1. Solving Network Coding Problems with Genetic Algorithmic Methods Anthony Kim Advisers: Muriel Medard and Una-May O’Reilly

2. What is network coding? Method to increase the throughput of information flow through networks by means of coding at intermediate nodes. The “butterfly” network A A B B A A B B A A B B Routing gives rates of 1/2 at each sink

3. What is network coding? Method to increase the throughput of information flow through networks by means of coding at intermediate nodes. The “butterfly” network A A B B A A B B Network coding gives rates of 1 A A A+B B B

4. Problem Description Given: –Acyclic graph G –Input processes & output processes –Connections between processes with desired rates Goal : –Determine whether or not all the connections are possible to satisfy, with network coding if necessary. z_2 y_1 z_1 y_2 Ex. Input processes: y_1, y_2 Output processes: z_1, z_2 Connections: {y_1, z_2, rate of 1}, {y_2, z_1, rate of 1}

5. Questions to ask Given a graph G with a set of desired connections, can we determine the feasibility of the problem? If the problem is feasible, can we find a network coding solution in a reasonable amount of time? If the problem is infeasible, can we relax constraints or introduce slight modification to the graph so that the problem now is feasible?

6. Project Network coding problems can be reformulated in terms of multivariate polynomials –Two sets of polynomials, P and Q –Goal: find, if possible, a solution over a finite field such that polynomials in P evaluate to 0 and ones in Q evaluate to nonzero. –In general, it is still a hard problem. Use genetic algorithms to solve the algebraic problem

7. An example (extended) Step 1: Compute the line graph z1,z2 y1 z3, z4 y2 y3 y4 21 4 3 8 6 5 9 7 Graph G Connections wanted: y1 ->z1, y2 ->z2, y3->z3, y4->z4

8. An example (extended) Step 1: Compute the line graph z1,z2 y1 z3, z4 y2 y3 y4 21 4 3 8 6 5 9 7 1 2 5 3 6 4 7 8 9 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 Graph G Line graph of G

9. An example (extended) Step 2: Compute matrices A, F, B and the transfer matrix M from the line graph

10. An example (extended) Step 2: Compute matrices A, F, B and the transfer matrix M from the line graph

11. An example (extended) Step 2: Compute matrices A, F, B and the transfer matrix M from the line graph

12. An example (extended) Step 3: Using the theorem, find polynomials in P and Q y1 y4 y2 y3 z1 z4 z2z3

13. An example (extended) Step 3: Using the theorem, find polynomials in P and Q y1 y4 y2 y3 z1 z4 z2z3 Submatrices in rectangle have to be nonsingular for decodability, i.e. the determinants have to be nonzero.

14. An example (extended) Step 3: Using the theorem, find polynomials in P and Q y1 y4 y2 y3 z1 z4 z2z3 Submatrices in rectangle have to be zero matrices for the zero interference condition.

15. An example (extended) Step 4: Solve the polynomial equations (P,Q). This is the hard part. Find a finite field F and values of x_i’s such that polynomials in Q evaluate to nonzero value in F and polynomials in P evaluate to zero in F.

16. An example (extended) Step 4: Solve the polynomial equations (P,Q). This is the hard part. It happens that this network coding problem is infeasible. There are conflicting variables that you can show by cases are not solvable. (For example, last four polynomials in P and 2 polynomials in Q.)

17. Polynomial equations are solvable by Groebner basis algorithms after reformulating the polynomial equations in terms of algebraic variety. ( Implementation available in Matlab, Maple, Mathematica, etc.) Computation can be doubly exponential in the problem size Experiments seems to show that Groebner basis algorithms can take a long time even on polynomials derived from network coding problems. Groebner Basis Maybe genetic algorithmic methods can help!

18. Current Progress Reasonably good understanding of where the polynomials come from and how they correspond to certain structures of the given graph A few graphs with their corresponding polynomials

19. References