CS 598: Topics in Graph Algorithms, Fall 2024

Course Summary

The course will cover selection of topics in graph algorithms with an emphasis on recent developments on fast algorithms for a variety of problems such as shortest paths, flows, cuts, and matchings. Structural results and connections to past ideas and results will also be discussed. This is an exploratory seminar-style course. A tentative list of topics (a superset of what we can cover).

  • New developments on shortest path algorithms with negative lengths, low-diameter graph decompositions, spanners etc
  • Global and Steiner mincut: tree packing, isolating cuts
  • Multiplicative weight updates and applications via fast data structures
  • Multicommodity flows, flow-cut gap, sparsest cut, cut-matching game
  • Expanders, well-linked sets, expander and well-linked decomposition, and applications
  • Making graphs look like trees: metrics, flows, separators
  • Graph Sparsification
  • Faster flow algorithms
  • Local algorithms for flows/cuts
  • Dynamic Data Structures for Matchings
  • Palette coloring and faster graph coloring algorithms
  • Connections to hypergraphs and submodularity
  • Gomory-Hu Trees

Administrative Information

Lectures: Tue, Thu 11-12.15pm in Siebel Center 0216.

Instructor:  Chandra Chekuri, 3228 Siebel Center, (chekuri at)

Office Hours (Chandra): Friday 1-2pm in 3228 Siebel, or by appointment

Grading Policy: 50% homework, 30% exams (2 x 15%), 20% project.

Attendance policy: at least 65% of lectures for a grade. The revolution will not be televised, it will be live.

Prerequisites: This is a graduate level class on recent developments. Background in algorithms at the level of CS 473 is necessary and expected. Familiarity with basics of linear programming and randomization, and knowledge of standard graph algorithms is necessary. Most important is mathematical maturity and comfort with formal proofs and arguments and interest in the topic. Consult the instructor if you have questions.

Mental health support at UofI: Diminished mental health, including significant stress, mood changes, excessive worry, substance/alcohol abuse, or problems with eating and/or sleeping can interfere with optimal academic performance, social development, and emotional wellbeing. The University of Illinois offers a variety of confidential services including individual and group counseling, crisis intervention, psychiatric services, and specialized screenings at no additional cost. If you or someone you know experiences any of the above mental health concerns, it is strongly encouraged to contact or visit any of the University's resources provided below. Getting help is a smart and courageous thing to do -- for yourself and for those who care about you.
Counseling Center: 217-333-3704, 610 East John Street, Champaign, IL 61820
McKinley Health Center: 217-333-2700, 1109 South Lincoln Avenue, Urbana, Illinois 61801
University wellness center
Please do not hesitate to contact the instructor if you need help or assistance.

Student code: All students are expected to be aware of the university's student code, in particular the academic integrity policies

CS Code of Conduct: See here for important information on the code of conduct guidelines of the Siebel School of CS.

References

Electronic versions of several books from Cambridge University, Springer, Wiley and other publishers are available free to Univ of Illinois students via the library.

 

 

Ed (sign up at link), Gradescope (R7KJRX)  


Homework

Lectures

  • Lecture 1: 8/27/2024, Introduction, MST, linear-time randomized algorithm

  • Lecture 2: 8/29/2024, Tree packing, fractional version of Tutte-Nash-Williams theorem, towards mincut
    • Draft notes
    • Paper of C, Quanrud, Xu
    • Karger's tree packing based mincut paper, different from his random contraction based paper with Stein
    • Comb opt/graph theory refs for tree/matroid packing

  • Lecture 3: 9/03/2024, Mincut and bounding (approximate) mincuts via tree packing

  • Lecture 4: 9/05/2024, Steiner mincut via Isolating Cuts

  • Lecture 5: 9/10/2024, Start of metric methods, s-t cut, Mulitcut via metric decomposition
    • Draft notes
    • Chapters 14, 15, 16 in Chandra's approximation algorithms notes and pointers there
    • CMU notes (see also older notes) on low-stretch spanning tree embeddings and pointers there

  • Lecture 6: 9/12/2024, Low-diameter decompositions and low-stretch metric tree embeddings
    • Draft notes
    • CMU notes (see also older notes) on low-stretch spanning tree embeddings and pointers there

  • Lecture 7: 9/17/2024, Sparsest cut
    • Draft notes
    • Books on approximation, Vazriani (Chapter 21), Williamson-Shmoys (Chapters 8, 15).
    • Matousek's lecture notes on metric embeddings
    • Survey on expanders

  • Lecture 8: 9/19/2024, Expander decomposition and well-linked decomposition

  • Lecture 9: 9/24/2024, Oblivious routing
    • Draft notes
    • Quanrud's notes (Chapter 10)
    • Williamson-Shmoys approximation book (Chapter 15 )

  • Lecture 10: 9/26/2024, Catch up lecture

  • Lectures 11/12: 10/01/2024 and 10/03/2024, Cut-based hierarchical decomposition
    • Draft notes
    • Video of talk by Harald Raecke on fast algorithm for decomposition

  • Lectures 13/14: 10/08/2024 and 10/10/2024, Beating Bellman-Ford for Shortest Paths with Negative Lengths

  • Lectures 15/16: 10/15/2024 and 10/17/2024, Shortest Paths with Negative Weights --- Near-linear time scaling algorithm