CS6842 - Algorithmic Algebra

Meetings  

• Theme 1 : Introduction & Motivation - 3 meetings

  • Meeting 01 : Mon, Jul 31, 04:50 pm-05:40 pm

  Introduction. Administrative Details. Story of Groups and Polynomials. <br> Graph Isomorphism Problem. Automorphisms. Structure. Permutations. Computing the Automorphism Group. Rigid Graphs Examples. Role of group theoretic algorithms in attempting on solutions to such problems.

  References
  Exercises
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  Introduction. Administrative Details. Story of Groups and Polynomials.
  Graph Isomorphism Problem. Automorphisms. Structure. Permutations. Computing the Automorphism Group. Rigid Graphs Examples. Role of group theoretic algorithms in attempting on solutions to such problems.
  

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  None

  • Meeting 02 : Wed, Aug 02, 02:00 pm-03:15 pm

  Primality Testing Problem. Formulation as a polynomial identity testing problem. Perfect Matching Problem. Formulation as a polynomial identity testing problem.

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  Reading

  Primality Testing Problem. Formulation as a polynomial identity testing problem. Perfect Matching Problem. Formulation as a polynomial identity testing problem.
  

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  • Meeting 03 : Thu, Aug 03, 03:25 pm-04:45 pm

  GraphAuto Problem. Groups. Operations, Generators. Subgroup, Coset structure, Lagrange's theorem. Size of the Generating Set. Reduction from GI to GA.

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  Exercises
  Reading

  GraphAuto Problem. Groups. Operations, Generators. Subgroup, Coset structure, Lagrange's theorem. Size of the Generating Set. Reduction from GI to GA.
  

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• Theme 2 : Algorithms for Permutations Groups - 19 meetings

  • Meeting 04 : Mon, Aug 07, 04:50 pm-05:40 pm

  Generating sets. Cyclic groups. Graph Automorphism Group. Automorphism problem. Any group has a generating set of size log |G|. Graph isomorphism. Colored Graph Isomorphism. Reductions.

  References
  Exercises
  Reading

  Generating sets. Cyclic groups. Graph Automorphism Group. Automorphism problem. Any group has a generating set of size log |G|. Graph isomorphism. Colored Graph Isomorphism. Reductions.
  

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  • Meeting 05 : Wed, Aug 09, 02:00 pm-03:15 pm

  GI reduces to GA using coloring. Reduction from GA to GI. The strategy. Tower of subgroups, coset representatives. Unique product representation. Orbit-Stabilizer Lemma. Computing the coset representatives using queries to GI. Using O(n^4) GI queries to solve GA.

  References
  Exercises
  Reading

  GI reduces to GA using coloring. Reduction from GA to GI. The strategy. Tower of subgroups, coset representatives. Unique product representation. Orbit-Stabilizer Lemma. Computing the coset representatives using queries to GI. Using O(n^4) GI queries to solve GA.
  

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  • Meeting 06 : Thu, Aug 10, 03:25 pm-04:40 pm

  Computational problems motivated by the previous algorithm. Order Computation, Membership Testing, Coset Representatives. Group Actions. Orbits, Orbit-Stabilizer Lemma. A reduction from membership testing to order finding. A strategy for order finding. Orbit computation problem.

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  Exercises
  Reading

  Computational problems motivated by the previous algorithm. Order Computation, Membership Testing, Coset Representatives. Group Actions. Orbits, Orbit-Stabilizer Lemma. A reduction from membership testing to order finding. A strategy for order finding. Orbit computation problem.
  

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  • Meeting 07 : Wed, Aug 16, 02:00 pm-03:15 pm

  Algorithm for computing orbits. Set Stabilizer Problem. Reduction from GA to SetStab. Group intersection problem. Reduction from SetStab to GroupInter.

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  Exercises
  Reading

  Algorithm for computing orbits. Set Stabilizer Problem. Reduction from GA to SetStab. Group intersection problem. Reduction from SetStab to GroupInter.
  

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  • Meeting 08 : Thu, Aug 17, 02:00 pm-03:15 pm

  Point-Stabilizer Problem. Recursive approach. The tower of subgroup idea. The hurdles. Natural attempts and counter examples.

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  Exercises
  Reading

  Point-Stabilizer Problem. Recursive approach. The tower of subgroup idea. The hurdles. Natural attempts and counter examples.
  

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  • Meeting 09 : Sat, Aug 19, 09:00 am-09:50 pm

  An algorithm for computing strong generating set with respect to the tower of stabilizer subgroups. Finding the Coset representatives.

  References
  Exercises
  Reading

  An algorithm for computing strong generating set with respect to the tower of stabilizer subgroups. Finding the Coset representatives.
  

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  • Meeting 10 : Mon, Aug 21, 04:50 pm-05:40 pm

  Schreier Generators. Reduction of generating sets. Algorithm for point-wise stabilizer problem. Membership Testing in Permutation Groups given by generators. Order finding.

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  Exercises
  Reading

  Schreier Generators. Reduction of generating sets. Algorithm for point-wise stabilizer problem. Membership Testing in Permutation Groups given by generators. Order finding.
  

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  • Meeting 11 : Wed, Aug 23, 02:00 pm-03:15 pm

  Jerrum's Filter. Linear sized generating sets.

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  Reading

  Jerrum's Filter. Linear sized generating sets.
  

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  • Meeting 12 : Thu, Aug 24, 02:00 pm-03:15 pm

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  Reading

  
  

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  • Meeting 13 : Mon, Aug 28, 04:50 pm-05:40 pm

  Normal Subgroups. Normalizer and Centralizers. Algorithmic problems in normal subgroups. Group intersection problem for special case. Tower of subgroups.

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  Exercises
  Reading

  Normal Subgroups. Normalizer and Centralizers. Algorithmic problems in normal subgroups. Group intersection problem for special case. Tower of subgroups.
  

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  • Meeting 14 : Wed, Aug 30, 02:00 pm-03:15 pm

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  • Meeting 15 : Thu, Aug 31, 02:00 pm-03:15 pm

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  • Meeting 16 : Mon, Sep 04, 04:50 pm-05:40 pm

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  • Meeting 17 : Wed, Sep 06, 02:00 pm-03:15 pm

  Color stabilizer. Structure forest. Minblock computation. Primitive actions. Examples. Algebraic Characterization. Computing the blocks of a transitive action.

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  Exercises
  Reading

  Color stabilizer. Structure forest. Minblock computation. Primitive actions. Examples. Algebraic Characterization. Computing the blocks of a transitive action.
  

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  • Meeting 18 : Thu, Sep 07, 02:00 pm-03:15 pm

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  Reading

  
  

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  • Meeting 19 : Sat, Sep 09, 02:00 pm-04:00 pm

  Color stabilizer problem. Divide and Conquer Methods. Sylow's theorem. Bounds on size of the groups when the action is primitive, faithful and transitive. Recursive algorithm. Solution for the recurrence.

  References
  Exercises
  Reading

  Color stabilizer problem. Divide and Conquer Methods. Sylow's theorem. Bounds on size of the groups when the action is primitive, faithful and transitive. Recursive algorithm. Solution for the recurrence.
  

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  • Meeting 20 : Mon, Sep 11, 04:40 pm-05:40 pm

  Trivalent Graph Isomorphism. Overview and basics. Tutte's theorem. Structure of the Automorphism group. Kernel, Image of the group homomorphism.

  References
  Exercises
  Reading

  Trivalent Graph Isomorphism. Overview and basics. Tutte's theorem. Structure of the Automorphism group. Kernel, Image of the group homomorphism.
  

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  • Meeting 21 : Wed, Sep 13, 02:00 pm-03:15 pm

  Computing the generating set of the image. Reduction from Trivalent GI to Restricted SetStab.

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  Exercises
  Reading

  Computing the generating set of the image. Reduction from Trivalent GI to Restricted SetStab.
  

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  • Meeting 22 : Thu, Sep 14, 02:00 pm-03:15 pm

  Combinatorial algorithms for graph isomorphism. Color valence of colored graphs. Bounded Color valence algorithm. Combining algebraic and Combinatorial Methods for GI. Coloring and color refinements. Stable coloring. Properties of stable coloring. Individualization of vertices. Approaching the general graph isomorphism problem using color refinements and vertex individualization. Notion of color valence. Runtime of the final algorithm assuming nice individualization sequences. Zemlyachenko's Trick to reach Bounded color valence case.

  References
  Exercises
  Reading

  Combinatorial algorithms for graph isomorphism. Color valence of colored graphs. Bounded Color valence algorithm. Combining algebraic and Combinatorial Methods for GI. Coloring and color refinements. Stable coloring. Properties of stable coloring. Individualization of vertices. Approaching the general graph isomorphism problem using color refinements and vertex individualization. Notion of color valence. Runtime of the final algorithm assuming nice individualization sequences. Zemlyachenko's Trick to reach Bounded color valence case.
  

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• Theme 3 : Algorithms for Polynomials - 21 meetings

  • Meeting 23 : Mon, Sep 18, 04:50 pm-05:40 pm

  Algorithmic problems on polynomials. Solving Polynomial Equations, Factorization, Identity testing problem. Rings, Zero-divisors, integral domains, Fields, Z_p.

  References
  Exercises
  Reading

  Algorithmic problems on polynomials. Solving Polynomial Equations, Factorization, Identity testing problem. Rings, Zero-divisors, integral domains, Fields, Z_p.
  

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  • Meeting 24 : Wed, Sep 20, 02:00 pm-03:15 pm

  Polynomial Rings. Formal definition of ideals and properties. Algorithmic problems to address. Solving polynomial equations. Ideals and Ideal Membership problem. Varieties and Ideals. Examples, basic observations.

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  Exercises
  Reading

  Polynomial Rings. Formal definition of ideals and properties. Algorithmic problems to address. Solving polynomial equations. Ideals and Ideal Membership problem. Varieties and Ideals. Examples, basic observations.
  

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  • Meeting 25 : Thu, Sep 21, 02:00 pm-03:15 pm

  Hilbert Basis theorem. Noetherian Rings. Ascending chain termination. Principal ideals. PID. Proof of the Hilbert Basis Theorem.

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  Exercises
  Reading

  Hilbert Basis theorem. Noetherian Rings. Ascending chain termination. Principal ideals. PID. Proof of the Hilbert Basis Theorem.
  

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  • Meeting 26 : Mon, Sep 25, 04:50 pm-05:40 pm

  Details of proof of Hilbert basis theorem. Ideal membership and Ideal equality problems. Simple cases first. Univariate case. Every ideal in F[x] is a principal ideal.

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  Exercises
  Reading

  Details of proof of Hilbert basis theorem. Ideal membership and Ideal equality problems. Simple cases first. Univariate case. Every ideal in F[x] is a principal ideal.
  

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  • Meeting 27 : Wed, Sep 27, 02:00 pm-03:15 pm

  Gcd of polynomials. Euclidean Algorithm and the correctness in terms of ideals generated. Solving the Ideal membership problem for uni-variate case. Towards generalizations. Term orders. Well-ordering. Example term orderings. Examples of term orderings.

  References
  Exercises
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  Gcd of polynomials. Euclidean Algorithm and the correctness in terms of ideals generated. Solving the Ideal membership problem for uni-variate case. Towards generalizations. Term orders. Well-ordering. Example term orderings. Examples of term orderings.
  

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  • Meeting 28 : Wed, Oct 04, 02:00 pm-03:15 pm

  Division algorithm with multiple multivariate polynomials. Proof of termination. Why division algorithm does not solve ideal membership problem. Need of a nice basis.

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  Exercises
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  Division algorithm with multiple multivariate polynomials. Proof of termination. Why division algorithm does not solve ideal membership problem. Need of a nice basis.
  

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  • Meeting 29 : Thu, Oct 05, 02:00 pm-03:15 pm

  Examples for remainder being non-zero depending on the order of the division. Definition of Grobner basis. Characterization of Grobner basis.

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  Exercises
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  Examples for remainder being non-zero depending on the order of the division. Definition of Grobner basis. Characterization of Grobner basis.
  

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  • Meeting 30 : Sat, Oct 07, 02:00 pm-03:30 pm

  Existence of Grobner basis for every ideal. Special case of monomial ideals first. Details of the general proof. When is a set not a Grobner basis? Looking for simplest counter examples. S-polynomials. Expressing general counter examples in terms S-polynomials. Proof of a special case first.

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  Existence of Grobner basis for every ideal. Special case of monomial ideals first. Details of the general proof. When is a set not a Grobner basis? Looking for simplest counter examples. S-polynomials. Expressing general counter examples in terms S-polynomials. Proof of a special case first.
  

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  • Meeting 31 : Wed, Oct 11, 02:00 pm-03:15 pm

  Main lemma : any violating example can be expressed in terms of S-polynomials. General case of the criterion. Buchberger's algorithm. Termination and Correctness. Minimal Grobner Basis. Size of the minimial Grobner basis is unique.

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  Main lemma : any violating example can be expressed in terms of S-polynomials. General case of the criterion. Buchberger's algorithm. Termination and Correctness. Minimal Grobner Basis. Size of the minimial Grobner basis is unique.
  

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  • Meeting 32 : Thu, Oct 12, 02:00 pm-03:15 pm

  Reduced Grobner basis. Reduction algorithm from minimal to reduced. Uniqueness of reduced Grobner basis.

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  Reduced Grobner basis. Reduction algorithm from minimal to reduced. Uniqueness of reduced Grobner basis.
  

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  • Meeting 33 : Sat, Oct 21, 02:00 pm-04:30 pm

  Applications of Grobner basis. Elimination Ideal. Ideal intersection problem. Polynomial maps. Kernel and Image. Integer programming. A special case. Reduction to the problem of testing membership in the image of a ring-homomorphism. A characterization of kernel in terms of elimination ideals. Checking whether a polynomials is in the image of the homomorphism. Observations about Grobner basis generated by the Buchberger algorithm in this case.

  References
  Exercises
  Reading	See textbook. Using the term ordering to solve the general optimization version of the integer programming problem. This will be a part of the course presentations.

  Applications of Grobner basis. Elimination Ideal. Ideal intersection problem. Polynomial maps. Kernel and Image. Integer programming. A special case. Reduction to the problem of testing membership in the image of a ring-homomorphism. A characterization of kernel in terms of elimination ideals. Checking whether a polynomials is in the image of the homomorphism. Observations about Grobner basis generated by the Buchberger algorithm in this case.
  

  References	:	None
  Reading	:	See textbook. Using the term ordering to solve the general optimization version of the integer programming problem. This will be a part of the course presentations.

  • Meeting 34 : Mon, Oct 23, 04:50 pm-05:40 pm

  Integral Domains, Units, Irreducibles, Primes, Primes are Irreducibles. All primes in an ID are irreducibles. Integral Domains with factorisation is a UFD if and only if all Irreducibles are primes.

  References
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  Integral Domains, Units, Irreducibles, Primes, Primes are Irreducibles. All primes in an ID are irreducibles. Integral Domains with factorisation is a UFD if and only if all Irreducibles are primes.
  

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  • Meeting 35 : Wed, Oct 25, 02:00 pm-03:15 pm

  All Principal Ideal Domains are Unique Facotorization Domains. Z, F[x] are all UFDs. Towards proving : if R is a UFD, so is R[x]. Gauss's Lemma (Statement). Factorization problem. Characterisation of irreducibility - Quotient ring is a field if and only if the polynomial is irreducible.

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  Exercises
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  All Principal Ideal Domains are Unique Facotorization Domains. Z, F[x] are all UFDs. Towards proving : if R is a UFD, so is R[x]. Gauss's Lemma (Statement). Factorization problem. Characterisation of irreducibility - Quotient ring is a field if and only if the polynomial is irreducible.
  

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  • Meeting 36 : Thu, Oct 26, 02:00 pm-03:15 pm

  Sizes of the quotient rings. Characteristic of a field. Field extensions. <br>Back to factorization problem. Formal derivatives and the reduction to Square free factorization.

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  Sizes of the quotient rings. Characteristic of a field. Field extensions.
  Back to factorization problem. Formal derivatives and the reduction to Square free factorization.
  

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  • Meeting 37 : Sat, Oct 28, 02:00 pm-04:00 pm

  Distinct degree factorization. A starting idea from Fermat's little theorem. Extracting product of linear factors. Generalization of the lemma and DDF algorithm.

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  Distinct degree factorization. A starting idea from Fermat's little theorem. Extracting product of linear factors. Generalization of the lemma and DDF algorithm.
  

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  • Meeting 38 : Mon, Oct 30, 04:50 pm-05:40 pm

  Berlekamp's algorithm. Chinese remindering - ideals version. Berlekamp Subalgebra. Characterization of dimension of the algebra when the polynomial is irreducible.

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  Exercises
  Reading

  Berlekamp's algorithm. Chinese remindering - ideals version. Berlekamp Subalgebra. Characterization of dimension of the algebra when the polynomial is irreducible.
  

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  • Meeting 39 : Wed, Nov 01, 02:00 pm-03:15 pm

  Dimension of the subalgebra and number of factors. Factorization algorithm assuming basis for the subalgebra. Finding a basis for the subalgebra. A quick overview of what we did in the factorization algorithms, and what we did not do.

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  Dimension of the subalgebra and number of factors. Factorization algorithm assuming basis for the subalgebra. Finding a basis for the subalgebra. A quick overview of what we did in the factorization algorithms, and what we did not do.
  

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  • Meeting 40 : Thu, Nov 02, 02:00 pm-03:15 pm

  Identity testing of univariate polynomials. Trivial algorithm. Faster randomized algorithm. Amplification of success probability. Identity testing of multivariate polynomials. Schwartz-Zippel Lemma.

  References
  Exercises
  Reading

  Identity testing of univariate polynomials. Trivial algorithm. Faster randomized algorithm. Amplification of success probability. Identity testing of multivariate polynomials. Schwartz-Zippel Lemma.
  

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  • Meeting 41 : Mon, Nov 06, 04:50 pm-05:40 pm

  Using Algebraic independence to do identity testing. Chen-Kao Algorithm. Error vs Running time Tradeoff.

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  Using Algebraic independence to do identity testing. Chen-Kao Algorithm. Error vs Running time Tradeoff.
  

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  • Meeting 42 : Wed, Nov 08, 02:00 pm-03:15 pm

  Identity testing via Chinese remaindering. Some naive attempts and comparison to Schwartz-Zippel Algorithms. Agrawal-Biswas's construction of the sample space using cyclotomic polynomials

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  Exercises
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  Identity testing via Chinese remaindering. Some naive attempts and comparison to Schwartz-Zippel Algorithms. Agrawal-Biswas's construction of the sample space using cyclotomic polynomials
  

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  • Meeting 43 : Thu, Nov 09, 02:00 pm-03:15 pm

  Agrawal-Biswas algorithm. Running and time and error bound. Extensions to multivariate case Randomized algorithm for primality testing - as an identity testing over Rings. Sample space size.

  References
  Exercises
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  Agrawal-Biswas algorithm. Running and time and error bound. Extensions to multivariate case Randomized algorithm for primality testing - as an identity testing over Rings. Sample space size.
  

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