Bilu-Linial Stable Instances of Max Cut and Minimum Multiway Cut

We investigate the notion of stability proposed by Bilu and Linial. We obtain an exact polynomial-time algorithm for $\gamma$-stable Max Cut instances with $\gamma \geq c\sqrt{\log n}\log\log n$ for some absolute constant $c > 0$. Our algorithm is robust: it never returns an incorrect answer; if the instance is $\gamma$-stable, it finds the maximum cut, otherwise, it either finds the maximum cut or certifies that the instance is not $\gamma$-stable. We prove that there is no robust polynomial-time algorithm for $\gamma$-stable instances of Max Cut when $\gamma < \alpha_{SC}(n/2)$, where $\alpha_{SC}$ is the best approximation factor for Sparsest Cut with non-uniform demands. Our algorithm is based on semidefinite programming. We show that the standard SDP relaxation for Max Cut (with $\ell_2^2$ triangle inequalities) is integral if $\gamma \geq D_{\ell_2^2\to \ell_1}(n)$, where $D_{\ell_2^2\to \ell_1}(n)$ is the least distortion with which every $n$ point metric space of negative type embeds into $\ell_1$. On the negative side, we show that the SDP relaxation is not integral when $\gamma < D_{\ell_2^2\to \ell_1}(n/2)$. Moreover, there is no tractable convex relaxation for $\gamma$-stable instances of Max Cut when $\gamma < \alpha_{SC}(n/2)$. That suggests that solving $\gamma$-stable instances with $\gamma =o(\sqrt{\log n})$ might be difficult or impossible. Our results significantly improve previously known results. The best previously known algorithm for $\gamma$-stable instances of Max Cut required that $\gamma \geq c\sqrt{n}$ (for some $c > 0$) [Bilu, Daniely, Linial, and Saks]. No hardness results were known for the problem. Additionally, we present an algorithm for 4-stable instances of Minimum Multiway Cut. We also study a relaxed notion of weak stability.Comment: 24 page

Matrix Minor Reformulation and SOCP-based Spatial Branch-and-Cut Method for the AC Optimal Power Flow Problem

Alternating current optimal power flow (AC OPF) is one of the most fundamental optimization problems in electrical power systems. It can be formulated as a semidefinite program (SDP) with rank constraints. Solving AC OPF, that is, obtaining near optimal primal solutions as well as high quality dual bounds for this non-convex program, presents a major computational challenge to today's power industry for the real-time operation of large-scale power grids. In this paper, we propose a new technique for reformulation of the rank constraints using both principal and non-principal 2-by-2 minors of the involved Hermitian matrix variable and characterize all such minors into three types. We show the equivalence of these minor constraints to the physical constraints of voltage angle differences summing to zero over three- and four-cycles in the power network. We study second-order conic programming (SOCP) relaxations of this minor reformulation and propose strong cutting planes, convex envelopes, and bound tightening techniques to strengthen the resulting SOCP relaxations. We then propose an SOCP-based spatial branch-and-cut method to obtain the global optimum of AC OPF. Extensive computational experiments show that the proposed algorithm significantly outperforms the state-of-the-art SDP-based OPF solver and on a simple personal computer is able to obtain on average a 0.71% optimality gap in no more than 720 seconds for the most challenging power system instances in the literature

New Formulation and Strong MISOCP Relaxations for AC Optimal Transmission Switching Problem

As the modern transmission control and relay technologies evolve, transmission line switching has become an important option in power system operators' toolkits to reduce operational cost and improve system reliability. Most recent research has relied on the DC approximation of the power flow model in the optimal transmission switching problem. However, it is known that DC approximation may lead to inaccurate flow solutions and also overlook stability issues. In this paper, we focus on the optimal transmission switching problem with the full AC power flow model, abbreviated as AC OTS. We propose a new exact formulation for AC OTS and its mixed-integer second-order conic programming (MISOCP) relaxation. We improve this relaxation via several types of strong valid inequalities inspired by the recent development for the closely related AC Optimal Power Flow (AC OPF) problem. We also propose a practical algorithm to obtain high quality feasible solutions for the AC OTS problem. Extensive computational experiments show that the proposed formulation and algorithms efficiently solve IEEE standard and congested instances and lead to significant cost benefits with provably tight bounds

Towards a better approximation for sparsest cut?

We give a new $(1+\epsilon)$-approximation for sparsest cut problem on graphs where small sets expand significantly more than the sparsest cut (sets of size $n/r$ expand by a factor $\sqrt{\log n\log r}$ bigger, for some small $r$; this condition holds for many natural graph families). We give two different algorithms. One involves Guruswami-Sinop rounding on the level-$r$ Lasserre relaxation. The other is combinatorial and involves a new notion called {\em Small Set Expander Flows} (inspired by the {\em expander flows} of ARV) which we show exists in the input graph. Both algorithms run in time $2^{O(r)} \mathrm{poly}(n)$. We also show similar approximation algorithms in graphs with genus $g$ with an analogous local expansion condition. This is the first algorithm we know of that achieves $(1+\epsilon)$-approximation on such general family of graphs