NDMI067 Flows, paths and cuts

Winter semester 2019/20

Petr Kolman

The lecture takes place on Wednesday from 12:20 in lecture room S8. .

Sylabus

Multicommodity flows generalize in a natural way the classical flow problem: instead of just a single source-destination pair, there are several such pairs but there is still just one network and all flows must fit in it. Multicommodity flows and their dual cut problems are extremely useful in the design of approximation algorithms for many different graph problems. The lecture will provide an overview of the most important result in this area and will ilustrate various techniques used in the design of approximation algorithms (e.g., LP relaxation, use of probability, geometry, SDP relaxation, greedy approach, ...).

• Approximate duality [10]
  • for sum multicommodity flow 
  • for concurrent multicomodity flow 
• Flows along short paths [9,3]
• Edge disjoint paths and packing problems [8]
• Algorithms for multicommodity flows
  • Column generation [4, chapter 3.6]
  • Lagrange Relaxation and greedy algorithm [5,6]
• Maximum Cut and semidefinite programming [7]
• The sparsest cut and semidefinite programming [2]

Covered topics

• October 9
  • Multicommodity flows and cuts - introduction, basic definitions. Approximation algorithm for minimum multicut based on LP relaxation [10], Sec. 5.2.
• October 16
  • Approximation algorithm for minimum multicut - analysis. Approximate duality. Optimality of the result.
• October 23
  • Concurrent multicomodity flow and sparsest cut, basic definitions, LP relaxation, approximation based on reduction to the min multicommodity cut.
• October 30
  • O(log k)-approximation algorithm for sparsest cut problem based on rounding the LP relaxation - Part I [10].
• November 6
  • O(log k)-approximation algorithm for sparsest cut problem based on rounding the LP relaxation - Part II. Metric embeddings.
• November 13
  • Lower bound on the metric embedding distortion.
  • Bisection and balanced cuts, pseudoapproximation for 1/3-balanced cut.
  • L-bounded flow and cuts - introduction. Finding a maximum L-bounded flow [3].
• November 20
  • Minimum cut linear arrangement and the balanced cut.
  • L-bounded cut problem. Approximation algorithms. Integrality gap of a natural LP relaxation.
• November 27
  • Semidefinite programming and Max Cut [7].
• December 4
  • Edge disjoint paths. Maximum multicommodity flow and the distance of terminals.
  • Flow number [9].
• December 11
  • Flow number. Balanced multicommodity flow. Shortenning lemma [9].
  • Greedy approximation for edge disjoint paths parametrized by the flow number.
• December 18
  • NP-hardness of the two edge disjoint path problem [11].
  • SDP and minimum balanced cut - description of the algorithm [2].
• January 8, 2020
  • SDP and minimum balanced cut - degustation: analysis of O(log n) pseudoapproximation [2].

Study material

1. D. W. Williamson, D. B. Shmoys.
The Design of Approximation Algorithms.
Cambridge University Press, 2011.

2. S. Arora, S. Rao, and U. Vazirani.
Expander flows, geometric embeddings, and graph partitionings.
Journal of the AMC, 56(2):5.1-5.37, 2009.

3. Baier, G., Erlebach, T., Hall, A., Köhler, E., Kolman P., Pangrác O., Schilling, H., and Skutella, M.
Length-bounded cuts and flows.
ACM Transactions on Algorithms, Volume 7, Issue 1, Article No. 4, 2010.

4. W. J. Cook, W. H. Cunningham, W. R. Pulleyblank, and A. Schrijver.
Combinatorial Optimization.
John Wiley, New York, 1997.

5. L. K. Fleischer.
Approximating fractional multicommodity flow independent of the number of commodities.
SIAM Journal on Discrete Mathematics, 13(4):505-520, 2000.

6. N. Garg and J. Konemann.
Faster and simpler algorithms for multicommodity flow and other fractional packing problems.
In Proc. of the 39th Annual Symposium on Foundations of Computer Science, 300-309, 1998.

7. M. X. Goemans and D. P. Williamson.
.878-approximation algorithms for MAX CUT and MAX 2SAT.
In Proc. of the Twenty-Sixth Annual ACM Symposium on the Theory of Computing, 422-431, 1994.

8. J. Kleinberg.
Approximation Algorithms for Disjoint Paths Problems.
PhD thesis, Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, 1996.

9. P. Kolman and C. Scheideler.
Improved bounds for the unsplittable flow problem.
In Proc. of the 13th ACM-SIAM Symposium on Discrete Algorithms, 184-193, 2002.

10. D. B. Shmoys.
Cut problems and their application to divide-and-conquer.
In D. S. Hochbaum, editor, Approximation Algorithms for NP-hard Problems, 192-235. PWS Publishing Company, 1997.

11. S. Fortune, J. Hopcroft and J. Wyllie
The directed subgraph homeomorphism problem.
Theoretical Computer Science, Volume 10, Issue 2, 1980.