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Introduction to Semidefinite Programming via MAX-CUT and SDP Application : Algorithms for Sparsest Cut

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A Randomized 0.878-Approximation Algorithm. SDP Relaxation. Opt(G) : Value of (1), i.e. the size of a maximum cut. A SDP whose value is an upper bound for OPT(G). – PowerPoint PPT presentation

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Title: Introduction to Semidefinite Programming via MAX-CUT and SDP Application : Algorithms for Sparsest Cut


1
Introduction to Semidefinite Programming via
MAX-CUT and SDP Application Algorithms for
Sparsest Cut
  • Bo-Young Kim
  • Applied Algorithm Lab, KAIST
  • 2011.6.13 ?

2
Contents
  • Introduction to SDP via MAX-CUT
  • Review MAX-CUT Problem Approximation Algorithm
  • 1st Algorithm A Naïve Algorithm Randomized
    0.5-Approximation
  • From LP to SDP (informal)
  • Cholesky Factorization
  • SDP (formal definition)
  • 2nd Algorithm Randomized 0.878-Approximation

3
Review MAX-CUT Problem
  • A graph G(V,E) is given.
  • A cut of G a pair (S, V \ S) for a nonzero
    subset S ? V.
  • Size of the cut
  • Def (Recall) MAX-CUT problem
  • Decision version NP-complete
  • Given G and n?N, is there a cut of sizen?
  • Garey, Johnson, and Stockmeyer (74, STOC)
  • Optimization version NP-hard

4
Review Approximation Algorithm
  • Def (Recall) Approximation Algorithm.
  • P a maximization(ref.minimization) problem
  • set of instances
  • A an algorithm that returns for every instances
    I ?
  • a feasible solution A(I) ? F(I).
  • is a function.
  • A is a -approximation algorithm for P if the
    following two properties hold.

  • (ref.
    .) From now on, only consider
    maximization
  • If is a constant c ? c-approximation
    algorithm. c 1. (ref. c 1 )
  • A randomized - approximation algorithm
  • Expected polynomial runtime

5
A Naïve Algorithm Randomized 0.5-Approximation
  • The above algorithm is a randomized
    0.5-approximation algorithm for the MAX-CUT
    problem.
  • Runs in polynomial time.

  • .
  • Deterministic 0.5-approximation,
    0.5(11/E)-approximation algorithm is possible.
  • Until 1994, no c-approximation algorithm could be
    found for cgt0.5.

6
From LP to SDP
  • Def (Recall) LP in equational form
  • where x,b,c are column vectors ( x,c?Rn, b?Rm)
    and A?Rmxn.
  • From LP to SDP (Brief Introduction)
  • Replace the vector space Rn by another real
    vector space vector space Sn of symmetric nxn
    matrices.
  • Replace ltx,ygtxTy over Rn by ltX,Ygt
    over Sn.
  • Replace the constraint x 0 by the constraint
  • Def (Recall) A symmetric matrix M is Positive
    Semidefinite All its eigenvalues are
    nonnegative.

7
Cholesky Factorization
  • Fact Let M?Sn. TFAE.
  • Cholesky Factorization of a positive semidefinite
    matrix M The computation of U that satisfies
    (iii). (Often needed in SDP.)
  • Outer product Cholesky Factorization Recursive,
    O(n3) operation for M ? Rnxn.
  • M( ) ? R1x1 ? U( ). a
    nonnegative eigenvalue.
  • Otherwise, M
  • Note . If not,
    ? Contradiction to (ii).

8
Cholesky Factorization
  • i)
  • Positive semidefinite
    again ? Recursively compute the decomposition.
  • ? .
  • ii) ? q0.
  • ? where
  • Polynomial time algorithm in a bit model.
    (Counting elementary operations)

9
Semidefinite Programs
  • Def Semidefinite Program(SDP) in equational form
  • where
    ,
    ,

  • , and is a linear
    operator.
  • As LP case, we say the SDP is feasible
  • if there is some feasible solution
  • The value of a feasible SDP
  • An optimal solution a feasible solution X s.t.
    .

10
A Randomized 0.878-Approximation Algorithm
  • Goemans and Williamson (94, STOC)
  • Fact ! SDP can be solved up to any desired
    accuracy e, where the runtime depends
    polynomially on the sizes of the input numbers,
    and on log(R/e), where R is the maximum size
    X of a feasible solution.
  • Formulating MAX-CUT problem as a constraint
    optimization problem
  • V1,2, ,n.
  • Variables x1, x2, ,xn ?-1,1.
  • Any assignment of these variables encodes a cut
    (S, V \ S).
  • Si ?V xi 1, V \ Si ?V xi -1.
  • Define ?
    the contribution of
    the pair i,j
  • to the size of the cut.

  • Reformulated MAX-CUT problem

  • (1)

11
A Randomized 0.878-Approximation Algorithm
  • SDP Relaxation
  • Opt(G) Value of (1), i.e. the size of a maximum
    cut.
  • A SDP whose value is an upper bound for OPT(G).
  • Let xi ? Sn-1x ?Rn x1, the (n-1)
    dimensional unit sphere.
  • Consider the problem
    (2)
  • Remark S0-1,1 can be embedded into Sn-1 via
    the mapping L x ? (0, ,0,x).
  • Let (x1, x2, ,xn) be a feasible solution of (1)
    with objective function value
  • ? (L(x1), L(x2), ,L(xn)) is a feasible solution
    of (2) with objective function value
  • ? (2) is a relaxation of (1).

12
A Randomized 0.878-Approximation Algorithm
  • xij xiTxj
  • (2) ? SDP
    (3)
  • xij xiTxj ? XUTU where U x1 x2
    xn .
  • by the condition (iii).
  • xi ? Sn-1 ? xii1.
  • Conversely,
  • X a feasible matrix for (3) ? the columns of
    any matrix U with XUTU is feasible vector
    solution of (2).
  • ? (2) ? (1).

13
A Randomized 0.878-Approximation Algorithm
  • (3) is feasible with the same finite value ?
    opt(G) as (2).
  • From the Fact! ? We can find in poly time a
    matrix X with objective function value at least
    ? e, for any egt0.
  • Recall We can compute U s.t. XUTU in poly time.
  • ? The columns x1, x2, , xn of U form an
    almost optimal solution of (2)

14
A Randomized 0.878-Approximation Algorithm
  • Rounding the vector solution
  • Mapping Sn-1 ? S0.
  • Choose p ? Sn-1 u.a.r.
  • Define
  • ? p partitions Sn-1 into a closed hemisphere
    Hx ? Sn-1 pTx0 and its complement.
  • ? Vectors in H ? 1 , vectors in complement ? -1.

15
A Randomized 0.878-Approximation Algorithm
  • Lemma Let xi, xj ? Sn-1. Then
  • Getting the bound
  • Expected number of cut edges
  • Know
  • Lemma For z ? -1,1,

(1-z)/2
arccos(z)/pi
16
A Randomized 0.878-Approximation Algorithm
  • By choosing e 510-4,

.
17
SDP Application Algorithms for Sparsest Cut
18
Sparsest Cut Problem
  • A weighted graph G(V,E) with positive edge
    weight(cost, capacity) is given.
  • Edge weight ce for every edge e ? E.
  • Vn.
  • A set of pairs of vertices (s1,t1), (s2,t2), ,
    (sk,tk) with associated demands Di between si
    and ti.
  • Def Sparsest Cut problem Minimize the
    sparsity of a cut S ? V
  • where
  • and

19
Sparsest Cut Problem - Example
  • The sparsest cut value 1.

Solid edges edges of the graph with weight
1. Dashed edges the demand edges of demand
value 1.
20
Sparsest Cut Problem - Example
  • The sparsest cut value 1.

3
3
Solid edges edges of the graph with weight
1. Dashed edges the demand edges of demand
value 1.
21
Relation with ncut and Expansion
  • Unit demands case The demands consist of all
    pairs and for all
  • ? Sparsest Cut Problem in unit demands case
  • Minimize node-normalized cut (ncut)
  • If S n/2, then n/2 S n,
  • ? the above problem
  • Minimize the expansion

22
LP Relaxation
  • Cut metric
  • Any n-point metric can be associated with vector
    in .
  • Reformulated Sparsest Cut Problem
  • (1) (equiv.)
  • where ? and is the weight
    of the edge between i and j,
  • and is the demand between i and j.

23
LP Relaxation
  • The positive cone generated by all cut metrics
  • From convexity, the optimum of (1) is achieved at
    an extreme point.
  • (2) (equiv.)
  • Prop d is l1-embeddable iff d is in CUTn.
  • ? (3)
    (equiv.)
  • (Relax l1 to all metric) ?
  • (4) (relaxed.)

24
LP Relaxation
  • LP
  • (5) (relaxed.)
  • Thm Suppose for each metric (V,d), there exists
    a metric µ µ(d) ? l1 such that
  • d(x,y) µ (x,y) a d(x,y) for all x,y
    ? V.
  • Then (5) has an integrality gap a .
  • Thm For all metrics d, there exists µ µ(d) ? l1
    such that aO(logn). Moreover, the number of
    dimensions needed O(log2n). (Approximate
    max-flow min-cut thm for multi-commodity flows)
  • Cor The LP relaxation of Sparsest Cut has
    integrality gap of O(logn).

25
LP Relaxation
  • Any metric in l1 can be written as a positive
    linear combination of cuts
  • ?
  • ? Simply pick the best cut S amongst the ones
    with non-zero aS in the cut-decomposition of µ.

26
SDP Relaxation
  • Tighter relaxation
  • LP minimizing over all cut metrics minimizing
    over all l1-metrics ? minimizing over all
    metrics.
  • SDP minimizing over all cut metrics
    minimizing over all l1-metrics ? minimizing over
    all l22 metrics.
  • If d ? l1 then d ? l22.
  • (Relax l1 to l22 -metric) ? (4)
    (relaxed.)
  • SDP
  • (5) (relaxed)

27
SDP Relaxation
Lemma (Structure Lemma)
  • O((logn)1/2) approximation in uniform case
  • SDP embedding lies on the unit ball ? Use
    Structure Lemma.
  • Pick S and T satisfying the Structure Lemma and
    cut at a random distance from S,
  • ? The expected total capacity crossing the cut

28
Reference
  • Introduction to SDP via MAX-CUT
  • Approximation Algorithms and Semidefinite
    Programming (http//www.ti.inf.ethz.ch/ew/courses/
    ApproxSDP09/), Jirí Matoušek, Bernd Gärtner.
  • SDP Application Algorithms for Sparsest Cut
  • Energy Models for Graph Clustering, Andreas
    Noack, Journal of Graph Algorithms and
    Applications, 2007.
  • Expander Flows, Geometric Embeddings and Graph
    Partitioning, Sanjeev Arora, Satish Rao, and
    Umesh Vazirani, 2007(Longer version of an ACM
    STOC 2004).
  • Scribing Note of CMU 18-854B Spring 2008
    Advanced Approximation Algorithms, Lecture 19,
    27(http//www.cs.cmu.edu/anupamg/adv-approx/),
    Lecturer Anupam Gupta
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