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Linear Programming Relaxations for MaxCut

Linear Programming Relaxations for MaxCut. Wenceslas 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.

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Linear Programming Relaxations for MaxCut

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  1. Linear Programming Relaxations for MaxCut Wenceslas Fernandez de la Vega Claire Kenyon-Mathieu

  2. Technique for approximation • IP formulation with 0-1 variables • LP relaxation  algorithm • Strengthen LP: add valid inequalities • Reduce integrality gap = •  Better approximation

  3. 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

  4. 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.

  5. 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!

  6. 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.

  7. 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.

  8. 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!

  9. 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

  10. … enhanced • Additional valid inequalities: x(a,b)+x(a,c)+…+x(d,e) ≤ 6 • We will prove that we still haveGap = 2. d a e b I cut at most 6 edges c

  11. Gap=2! • Graph: sparse random, altered for large girth. • MaxCut ≈|E|/2 w.h.p. • To definex(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|

  12. Core of proof: feasibility • (x(i,j)) satisfies everyconstraint: 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.

  13. Defining x(i,j)

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

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

  16. 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.

  17. 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.

  18. That’s it • The end

  19. 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.

  20. Proof(1/1) based on [AFKK]

  21. 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).

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