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

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 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) Gap = 2 ij 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 e b 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)

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

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

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).