Linear Programming Relaxations for MaxCut

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Linear Programming Relaxations for MaxCut

• Wenceslas Fernandez de la Vega
• Claire Kenyon-Mathieu

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Technique for approximation

• IP formulation with 0-1 variables
• LP relaxation ? algorithm
• Strengthen LP add valid inequalities
• Reduce integrality gap
• ? Better approximation

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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
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Lift and Project (LP)

• 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.

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LP and int gaps

• Vertex cover KG98,AB,L02,C02STT06
• Max 3 SAT, Set cover, Hypergraph vertex cover
  BOGH03,AAT05
• Here Maxcut
• Because Theory people like Maxcut!

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LP for MaxCut

• LP relaxation has gap2 PT94
• Thm here gap is still 2 even after log(n)c
  rounds of Sherali-Adams LP
• Thm STT (for another LP) gap is still 2 even
  after a linear number of rounds of
  Lovasz-Shrijver LP.
• The moral for MaxCut, SDP is better than LP,
  even if the LPs are greatly enhanced.

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Questions

• Definition of LP?
• Differences Lovasz-Shrijver vs. Sherali-Adams vs.
  others?
• SDP variant of LP?
• Compare proof to other lower bound proofs for
  LP?
• No answers in this talk.

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What I like about this work

• Not the result somewhat unsurprising
• Not 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!

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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
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enhanced

• Additional valid inequalities
• x(a,b)x(a,c)x(d,e) 6
• We will prove that we still have Gap 2.

d
a
e
b
I cut at most 6 edges
c
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• Graph sparse random, altered for large girth.
• MaxCut E/2 w.h.p.
• To define x(i,j) threshold T.
• if distance gt 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

Gap2!
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Core of proof feasibility

• (x(i,j)) satisfies every constraint let S be the
  vertices involved in ax-b?0.
• 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.

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Defining x(i,j)

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Defining y(i,j) when Si,j,k,u,v

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

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Positive results

• Without SDP, is LP actually useful?
• Thm here in dense graphs, gap1 after O(1)
  rounds of Sherali-Adams LP
• Note this is not surprising since there already
  exist at least 3 PTAS for MaxCut in dense graphs.

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Conclusion

• LP is potentially an attractive alternative to
  ad hoc fumbling with existing LPs
• Unfortunately, most results so far are negative
  if we dont use SDP.
• To justify continued work on LP, we need some
  positive results for some problem, find a new,
  better approximation algorithm by using LP
  explicitly and voluntarily.

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Thats it

• The end

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Makespan minimization

• Independent jobs, m parallel machines
• LP x(i,j) indicates whether job j goes on
  machine i, and tmakespan.
• Constraints
• Every job must go on some machine
• Makespan greater than load on each machine
• Unbounded gap
• Add makespanp(j) for every job reduces gap to
  2, but this does not appear in LP until after m
  rounds.

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Proof(1/1) based on AFKK

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(No Transcript)
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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 lt 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).