Flows, Paths and Cuts (NDMI067)

Winter semester 2023/24

Petr Kolman

The weekly lecture takes place every Friday from 12:20 pm in the lecture hall S6 in the MFF UK building at Malostranske namesti. We start on October 6, 2023.

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, 2023
  • Multicommodity flows and cuts - introduction, basic definitions.
  • Approximation algorithm for minimum multicut based on LP relaxation - sketch.
• October 13
  • Approximation algorithm for minimum multicut based on LP relaxation - details, analysis.
  • Approximate duality of multicommodity flows and cuts [10], Sec. 5.2.
  • Optimality of the bound on the approximate duality of sum maximum multicommodity flow and minimum multicommodity cut.
• October 20
  • Concurrent multicommodity flow and sparsest cut, basic definitions, LP relaxation [10].
• October 27
  • O(log k)-approximation algorithm for sparsest cut problem based on rounding the LP relaxation.
• November 3
  • O(log k)-approximation algorithm for sparsest cut problem based on rounding the LP relaxation (contd.).
• November 10
  • Applications of the Sparsest cut.
    • Metric embeddings.
    • Graph expansion.
    • Bisection and balanced cuts, pseudoapproximation for 1/3-balanced cut.
• November 17
  • Public holiday - no lecture.
• November 24
  • Minimum cut linear arrangement and the balanced cut.
  • Crossing number and the balanced cut.
  • Semidefinite programming (SDP) and the Max cut [7].
• December 1
  • Edge disjoint paths problem.
  • O(n^{2/3})-approximation.
  • Trade-off between source-sink distance and the size of the max flow.
• December 8
  • Flow number of a graph [9].
  • Shortening Lemma and the Flow Number [9].
• December 15
  • Shortening Lemma - proof [9].
  • Edge disjoint paths problem and paremetrization by the Flow number - O(F)-approximation.
• January 5, 2024
  • NP-hardness of the two edge disjoint directed paths problem [11].
  • Hardness of approximation of the edge disjoint directed paths problem [11].
  • L-bounded flows and cuts - introduction. [3].
• January 12
  • L-bounded cuts - approximations. Integrality gap of a natural LP relaxation.
  • Multiplicative weight update method: Combinatorial FPTAS for L-bounded flow [12].

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.