On Approximation of Robust Max-Cut and Related Problems using Randomized Rounding Algorithms

TL;DR

This paper extends the Goemans–Williamson SDP-relaxation and randomized rounding framework for Max-Cut to robust and distributionally robust variants, proving that the classic $0.878$-approximation bound transfers to these uncertain settings. It defines robust and distributionally robust counterparts Val(RP-R) and Val(DRP-R) with corresponding SDP formulations Val(RP-SD) and Val(DRP-SD), and provides analysis showing the rounding still achieves the same performance guarantee via the standard arccos-based rounding analysis. The results further generalize to related problems such as Max-DiCut and Max k-CSP, and the authors explore tractability under polyhedral, ellipsoidal, and Wasserstein uncertainty, offering practical SDP and SIP reformulations. Overall, the work demonstrates that robust and distributionally robust max-cut problems can be solved with the same high-quality guarantees as the nominal problem, broadening applicability to uncertainty-rich optimization tasks.

Abstract

Goemans and Williamson proposed a randomized rounding algorithm for the MAX-CUT problem with a 0.878 approximation bound in expectation. The 0.878 approximation bound remains the best-known approximation bound for this APX-hard problem. Their approach was subsequently applied to other related problems such as Max-DiCut, MAX-SAT, and Max-2SAT, etc. We show that the randomized rounding algorithm can also be used to achieve a 0.878 approximation bound for the robust and distributionally robust counterparts of the max-cut problem. We also show that the approximation bounds for the other problems are maintained for their robust and distributionally robust counterparts if the randomization projection framework is used.

Theorems & Definitions (15)

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