NDMI067 Flows, paths and cuts

Winter semester 2023/24

Petr Kolman

The weekly lecture takes place every Thursday from 2:00 pm in the KAM corridor on the second floor of the MFF UK building at Malostranske namesti.

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 (and cuts) along short paths  [9,12,3]
• 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]
• Unique Games Hardness of Maximum Cut [13]
• The sparsest cut and semidefinite programming [2]

Covered topics

• October 6, 2022
  • Multicommodity flows and cuts - introduction, basic definitions.
  • Approximation algorithm for minimum multicut based on LP relaxation.
  • Approximate duality [10], Sec. 5.2.
• October 13
  • Optimality of the bound on the approximate duality of sum maximum multicommodity flow and minimum multicommodity cut.
  • Concurrent multicommodity flow and sparsest cut, basic definitions, LP relaxation.
  • O(log k)-approximation algorithm for sparsest cut problem based on rounding the LP relaxation - Part I [10].
• October 20
  • O(log k)-approximation algorithm for sparsest cut problem based on rounding the LP relaxation - Part II. Metric embeddings.
• October 27
  • O(log k)-approximation algorithm for sparsest cut problem based on rounding the LP relaxation - Part III. Main proofs.
  • Lower bound on the metric embedding distortion.
• November 3
  • Approximation of the graph expansion.
  • Bisection and balanced cuts, pseudoapproximation for 1/3-balanced cut.
  • Minimum cut linear arrangement and the balanced cut.
  • Crossing number and the balanced cut.
• November 10
  • Semidefinite programming (SDP) and Max cut [7].
  • SDP and Minimum balanced cut [2].
• November 17
  • Public holiday - no lecture.
• November 24
  • Analysis of an SDP-algorithm for balanced cut [2].
• December 1
  • Edge disjoint path problem and paremetrization by the Flow number.
  • Flow number of a graph [9].
• December 8
  • Shortening Lemma and the Flow Number [9].
  • L-bounded flows and cuts - introduction. Finding a maximum L-bounded flow by LP [3],
• December 15
  • Multiplicative weight update method: Combinatorial FPTAS for L-bounded flow [12].
• January 5, 2023 notes
  • Unique Games Hardness of Max Cut [13].

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.

12. K. Kateřina Altmanová and P. Kolman and J. Voborník
On Polynomial-Time Combinatorial Algorithms for Maximum L-Bounded Flow.
Journal of Graph Algorithms and Applications, Volume 24, No 3, 2020.

13. S. Khot, G. Kindler, E. Mossel and R. O'Donnell
Optimal Inapproximability Results for MAX‐CUT and Other 2‐Variable CSPs?.
SIAM Journal on Computing, Volume 37, Issue 1, 2007.