Linear Programming
Relaxations for MaxCut
Wenceslas Fernandez de la Vega
Claire Kenyon-Mathieu
Technique for approximation
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IP formulation with 0-1 variables
LP relaxation  algorithm
Strengthen LP: add valid inequalities
Reduce integrality gap =
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 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 LovaszShrijver 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. SheraliAdams 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 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!
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)
i
j
• Gap = 2
k
… enhanced
• Additional valid inequalities:
a
b
d
e
I cut at most 6 edges
c
x(a,b)+x(a,c)+…+x(d,e) ≤ 6
• We will prove that we still have Gap = 2.
• Graph: sparse random, altered for large
girth.
Gap=2!
•
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|
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.
Defining x(i,j)
•
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Defining y(i,j) when S={i,j,k,u,v}
•
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Coupling x(i,j) and y(i,j)
•
QuickTime™ and a
TIFF (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 machine
Makespan 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).