Integrality Gaps for Sparsest Cut and Minimum Linear Arrangement Problems

1. Integrality Gaps for Sparsest Cut and Minimum Linear Arrangement Problems Nikhil R. Devanur Subhash A. Khot Rishi Saket Nisheeth K. Vishnoi

2. Sparsest Cut Problem (SCP) and b-Balanced Cuts (BSP) • Given undirected graph G=(V,E), find subset of nodes S, |S|<|V|/2 that minimizes |E(S, V\S)| / |S|·|V\S| • b-Balanced cuts ensure that S and V\S are at least bn in size, where 0≤b≤1/2. • b-Balanced Separator Problem (BSP) satisfies both conditions

3. Previously known results • An f(n)-approximation algorithm for SCP can be applied iteratively to obtain O(f(n)) approximation algorithm for BSP • [Leighton-Rao, JACM 1999] a linear-programming relaxation produces O(log n) approximation to SCP.

4. Linear Programming (LP) Review • Given matrix A, and vectors b and c, find x • Maximize cT·x • Subject to A·x≤b, x≥0 • NP-hard to find optimal integral solution • Relatively easy to find a fractional solution • Simplex method, Ellipsoid method • Approximation results by rounding fractional x • Lower bound of the approximation factor is sometimes called “integrality gap”

5. Semidefinite Programming (SDP) • Find X that maximizes ∑cij∙xij • Subject to • ∑aijk∙xij = bk • X is a symmetric and positive semidefinite matrix • Equivalent to vector programming (VP) • Find set of vectors V • X=VTV  xij=vi∙vj • Often SDP approximates better than LP

6. SDP references • M. Goemans and D. Williamson • MAXCUT algorithm [1995] • Extensions to MAX3SAT and MAXDICUT • D. Williamson • Great lecture notes on SDP • Comprehensive website on SDP • http://www-user.tu-chemnitz.de/~helmberg/semidef.html • List of papers maintained by Farid Alizadeh • http://rutcor.rutgers.edu/~alizadeh/Sdppage/papers.html

7. LP Useful dual problems Linear functions Fractional solution which has to be rounded Simplex and ellipsoid methods are poly-time SDP Same Non-linear functions Usually a vector solution which has to be matched Interior point or general convex optimization algorithms, also poly-time but with large constants Difference between LP and SDP

8. SDP results for graph partitioning • Arora, Rao, and Vazirani. Expander flows, geometric embeddings and graph partitioning. STOC 2004. • An SDP relaxation of the problem gives O(sqrt(log n)) approximation • ARV-conjecture • Standard SDP relaxation can give constant factor approximation

9. Devanur, et al. results • The standard SDP relaxations of BSP with the triangle inequality constraint have an integrality gap at least Ω(log log n) • Ω(log log n) lower bound for BSP • Implies the bound for SCP • Similar bound for Minimum Linear Arrangement Problem • Find a bijection π : V -> {1, …, n} that minimizes ∑e=(u,v) |π(u)-π(v)|

10. SDP relaxation for SCP • How to encode any cut of the graph. • If node i is left of the cut, set it equal to some vector w. Otherwise, set it to –w.

11. SDP relaxation for SCP (con’t) • The following objective function and constraints are equal to the sparsity value.

12. Algorithm for SCP • Solve the SDP • Choose w • Obtain a plain orthogonal to w • For all nodes i whose vi is on w side of the plane, place them in S • For all other nodes, place them in V\S

13. SDP relaxation for BSP - Main Theorem • There are absolute constants c1, c2> 0 such that, for every large enough n there exists a multi-graph G(V;E) on n vertices, and a vector assignment i->vi for every i in V s.t. • Every (1/3, 2/3) balanced cut must contain at least c1∙|E|∙(log log n / log n) • The vector assignment gives a low SDP objective value < c2∙|E|∙(1/log n) • Vectors are well-separated • Δ-inequality on the vectors holds

14. SDP relaxation for BSP (con’t) • Value of the b-Balanced sparsest cut is given by the following objective function

15. Questions and Comments