Linear Programming Relaxations for MaxCutWenceslas Fernandez de la Vega

Claire Kenyon-Mathieu

Technique for approximation

IP formulation with 0-1 variables LP relaxation algorithm Strengthen LP: add valid inequalities Reduce integrality gap =

Better approximation

Example: Min Cost Perfect (non-bipartite) Matching

Unbounded gap LP:Edge e is taken with probability x(e)

Every vertex has exactly one adjacent edge

[Edmonds 1965] Reduce gap to 1 by adding:Every odd vertex set has at least one edge to the

outside

outside

Lift and Project (L&P)

[BCC, LS, SA, L]

Systematic way to strengthen LPs. Rounds: • After 0 rounds: basic LP• After k rounds: contains all valid inequalities

with support k • After n rounds: IP

Poly-time solvable for any fixed k.

L&P and int gaps

• Vertex cover [KG’98,AB,L’02,C’02STT’06]• Max 3 SAT, Set cover, Hypergraph vertex

cover [BOGH+03,AAT05]

Here: Maxcut

Because: Theory people like Maxcut!

L&P for MaxCut

• LP relaxation has gap=2 [PT’94]

• Thm [here]: gap is still 2 even after log(n)ˆc rounds of Sherali-Adams L&P

• Thm [STT]: (for another LP) gap is still 2 even after a linear number of rounds of Lovasz-Shrijver L&P.

• The moral: for MaxCut, SDP is better than LP, even if the LPs are greatly enhanced.

Questions

• Definition of L&P?

• Differences Lovasz-Shrijver vs. Sherali-Adams vs. others?

• SDP variant of L&P?

• Compare proof to other lower bound proofs for L&P?

No answers in this talk.

What I like about this work

Not the result: somewhat unsurprisingNot the “broader impacts”…The proof: Relatively clean: few short

calculations, all driven by intuition

Next: some key ideas for a simple case

No need to know about lift and project!

MaxCut LP relaxation…

• x(i,j) indicates whether {i,j} crosses the cut

x(i,j)+x(j,k)+x(k,i) ≤ 2 x(i,j) ≤ x(j,k)+x(k,i)

• Gap = 2

i j

k

… enhanced

• Additional valid inequalities:

x(a,b)+x(a,c)+…+x(d,e) ≤ 6• We will prove that we still have Gap = 2.

I cut at most 6 edges

a

c

eb

d

• Graph: sparse random, altered for large girth.

• MaxCut ≈|E|/2 w.h.p.• To define x(i,j): threshold T. if distance > T then x(i,j)=1/2; else, construct a random labeling on the

shortest path, and let x(i,j)=Pr(labels differ).

• Such that x(i,j)=1- for i and j adjacent FRAC ≈ |E|

Gap=2!

Core of proof: feasibility

• (x(i,j)) satisfies every constraint: let S be the vertices involved in .

• Define a distribution over labels of S only, and let y(i,j)=Pr(labels differ).

• y is a fractional cut, and constraint is valid inequality, so by definition ay-b ≥ 0: no calculations needed for this!

• Observe that y(i,j) ≈ x(i,j)• Thus: ax-b ≈ ay-b ≥ 0.

Defining x(i,j)

• QuickTime™ and a

TIFF (Uncompressed) decompressorare needed to see this picture.

Defining y(i,j) when S={i,j,k,u,v}

• QuickTime™ and a

TIFF (Uncompressed) decompressorare needed to see this picture.

Coupling x(i,j) and y(i,j)

•

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

Positive results

Without SDP, is L&P actually useful?

Thm [here]: in dense graphs, gap~1 after O(1) rounds of Sherali-Adams L&P

Note: this is not surprising since there already exist at least 3 PTAS for MaxCut in dense graphs.

Conclusion

• L&P is potentially an attractive alternative to ad hoc fumbling with existing LPs

• Unfortunately, most results so far are negative if we don’t use SDP.

• To justify continued work on L&P, we need some positive results: for some problem, find a new, better approximation algorithm by using L&P explicitly and voluntarily.

That’s it

• The end

Makespan minimization

• Independent jobs, m parallel machines• LP: x(i,j) indicates whether job j goes on

machine i, and t=makespan.Constraints:Every job must go on some machineMakespan greater than load on each machine

• Unbounded gap• Add: “makespan≥p(j) for every job” reduces

gap to 2, but this does not appear in L&P until after m rounds.

Proof(1/1) based on [AFKK]

Proof(4/4)

• Given S set of 5 vertices or less, define (y(i,j)) over cuts of S

• Subgraph H(S)={edges on some i-to-j path with i,j in S and distance < T}

• Large girth H(S) is a forest• Remove each edge of H(S) w.p. 2 independently;

In each connected component, label vertices alternating 1 and 0 from a random starting point

Set Y(i,j)=1 iff i and j have different labels. set y(i,j)=Expectation of Y(i,j).