 + x2 + x4 − 3 = 0, x25 − x5 = 0

Proposition: Let G be a graph, k an integer, encoded as the above (n + m + 1)system of equations. Then this system has a solution over C if and only if G has astable set of size k . Bijection between stable sets of size k and solutions of theequations.

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Graph coloring: Given a graph G , and an integer k , can the vertices becolored with k colors in such a way that no two adjacent vertices are thesame color?

Is the Petersen Graph 3-colorable?

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Graph Coloring modeled by a Polynomial System

one variable per vertex

vertex polynomials: For every vertex i = 1, . . . , n,

xki − 1 = 0

edge polynomials: For every edge (i , j) ∈ E (G ),

xk−1i + xk−2

i xj + · · ·+ xixk−2j + xk−1

j = 0

Proposition:(1988 D. Bayer) Let G be a graph, k an integer, then the systemof equations has a solution over C if and only if G is k-colorable. Moreover,the number of k-colorings is equal to the number of solutions divided by k!.

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Example: Petersen Graph Polynomial System of Equations

Figure: Decision Question: Is the Petersen graph 3-colorable?

x31 − 1 = 0, x3

2 − 1 = 0, x21 + x1x2 + x2

2 = 0, x21 + x1x5 + x2

5 = 0

x33 − 1 = 0, x3

4 − 1 = 0, x21 + x1x6 + x2

6 = 0, x22 + x2x3 + x2

3 = 0

x35 − 1 = 0, x3

6 − 1 = 0, x22 + x2x7 + x2

7 = 0, x23 + x3x8 + x2

8 = 0

x37 − 1 = 0, x3

8 − 1 = 0, · · · · · · · · · · · ·x39 − 1 = 0, x3

10 − 1 = 0, x27 + x7x9 + x2

9 = 0, x28 + x8x10 + x2

10 = 0

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Other algebraic ways to think about colorability

Definition: Let G be a graph with vertices V = {1, . . . , n} and edges E . Thegraph polynomial of G is

fG =∏

{i,j}∈E , i<j

(xi − xj).

Theorem: (1990 Kleitman Lovasz) Let H(n, k) be the set of all graphs with nvertices consisting of a clique of size k + 1 and all other n − k + 1 verticesisolated. The graph G on n vertices is not k-colorable if and only if

fG =∑

H∈H(n,k)

αH fH

where αH are polynomials.

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Polynomials are expressive: Largest k-colorable subgraph

A graph G has a k-colorable subgraph with R edges if and only if the followingsystem of equations has a solution:∑

{i,j}∈E(G)

yij − R = 0,

For each vertex i ∈ V (G ):xki = 1,

For each edge {i , j} ∈ E (G ):

y2ij − yij = 0, yij

(xk−1i + xk−2

i xj + · · ·+ xk−1j

)= 0.

Many other interesting encodings: e.g., existence of k-cycle in a graph, largestplanar subgraph, graph isomorphism problem, etc.

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Applications: Proving theorems and characterizations

(Lovasz-Schrijver 1990) A graph is t-perfect: A linear form f (z) ≥ 0 for allincidence vectors of stable sets if and only if there exist polynomials gi , ofdegree ≤ t, such that

f = g21 + . . . g2

k +∑

aijxixj +∑

bi (x2i − xi )

Proposition: If there exist an integer constant t for which the graph G ist-perfect then the stable set polytope of G is the projection of a polytopewhose number of facets is O(mn) (number of edges and nodes). facets.

(Hillar-Windfeldt 2008) An algebraic characterization for when a graph isuniquely k-colorable. The number of k-colorings equals dimension of quotientring.

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A big important issue...

Noga Alon 2000: “Is it possible to modify the algebraic proofs given here so thatthey yield efficient ways of solving the corresponding algorithmic problems? Itseems likely that such algorithms do exists. ”

To answer we will go back 120 years!!!

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TAPAS 4 & 5Non-Linear Integer Optimization over Polytopes

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At the beginning there was...

Linear programs

max c>x

s.t. Ax ≤ b

max c>

Easy(polynomial-time

solvable)

Special integer programs

max c>x

s.t. Ax ≤ b

all xi integer

Matrix A is SPECIAL!

Medium(can be easy or hard!)

Network problemsFixed dimension

knapsacks0-1 matrices

Integer programs

max c>x

s.t. Ax ≤ b

all xi integer

max c>

Hard(NP-hard)

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Linear programs

max c>x

s.t. Ax ≤ b

max c>


solvable)


max c>x

s.t. Ax ≤ b

all xi integer





Integer programs

max c>x

s.t. Ax ≤ b

all xi integer

max c>

Hard(NP-hard)

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Linear programs

max c>x

s.t. Ax ≤ b

max c>


solvable)


max c>x

s.t. Ax ≤ b

all xi integer





Integer programs

max c>x

s.t. Ax ≤ b

all xi integer

max c>

Hard(NP-hard)

() July 21, 2014 47 / 55


Linear programs

max c>x

s.t. Ax ≤ b

max c>


solvable)


max c>x

s.t. Ax ≤ b

all xi integer





Integer programs

max c>x

s.t. Ax ≤ b

all xi integer

max c>

Hard(NP-hard)

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How about polyhedral constraints non-linear objective??

Let f be a multivariate polynomial function,

max f(x)

s.t. Ax ≤ b

Hard(NP-hard)

Special programs

max f(x)

s.t. Ax ≤ b

all xi integer


???We study TWO

special cases

max f(x)

s.t. Ax ≤ b

all xi integer

Hard(NP-hard)

() July 21, 2014 48 / 55



max f(x)

s.t. Ax ≤ b

Hard(NP-hard)

Special programs

max f(x)

s.t. Ax ≤ b

all xi integer


???We study TWO

special cases

max f(x)

s.t. Ax ≤ b

all xi integer

Hard(NP-hard)

() July 21, 2014 48 / 55



max f(x)

s.t. Ax ≤ b

Hard(NP-hard)

Special programs

max f(x)

s.t. Ax ≤ b

all xi integer


???We study TWO

special cases

max f(x)

s.t. Ax ≤ b

all xi integer

Hard(NP-hard)

() July 21, 2014 48 / 55



max f(x)

s.t. Ax ≤ b

Hard(NP-hard)

Special programs

max f(x)

s.t. Ax ≤ b

all xi integer


???

We study TWOspecial cases

max f(x)

s.t. Ax ≤ b

all xi integer

Hard(NP-hard)

() July 21, 2014 48 / 55



max f(x)

s.t. Ax ≤ b

Hard(NP-hard)

Special programs

max f(x)

s.t. Ax ≤ b

all xi integer


???We study TWO

special cases

max f(x)

s.t. Ax ≤ b

all xi integer

Hard(NP-hard)

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Problem type

max f (x1, . . . , xd)

subject to (x1, . . . , xd) ∈ P ∩ Zd ,

where

P is a polytope (boundedpolyhedron) given by linearconstraints,

f is a (multivariate)polynomial functionnon-negative over P ∩ Zd ,

the dimension d is fixed.

Prior Work

Integer Linear Programming can besolved in polynomial time

(H. W. Lenstra Jr, 1983)

Convex polynomials f can beminimized in polynomial time

(Khachiyan and Porkolab, 2000)

WHAT CAN BE PROVED IN THISCASE??

Lemma Optimizing an arbitrary degree-4polynomial f over the lattice points of apolygon is NP-hard

NP-complete to decide whether, giventhree positive integers a, b, c , there existsa positive integer x < c such that x2 iscongruent with a modulo b.

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Problem type

max f (x1, . . . , xd)


where




Prior Work








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Problem type

max f (x1, . . . , xd)


where




Prior Work








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Problem type

max f (x1, . . . , xd)


where




Prior Work








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Problem type

max f (x1, . . . , xd)


where




Prior Work








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Problem type

max f (x1, . . . , xd)


where




Prior Work








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Theorem (FPTAS for Integer Polynomial Maximization)JDL, Hemmecke, Koppe, Weismantel, 2006

Let the dimension d be fixed. There exists an algorithm whose input data area polytope P ⊂ Rd , given by rational linear inequalities, anda polynomial f ∈ Z[x1, . . . , xd ] with integer coefficients and maximum totaldegree D that is non-negative on P ∩ Zd

with the following properties.1 For a given k , it computes in running time polynomial in k, the encoding size

of P and f , and D lower and upper bounds Lk ≤ f (xmax) ≤ Uk satisfying

Uk − Lk ≤(

k

√|P ∩ Zd | − 1

)· f (xmax).

2 For k = (1 + 1/ε) log(|P ∩ Zd |), the bounds satisfy

Uk − Lk ≤ ε f (xmax),

and they can be computed in time polynomial in the input size, the totaldegree D, and 1/ε.

3 By iterated bisection of P ∩ Zd , it constructs a feasible solution xε ∈ P ∩ Zd

with ∣∣f (xε)− f (xmax)∣∣ ≤ εf (xmax).

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Uk − Lk ≤(

k

√|P ∩ Zd | − 1

)· f (xmax).






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Uk − Lk ≤(

k

√|P ∩ Zd | − 1

)· f (xmax).






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Uk − Lk ≤(

k

√|P ∩ Zd | − 1

)· f (xmax).






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Key Idea: Represent Sets of Lattice Points as RationalFunction

Given K ⊂ Rd we define the sum

f (K ) =∑

α∈K∩Zd

zα11 zα2

2 . . . zαnn .

Think of the lattice points as monomials!!! EXAMPLE: (7, 4,−3) is z71 z4

2 z−33 .

When K is a rational convex polyhedron, i.e. K = {x ∈ Rn|Ax = b, Bx ≤ b′},where A, B are integral matrices and b, b′ are integral vectors, The generatingfunction f (K ), and thus ALL the lattice points of the polyhedron K , can beencoded in a sum of rational functions!

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Example

Let P be the square with vertices V1 = (0, 0), V2 = (5000, 0), V3 = (5000, 5000),and V4 = (0, 5000).

The generating function f (P) has over 25,000,000 monomials,f (P) = 1 + z1 + z2 + z1

1 z22 + z2

1 z2 + · · ·+ z50001 z5000

2 ,

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But it can be written using only four rational functions

1

(1− z1) (1− z2)+

z15000

(1− z1−1) (1− z2)

+z2

5000

(1− z2−1) (1− z1)

+z1

5000z25000

(1− z1−1) (1− z2

−1)

Also, f (tP, z) is

1

(1− z1) (1− z2)+

z15000·t

(1− z1−1) (1− z2)

+z2

5000·t

(1− z2−1) (1− z1)

+z1

5000·tz25000·t

(1− z1−1) (1− z2

−1)

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Theorem: Convex Integer Optimization on TransportationPolytopesJDL, Hemmecke, Onn, Rothblum, Weismantel, 2009

Problem: Convex function c : Rd −→ R, find a nonnegative integer vectorx ∈ Nn maximizing

max {c(w1x , . . . ,wdx) : Ax = b, x ∈ Nn} .

INTERPRETATION: Given d linear objective functions w1, . . . ,wd , want tomaximize their “convex balancing” c(w1x , . . . ,wdx) over feasible lattice integerpoints.

Theorem (convex balancing on transportation problems) For any fixed d , p thereis a polynomial oracle-time algorithm that, given n, arrays w1, . . . ,wd ∈ Zp×n, andconvex c : Rd −→ R given by comparison oracle, solves the convex integertransportation problem with p many suppliers.

max{ c(w1x , . . . ,wdx) : x ∈ Np×n ,

p∑i

xi,j = zj ,

n∑j

xi,j = vi}

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Gracias

Thank you

Danke

Merci

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Algebraic-Geometric ideas in Discrete Optimization ...This book presents recent advances in the mathematical theory of discrete optimization, particularly those supported by methods