graph coloring ideals
play

Graph-coloring ideals Nullstellensatz certificates, Grbner bases - PowerPoint PPT Presentation

Feb 04, 2024 •34 likes •804 views

Graph-coloring ideals Nullstellensatz certificates, Grbner bases for chordal graphs, and hardness of Grbner bases David Rolnick Massachusetts Institute of Technology drolnick@mit.edu Joint work with Jess De Loera, Susan Margulies,

  1. Graph-coloring ideals Nullstellensatz certificates, Gröbner bases for chordal graphs, and hardness of Gröbner bases David Rolnick Massachusetts Institute of Technology drolnick@mit.edu Joint work with Jesús De Loera, Susan Margulies, Michael Pernpeintner, Eric Riedl, Gwen Spencer, Despina Stasi, Jon Swenson International Symposium on Symbolic and Algebraic Computation Bath, July 2015 David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 1 / 22

  2. Overview David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 2 / 22

  3. Graph-coloring Graph-coloring problem: Proper coloring: no two neighboring vertices the same color Is there a proper coloring with k colors? David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 3 / 22

  4. Graph-coloring Graph-coloring problem: Proper coloring: no two neighboring vertices the same color Is there a proper coloring with k colors? Approach: k -coloring ⇔ system of polynomial equations Solve the system or prove unsolvable David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 3 / 22

  5. The coloring ideal Graph G = ( V , E ) Variable x i for each vertex i ∈ V Coloring ideal I k ( G ) generated by: ν i ( x ) := x k Vertex polynomials i − 1 , ∀ i ∈ V x k i − x k j Edge polynomials η ij ( x ) := ∀ ij ∈ E , x i − x j David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 4 / 22

  6. The coloring ideal Graph G = ( V , E ) Variable x i for each vertex i ∈ V Coloring ideal I k ( G ) generated by: ν i ( x ) := x k Vertex polynomials i − 1 , ∀ i ∈ V x k i − x k j Edge polynomials η ij ( x ) := ∀ ij ∈ E , x i − x j Solutions x ⇔ proper k -colorings [Bayer 1982] David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 4 / 22

  7. The coloring ideal Graph G = ( V , E ) Variable x i for each vertex i ∈ V Coloring ideal I k ( G ) generated by: ν i ( x ) := x k Vertex polynomials i − 1 , ∀ i ∈ V x k i − x k j Edge polynomials η ij ( x ) := ∀ ij ∈ E , x i − x j Solutions x ⇔ proper k -colorings [Bayer 1982] Need tool for finding solutions to a polynomial system David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 4 / 22

  8. Gröbner bases Polynomial ideal I Leading terms of f ∈ I form leading term ideal L ( I ) Gröbner basis: g 1 , g 2 , . . . , g m ∈ I , leading terms generate L ( I ) Implies that { g i } form basis for I David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 5 / 22

  9. Gröbner bases Polynomial ideal I Leading terms of f ∈ I form leading term ideal L ( I ) Gröbner basis: g 1 , g 2 , . . . , g m ∈ I , leading terms generate L ( I ) Implies that { g i } form basis for I Example: I = � x + z , x + y � with lexicographic monomial order x > y > z { x + z , y − z } is a Groebner basis for I Check: � x , y � = L ( I ) David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 5 / 22

  10. Gröbner bases Polynomial ideal I Leading terms of f ∈ I form leading term ideal L ( I ) Gröbner basis: g 1 , g 2 , . . . , g m ∈ I , leading terms generate L ( I ) Implies that { g i } form basis for I Example: I = � x + z , x + y � with lexicographic monomial order x > y > z { x + z , y − z } is a Groebner basis for I Check: � x , y � = L ( I ) Gröbner basis ⇒ solutions to ideal David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 5 / 22

  11. Gröbner basis of coloring ideal 3 3 3 3 3 1 1 1 1 1 4 4 4 4 4 2 2 2 2 2 David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 6 / 22

  12. Gröbner basis of coloring ideal 3 3 3 3 3 1 1 1 1 1 4 4 4 4 4 2 2 2 2 2 Coloring ideal for k = 3: � x 3 1 − 1 , x 3 2 − 1 , x 3 3 − 1 , x 3 4 − 1 , x 2 1 + x 1 x 2 + x 2 2 , x 2 1 + x 1 x 3 + x 2 3 , x 2 1 + x 1 x 4 + x 2 4 , x 2 2 + x 2 x 3 + x 2 3 , x 2 3 + x 3 x 4 + x 2 4 � David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 6 / 22

  13. Gröbner basis of coloring ideal 3 3 3 3 3 1 1 1 1 1 4 4 4 4 4 2 2 2 2 2 Coloring ideal for k = 3: � x 3 1 − 1 , x 3 2 − 1 , x 3 3 − 1 , x 3 4 − 1 , x 2 1 + x 1 x 2 + x 2 2 , x 2 1 + x 1 x 3 + x 2 3 , x 2 1 + x 1 x 4 + x 2 4 , x 2 2 + x 2 x 3 + x 2 3 , x 2 3 + x 3 x 4 + x 2 4 � Gröbner basis: David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 6 / 22

  14. Gröbner basis of coloring ideal 3 3 3 3 3 1 1 1 1 1 4 4 4 4 4 2 2 2 2 2 Coloring ideal for k = 3: � x 3 1 − 1 , x 3 2 − 1 , x 3 3 − 1 , x 3 4 − 1 , x 2 1 + x 1 x 2 + x 2 2 , x 2 1 + x 1 x 3 + x 2 3 , x 2 1 + x 1 x 4 + x 2 4 , x 2 2 + x 2 x 3 + x 2 3 , x 2 3 + x 3 x 4 + x 2 4 � Gröbner basis: 2 + x 3 x 4 + x 4 3 − 1 } 2 , x 4 { x 1 + x 3 + x 4 , x 2 − x 4 , x 3 David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 6 / 22

  15. Gröbner basis of coloring ideal 3 3 3 3 3 1 1 1 1 1 4 4 4 4 4 2 2 2 2 2 Coloring ideal for k = 3: � x 3 1 − 1 , x 3 2 − 1 , x 3 3 − 1 , x 3 4 − 1 , x 2 1 + x 1 x 2 + x 2 2 , x 2 1 + x 1 x 3 + x 2 3 , x 2 1 + x 1 x 4 + x 2 4 , x 2 2 + x 2 x 3 + x 2 3 , x 2 3 + x 3 x 4 + x 2 4 � Gröbner basis: 2 + x 3 x 4 + x 4 3 − 1 } 2 , x 4 { x 1 + x 3 + x 4 , x 2 − x 4 , x 3 David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 6 / 22

  16. Gröbner basis of coloring ideal 3 3 3 3 3 1 1 1 1 1 4 4 4 4 4 2 2 2 2 2 Coloring ideal for k = 3: � x 3 1 − 1 , x 3 2 − 1 , x 3 3 − 1 , x 3 4 − 1 , x 2 1 + x 1 x 2 + x 2 2 , x 2 1 + x 1 x 3 + x 2 3 , x 2 1 + x 1 x 4 + x 2 4 , x 2 2 + x 2 x 3 + x 2 3 , x 2 3 + x 3 x 4 + x 2 4 � Gröbner basis: 2 + x 3 x 4 + x 4 3 − 1 } 2 , x 4 { x 1 + x 3 + x 4 , x 2 − x 4 , x 3 David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 6 / 22

  17. Gröbner basis of coloring ideal 3 3 3 3 3 1 1 1 1 1 4 4 4 4 4 2 2 2 2 2 Coloring ideal for k = 3: � x 3 1 − 1 , x 3 2 − 1 , x 3 3 − 1 , x 3 4 − 1 , x 2 1 + x 1 x 2 + x 2 2 , x 2 1 + x 1 x 3 + x 2 3 , x 2 1 + x 1 x 4 + x 2 4 , x 2 2 + x 2 x 3 + x 2 3 , x 2 3 + x 3 x 4 + x 2 4 � Gröbner basis: 2 + x 3 x 4 + x 4 3 − 1 } 2 , x 4 { x 1 + x 3 + x 4 , x 2 − x 4 , x 3 David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 6 / 22

  18. Gröbner basis of coloring ideal 3 3 3 3 3 1 1 1 1 1 4 4 4 4 4 2 2 2 2 2 Coloring ideal for k = 3: � x 3 1 − 1 , x 3 2 − 1 , x 3 3 − 1 , x 3 4 − 1 , x 2 1 + x 1 x 2 + x 2 2 , x 2 1 + x 1 x 3 + x 2 3 , x 2 1 + x 1 x 4 + x 2 4 , x 2 2 + x 2 x 3 + x 2 3 , x 2 3 + x 3 x 4 + x 2 4 � Gröbner basis: 2 + x 3 x 4 + x 4 3 − 1 } 2 , x 4 { x 1 + x 3 + x 4 , x 2 − x 4 , x 3 David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 6 / 22

  19. Computing Gröbner bases Buchberger’s algorithm works but is slow Computation of Gröbner bases is EXPSPACE-complete [Kühnle and Mayr 1996] Even hard to write down: maximum degree can be large Mayr, Ritscher (2010): upper bound on maximum degree for r -dimensional ideal, n generators of degree d : � 2 r � 1 2 d n − r + d 2 Ritscher (2009): example attaining maximum degree d n In practice, special cases often tractable David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 7 / 22

  20. Chordal graph algorithm Definition A graph is chordal if it has no induced cycle of length ≥ 4 . David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 8 / 22

  21. Chordal graph algorithm Definition A graph is chordal if it has no induced cycle of length ≥ 4 . David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 8 / 22

  22. Chordal graph algorithm Definition A graph is chordal if it has no induced cycle of length ≥ 4 . David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 8 / 22

  23. Chordal graph algorithm Definition A graph is chordal if it has no induced cycle of length ≥ 4 . David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 8 / 22

  24. Chordal graph algorithm Definition A graph is chordal if it has no induced cycle of length ≥ 4 . Chordal graphs admit a perfect elimination ordering: When a vertex is added, its neighborhood forms a clique 3 6 5 1 7 8 4 2 David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 8 / 22

  25. Chordal graph algorithm Definition A graph is chordal if it has no induced cycle of length ≥ 4 . Chordal graphs admit a perfect elimination ordering: When a vertex is added, its neighborhood forms a clique 1 David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 8 / 22

  26. Chordal graph algorithm Definition A graph is chordal if it has no induced cycle of length ≥ 4 . Chordal graphs admit a perfect elimination ordering: When a vertex is added, its neighborhood forms a clique 1 2 David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 8 / 22

  27. Chordal graph algorithm Definition A graph is chordal if it has no induced cycle of length ≥ 4 . Chordal graphs admit a perfect elimination ordering: When a vertex is added, its neighborhood forms a clique 3 1 2 David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 8 / 22

  28. Chordal graph algorithm Definition A graph is chordal if it has no induced cycle of length ≥ 4 . Chordal graphs admit a perfect elimination ordering: When a vertex is added, its neighborhood forms a clique 3 1 4 2 David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 8 / 22

  29. Chordal graph algorithm Definition A graph is chordal if it has no induced cycle of length ≥ 4 . Chordal graphs admit a perfect elimination ordering: When a vertex is added, its neighborhood forms a clique 3 5 1 4 2 David Rolnick (MIT) Graph-coloring ideals ISSAC 2015 8 / 22

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

Graph Coloring Independent Set and Coloring k -Coloring : 3-Coloring IndependentSet Instance:

Graph Coloring Independent Set and Coloring k -Coloring : 3-Coloring IndependentSet Instance:

CS500 CS500 Graph Coloring Independent Set and Coloring k -Coloring : 3-Coloring IndependentSet Instance: A graph G . Question: Can the vertices of G be colored with k colors such that no two vertices of the same color are adjacent? (Note

226 views • 4 slides
GRAPH COLORING ON THE GPU AND SOME TECHNIQUES TO IMPROVE LOAD IMBALANCE SHUAI CHE, GREGORY

GRAPH COLORING ON THE GPU AND SOME TECHNIQUES TO IMPROVE LOAD IMBALANCE SHUAI CHE, GREGORY

GRAPH COLORING ON THE GPU AND SOME TECHNIQUES TO IMPROVE LOAD IMBALANCE SHUAI CHE, GREGORY RODGERS, BRAD BECKMANN, STEVE REINHARDT AMD SPEAKER: DIBYENDU DAS GRAPH COLORING Graph coloring is a key building block for many graph applications

839 views • 15 slides
Graph coloring Simone Campanoni simonec@eecs.northwestern.edu Outline Graph coloring

Graph coloring Simone Campanoni simonec@eecs.northwestern.edu Outline Graph coloring

Graph coloring Simone Campanoni simonec@eecs.northwestern.edu Outline Graph coloring Heuristics L2c Graph coloring task Input : the interference graph Output: the interference graph where each node has a color Task:

881 views • 25 slides
Today. Graph Coloring. Planar graphs and maps. Given G = ( V , E ) , a coloring of a G assigns

Today. Graph Coloring. Planar graphs and maps. Given G = ( V , E ) , a coloring of a G assigns

Today. Graph Coloring. Planar graphs and maps. Given G = ( V , E ) , a coloring of a G assigns colors to vertices V Planar graph coloring map coloring. where for each edge the endpoints have different colors. Planar Five Color theorem.

540 views • 4 slides
Sublinear Algorithms for Graph Coloring Sanjeev Khanna University of Pennsylvania Joint work

Sublinear Algorithms for Graph Coloring Sanjeev Khanna University of Pennsylvania Joint work

Sublinear Algorithms for Graph Coloring Sanjeev Khanna University of Pennsylvania Joint work with Sepehr Assadi (Princeton) and Yu Chen (Penn). Graph Coloring A -coloring of a graph , assigns each vertex a color from the

1.05k views • 58 slides
Advanced graph coloring Simone Campanoni simonec@eecs.northwestern.edu A coloring algorithm

Advanced graph coloring Simone Campanoni simonec@eecs.northwestern.edu A coloring algorithm

Advanced graph coloring Simone Campanoni simonec@eecs.northwestern.edu A coloring algorithm Algorithm: 1. Repeatedly select a node and remove it from the graph, putting it on top of a stack 2. When the graph is empty, rebuild it Select a

512 views • 29 slides
Graph Coloring Graph Coloring CSE, IIT KGP K- - coloring coloring K A A k k- -coloring

Graph Coloring Graph Coloring CSE, IIT KGP K- - coloring coloring K A A k k- -coloring

Graph Coloring Graph Coloring CSE, IIT KGP K- - coloring coloring K A A k k- -coloring coloring of G is a labeling of G is a labeling f:V(G) f:V(G) {1, {1, ,k}. ,k}. The labels are The labels are colors colors

2.83k views • 14 slides
Parameterized Maximum Path Coloring Michael Lampis September 9, 2011 1 / 17 Path Coloring

Parameterized Maximum Path Coloring Michael Lampis September 9, 2011 1 / 17 Path Coloring

Parameterized Maximum Path Coloring Michael Lampis September 9, 2011 1 / 17 Path Coloring Definition Path Coloring Path Coloring Example Known results Input : A graph G and a multi-set of paths on that graph Edge slicing

704 views • 47 slides
Obstructions against 3-coloring graphs without long induced paths Oliver Schaudt Universit at

Obstructions against 3-coloring graphs without long induced paths Oliver Schaudt Universit at

Obstructions against 3-coloring graphs without long induced paths Oliver Schaudt Universit at zu K oln & RWTH Aachen with Maria Chudnovsky, Jan Goedgebeur, and Mingxian Zhong Graph coloring Graph coloring a k-coloring is an

588 views • 45 slides
3-coloring graphs without induced paths on seven vertices Oliver Schaudt Universit at zu K

3-coloring graphs without induced paths on seven vertices Oliver Schaudt Universit at zu K

3-coloring graphs without induced paths on seven vertices Oliver Schaudt Universit at zu K oln, Institut f ur Informatik with Flavia Bonomo, Maria Chudnovsky, Peter Maceli, Maya Stein, and Mingxian Zhong Graph coloring Graph coloring

598 views • 56 slides
On-line Graph Coloring Iwona Cie slik Algorithmics Research Group, Jagiellonian University,

On-line Graph Coloring Iwona Cie slik Algorithmics Research Group, Jagiellonian University,

On-line coloring Competitiveness Results Example On-line Graph Coloring Iwona Cie slik Algorithmics Research Group, Jagiellonian University, Cracow, Poland Chambry, June 2005 Iwona Cie slik On-line Graph Coloring On-line coloring

981 views • 55 slides
Coloring graphs without long induced paths Oliver Schaudt Universit at zu K oln & RWTH

Coloring graphs without long induced paths Oliver Schaudt Universit at zu K oln & RWTH

Coloring graphs without long induced paths Oliver Schaudt Universit at zu K oln & RWTH Aachen with Flavia Bonomo, Maria Chudnovsky, Jan Goedgebeur, Peter Maceli, Maya Stein, and Mingxian Zhong Graph coloring Graph coloring a

921 views • 90 slides
Graph coloring Kevin Ly September 20, 2019 Itinerary Introduction to simple graphs A

Graph coloring Kevin Ly September 20, 2019 Itinerary Introduction to simple graphs A

Graph coloring Kevin Ly September 20, 2019 Itinerary Introduction to simple graphs A brief game of matching The coloring problem and DSATUR Networkx demo Graphs primer A simple graph G consists of a set V and a set of pairs E .

524 views • 23 slides
These slides are intended for those familiar with graphs and graph coloring. 1 Slide 1 Ordinarily

These slides are intended for those familiar with graphs and graph coloring. 1 Slide 1 Ordinarily

These slides are intended for those familiar with graphs and graph coloring. 1 Slide 1 Ordinarily a graph G is considered to be (properly) colored when each vertex has one color, and the colors of neighboring vertices are distinct. The smallest

686 views • 29 slides
Graph homotopy, ideals of finite varieties and a surprising duality William J. Martin Department

Graph homotopy, ideals of finite varieties and a surprising duality William J. Martin Department

Ideals Discrete Homotopy Duality Graph homotopy, ideals of finite varieties and a surprising duality William J. Martin Department of Mathematical Sciences and Department of Computer Science Worcester Polytechnic Institute Modern Trends in

593 views • 45 slides
Some Pretty Edge Coloring Conjectures Rong Luo Department of Mathematics West Virginia

Some Pretty Edge Coloring Conjectures Rong Luo Department of Mathematics West Virginia

Some Pretty Edge Coloring Conjectures Rong Luo Department of Mathematics West Virginia University Rong Luo (WVU) Some Pretty Edge Coloring Conjectures 1 / 1 Introduction A simple graph is a graph without multiple edges while a graph means

1.37k views • 122 slides
What is the core distribution of a graph telling us? Sonja Petrovi c Illinois Institute of

What is the core distribution of a graph telling us? Sonja Petrovi c Illinois Institute of

What is the core distribution of a graph telling us? Sonja Petrovi c Illinois Institute of Technology, Chicago Joint work with: Vishesh Karwa (Carnegie Mellon / Harvard), Michael Pelsmajer (IIT), Despina Stasi (Univ. of Cyprus / IIT) Dane

266 views • 15 slides
Algebraic Applications of the Theory of Violator Spaces Dane Wilburne Illinois Institute of

Algebraic Applications of the Theory of Violator Spaces Dane Wilburne Illinois Institute of

Algebraic Applications of the Theory of Violator Spaces Dane Wilburne Illinois Institute of Technology Joint work with: Jes us De Loera (UC Davis) Sonja Petrovi c (IIT) Despina Stasi (IIT) North Dakota State University AMS Central

1.18k views • 14 slides
Welcome to Hastings Community Network 25th January 2019 Social Prescribing connecting to the

Welcome to Hastings Community Network 25th January 2019 Social Prescribing connecting to the

Welcome to Hastings Community Network 25th January 2019 Social Prescribing connecting to the right support across Hastings and St Leonards One of eight Locality Networks supporting community resilience across East Sussex Welcome and Setting

504 views • 26 slides
PARADIME PARADIME Parametrizable Domain-Adaptive Information and Message Extraction Adapting

PARADIME PARADIME Parametrizable Domain-Adaptive Information and Message Extraction Adapting

PARA DIME PARA DIME PARADIME PARADIME Parametrizable Domain-Adaptive Information and Message Extraction Adapting the SMES System to a New Domain Gnter Neumann and Thierry Declerck 1 1 PARADIME: Source: TD & GN Goals of the PARADIME

896 views • 20 slides
Whats in your wallet?! Lara Pudwell Valparaiso University January 27, 2017 Whats in your

Whats in your wallet?! Lara Pudwell Valparaiso University January 27, 2017 Whats in your

The Coin Keeper The Simple Spender The Gamblers The Small Spender The Whole Shebang ...and More Fun Whats in your wallet?! Lara Pudwell Valparaiso University January 27, 2017 Whats in your wallet?! Lara Pudwell The Coin Keeper The

1.19k views • 91 slides
A Direct Proof of the Strong HananiTutte Theorem on the Projective Plane ric Colin de

A Direct Proof of the Strong HananiTutte Theorem on the Projective Plane ric Colin de

A Direct Proof of the Strong HananiTutte Theorem on the Projective Plane ric Colin de Verdire 1 Vojtch Kalua 2 Pavel Patk 3 Zuzana Patkov 3 Martin Tancer 2 1 Dpartement dinformatique, cole normale suprieure, Paris and

986 views • 69 slides
Counting and Locating Multiple Solutions of Estimating Equations Speaker: Donald Richards (Penn

Counting and Locating Multiple Solutions of Estimating Equations Speaker: Donald Richards (Penn

Counting and Locating Multiple Solutions of Estimating Equations Speaker: Donald Richards (Penn State University) This talk is based on joint work with: Despina Stasi (Penn State University) Elizabeth Gross (NC State University) Sonja

322 views • 18 slides
Wireless extensions to the PlanetLab infrastructure Giovanni Di Stasi Consorzio

Wireless extensions to the PlanetLab infrastructure Giovanni Di Stasi Consorzio

Wireless extensions to the PlanetLab infrastructure Giovanni Di Stasi Consorzio Interuniversitario Nazionale per lInformatica CINI Universit di Napoli Federico II 1 Onelab Information Day - Napoli, 22/07/ 2008 Onelab goals

349 views • 30 slides

More recommend