CSE550 Combinatorial Algorithms/Intractability
From esoterum.org
Resources
- -Unimodularity ensures that the solution to an LP will always be integer if all of the costs and constraints are also integer
- Linear Programming animation (simplex method)
- List of LP solvers (including NEOS)
- Integer Linear Programming Tutorial
- Interger Linear Programming Tutorial (CMU)
- Opensource Algorithm Code, Zuse Institute
- Lectures from the University of Freiburg
- > List of NP-Hard problems
- > The Algorithm Design Manual, Steven S. Skiena, Department of Computer Science State University of New York (Online)
- Good article on intractability from Princeton
- David S. Johnson, "The NP-Completeness Column: An Ongoing Guide", J. Algorithms 5, 147-160 (1984)
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
- Randeep Bhatia · Murali Kodialam · T. V. Lakshman, "Finding disjoint paths with related path costs", Springer Science+Business Media, LLC 2006
HW 7
- 1.
- Optimization Theory By Hubertus Th. Jongen, Klaus Meer, Eberhard Triesch, partial search result on Google book search
- Solution to part (a),(b)
- Solution to part (a),(b)
- Additional infor (a),(b), possible references for (c)
- -Theorem 1.2 (Kumar and Li, 2002) Any asymmetric TSP on n locations can be reducedto a symmetric TSP on 2n locations
Midterm
Q1
- Bin Zhang, Julie Ward, Qi Feng, "Simultaneous Parametric Maximum Flow Algorithm with Vertex Balancing", HP Laboratories Palo Alto, June 28, 2005
- J. M. W. Rhys, "A Selection Problem of Shared Fixed Costs and Network Flows", Management Science, Vol. 17, No. 3, Theory Series (Nov., 1970), pp. 200-207
Q4
Final
Q2
Q2
Q5
- M. R. Garey; R. L. Graham; D. S. Johnson; D. E. Knuth, "Complexity Results for Bandwidth Minimization", SIAM Journal on Applied Mathematics, Vol. 34, No. 3., May, 1978
- David Muradian, "The bandwidth minimization problem for cyclic caterpillars with hair length 1 is NP-complete", Theoretical Computer Science 307, 2003
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).
- -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.
- GNU Linear Programming Kit guide from IBM
- GLPsol Tutorial
- Practical Optimization: A Gentle Introduction
- Robert Fourer, AMPL: "A Mathematical Programming Language"
- LP formulations (covering, packing, partition) (non-linear information)
- Dan Bienstock, "Some Generalized Max-Flow Min-Cut Problems in the Plane" Mathematics of Operations Research, Vol. 16, No. 2. (May, 1991), pp. 310-333.
- Linear Programming and set covering problems (based on notes from Tardos)
