Characterizing QUBO Reformulations of the Max-k-Cut Problem for Quantum Computing

TL;DR

This paper characterizes minimal penalty coefficients for QUBO reformulations of the max-k-cut problem, aiding quantum algorithms in solving combinatorial optimization problems more effectively.

Contribution

It provides closed-form formulas for tight penalty coefficients in two QUBO reformulations of max-k-cut, depending on graph vertex degrees.

Findings

Derived explicit formulas for penalty coefficients based on vertex degrees.

Validated theoretical results with examples and numerical experiments.

Abstract

Quantum computing offers significant potential for solving NP-hard combinatorial (optimization) problems that are beyond the reach of classical computers. One way to tap into this potential is by reformulating combinatorial problems as a quadratic unconstrained binary optimization (QUBO) problem. The solution of the QUBO reformulation can then be addressed using adiabatic quantum computing devices or appropriate quantum computing algorithms on gate-based quantum computing devices. In general, QUBO reformulations of combinatorial problems can be readily obtained by properly penalizing the violation of the problem's constraints in the original problem's objective. However, characterizing tight (i.e., minimal but sufficient) penalty coefficients for this purpose is important and non-trivial for enabling the solution of the resulting QUBO in current and near-term quantum computing devices. Along these lines, we present closed-form characterizations of tight penalty coefficients for two distinct QUBO reformulations of the max $k$-cut problem whose values depend on the (weighted) degree of the vertices of the graph defining the problem. These findings contribute to the ongoing effort to make quantum computing a viable tool for solving combinatorial problems at scale. We support our theoretical results with illustrative examples and simple numerical results.