CSE550 Combinatorial Algorithms/Intractability

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Resources

-Unimodularity ensures that the solution to an LP will always be integer if all of the costs and constraints are also integer

HW 6

1. 2-SAT is in NP
2. A sub-optimal solution to TSP is a Hamiltonian Cycle.
3. 3SAT reduction to NAESAT
4. Finding disjoint paths with different path-costs: Complexity and algorithms

HW 7

1.
-Theorem 1.2 (Kumar and Li, 2002) Any asymmetric TSP on n locations can be reducedto a symmetric TSP on 2n locations

Midterm

Q1

Q4

Final

Q2

Q3

Q4

Q5

Project

2. Linear program formulation and solving. You can examine one or more linear programming formulations for a speci�c problem. This should be done by using a free solver, such as GLPK and a modeling language such as AMPL or the subset of AMPL that comes with GLPK. (If you have access to CPLEX and/or real AMPL, that is also perfectly fine with me.) Your goal in this might be to examine and compare the solution times for several formulations of a problem (as in the mincut example), or to study the tightness of a relaxation (as in the case of Steiner trees and edge coloring). Some suggestions for this type of project:

-Comparing minimum cut formulations (standard cut covering, polynomial-size directed flow formulation, compact formulation by Carr et al.).
-Bidirected formulation for the Steiner tree problem (Rajagopalan-Vazirani).
-Asymmetric TSP (Charikar, Goemans, Karloff).
Moses Charikar, Michel X. Goemans, Howard Karloff, "On the Integrality Ratio for Asymmetric TSP", Annual IEEE Symposium on Foundations of Computer Science, 2004
-Matching-based LP relaxation of edge-coloring gap should be an additive 1! There is a paper by Jeff Kahn, but it is somewhat difficult.

GLPsol and LP

-Chapter 7. LP in Practice
-> Chapter 10. Network Flow Programming

Min-Cut Max-Flow

TSP