Advanced Approximation Algorithms
## Course descriptionThe course will cover several advanced topics loosely related to the area of Approximation Algorithms. A tentative list of topics includes: Graph Minor Theory; routing problems; iterative rounding technique; Lovász Local Lemma (with a constructive proof); oblivious routing and related graph decompositions; cut- and flow- graph sparsifiers; Semidefinite Programming-based approximation algorithms; and hierarchies of convex relaxations. Course requirements: 3 homework assignments.## Topics Covered and Additional Reading Material- 3/28: Introduction to Graph Minor Theory; definitions of several types of graph minors; Wagner's conjecture; forbidden minors and Structure Theorem for them. Additional material:
- A survey of László Lovász
- David Johnson's column
- Jeff Erickson's lecture notes
- 3/30: Forbidden minors continued: Separator Theorem and its use in obtaining PTAS for Independent Set in minor-free graphs; minor-closed family of graphs; obstructions; negative result of Fellows and Langston on computing obstructions. Additional material:
- Separator theorem for minor-free graphs: the Alon-Seymour-Thomas paper; an exposition of the proof in lecture notes of Uri Feige; another proof with somewhat different parameters by Plotkin, Rao and Smith.
- The Fellows-Langston paper (see Theorem 16).
- Paul Seymour's survey on Hadwiger's conjecture
- 4/4: Treewidth: definition and basic properties. Algorithms for coloring and Node-Disjoint Paths in bounded-treewidth graphs. Additional reading:
- Lecture notes by Uri Feige
- A survey by Bruce Reed (the survey mostly focuses on brambles and tangles which we did not cover, but also talks about treewidth and its properties).
- A survey of Robertson and Seymour on the algorithm for the Node-Disjoint Paths problem (in a book called "Paths, Flows, and VLSI-Layout").
- 4/6: Definition of pathwidth; Baker's technique (including for excluded-minor graphs) and its uses for designing 2-approximation for Coloring and PTAS for weighted Independent Set; proof of Baker's decomposition for planar graphs. Additional reading:
- Baker's original paper
- An exposition of the proof in Uri Feige's lecture notes
- The paper proving the partitioning theorem for excluded-minor graphs.
- 4/11: Different versions of sparsest cut (both in edge-cut and vertex-cut settings); definition of expanders; several types of well-linkedness and connections to sparsest cut. Additional reading:
- There are extensive discussions on sparsest cut in most books / lecture notes on approximation algorithms. See for example Vazirani's book, and Williamson-Shmoys book.
- Node well-linkedness is a standard concept in graph theory; its relationship to treewidth is explored e.g. in Bruce Reed's survey.
- The weaker edge-based well-linkedness concept was mainly used in work on routing problems (see e.g. this paper and this paper). The boosting theorem that connects the two well-linkedness notions was proved in this paper.
- 4/13: proof of the theorem connecting node-well-linkedness to treewidth; proof of the boosting theorem for well-linkedness.
- 4/18: statement of the Excluded Grid theorem for general and excluded-minor graphs; its uses in bidimensionality theory; high-level description of Robertson-Seymour's algorithm for Node-Disjoint Paths.
Additional reading:
- A paper that shows linear dependence between treewidth and grid-minor size in excluded-minor graphs.
- A survey on bidimensionality theory.
- An outline of Robertson and Seymour of their algorithm for the disjoint paths problem in this book.
- Best current proof of the Flat Wall Theorem.
- A simpler proof of the Unique Linkage Theorem (also explains how this theorem is used in the algorithm).
- 4/20: Proof of the Excluded Grid Theorem, based on this paper.
- 4/25: Iterative rounding technique, its use for finding exact algorithm for weighted bipartite matching; started bounded-degree spanning trees.
Additional reading:
- Lecture notes of Ola Svensson.
- 4/27: Bounded-degree spanning trees. Additional reading:
- The paper of Singh and Lau.
- Chapter 11 in Williamson-Shmoys book.
- Lecture notes of Ola Svensson.
- A book of Lau, Ravi and Singh on iterative methods.
- 5/2: End of bounded-degree spanning trees; start survivable network design.
- 5/4: Survivable network design. Additional reading:
- The paper of Kamal Jain.
- Chapter 11 in Williamson-Shmoys book.
- Lecture notes of Chandra Chekuri.
- A book of Lau, Ravi and Singh on iterative methods.
- 5/9: Proof of algorithmic Lovasz Local Lemma, based on this paper of Moser and Tardos.
- 5/11: Guest lecture by Madhur Tulsiani on Sum of Squares. Additional reading:
- 5/16: Poof of Lovasz Local Lemma - continued. Application to packet routing, based on this paper.
- 5/18 Application of Asymmetric Lovasz Local Lemma to frugal coloring. Congestion minimization: randomized rounding, constant approximation for grid graphs, negative example.
Additional reading:
- Alistair Sinclair's lecture notes discuss the frugal coloring application
- Randomized Rounding technique is due to Raghavan and Thompson. See also lecture notes by James Lee
- A paper related to congestion minimization in grids.
- A better algorithm for congestion minimization when all paths are short via LLL (together with the negative example) can be found in this paper
- This famous paper of Leighton and Rao gives many important results, and also shows that one can get near-optimal routing on expanders via short paths. If there is one paper you want to study on cuts an flows, it should be this one.
- 5/23
Oblivious routing: based on this paper by Harald Räcke. Additional reading:
- Chapter 15 in Williamson-Shmoys book.
- An interesting interpretation of the result in Andersen-Feige paper.
- Another paper of Räcke with slightly different guarantees, but the routing is based on a single tree. This paper also gives a hierarchical decomposition of trees into "expander of expanders"-like structure, that has found many other uses.
- 5/25 Approximation of minimum bisection via oblivious routing; started routing problems (Node-Disjoint Paths/Edge-Disjoint Paths): greedy algorithm; multicommodity-flow LP and its integrality gap; hardness of approximation for directed graphs; overview of known results for various special cases.
- 6/1 Edge-Disjoint Paths on expanders; polylog(n)-approximation for Edge-Disjoint Paths in general graphs with constant congestion. Based on these papers.
- Homework 1 Due: Apr 27 in class.
- Homework 2 Due: May 16 in class.
- Homework 3 Due: June 1 in class. There will be no extensions!
## Course requirementsThere will be 3 homework assignments.## Homework Assignments |