Sherali - Adams Strikes Back.

Let G be any n-vertex graph whose random walk matrix has its nontrivial eigenvalues bounded in magnitude by 1/root Delta (for example, a random graph G of average degree Theta(Delta) typically has this property). We show that the exp(c log n/log Delta)-round Sherali-Adams linear programming hierarchy certifies that the maximum cut in such a G is at most 50.1% (in fact, at most 1/2 + 2(-Omega(c))). For example, in random graphs with n(1.01) edges, O(1) rounds suffice; in random graphs with n.polylog(n) edges, n(O(1/ log log n)) = n(o(1)) rounds suffice. Our results stand in contrast to the conventional beliefs that linear programming hierarchies perform poorly for MAX-CUT and other CSPs, and that eigenvalue/SDP methods are needed for effective refutation. Indeed, our results imply that constant-round Sherali-Adams can strongly refute random Boolean k-CSP instances with n([k/2]+delta) constraints; previously this had only been done with spectral algorithms or the SOS SDP hierarchy.