Characterization of QUBO reformulations for the maximum $k$-colorable subgraph problem

Quantum devices can be used to solve constrained combinatorial optimization (COPT) problems thanks to the use of penalization methods to embed the COPT problem's constraints in its objective to obtain a quadratic unconstrained binary optimization (QUBO) reformulation of the COPT. However, the particular way in which this penalization is carried out, affects the value of the penalty parameters, as well as the number of additional binary variables that are needed to obtain the desired QUBO reformulation. In turn, these factors substantially affect the ability of quantum computers to efficiently solve these constrained COPT problems. This efficiency is key towards the goal of using quantum computers to solve constrained COPT problems more efficiently than with classical computers. Along these lines, we consider an important constrained COPT problem; namely, the maximum $k$-colorable subgraph (M$k$CS) problem, in which the aim is to find an induced $k$-colorable subgraph with maximum cardinality in a given graph. This problem arises in channel assignment in spectrum sharing networks, VLSI design, human genetic research, and cybersecurity. We derive two QUBO reformulations for the M$k$CS problem, and fully characterize the range of the penalty parameters that can be used in the QUBO reformulations. Further, one of the QUBO reformulations of the M$k$CS problem is obtained without the need to introduce additional binary variables. To illustrate the benefits of obtaining and characterizing these QUBO reformulations, we benchmark different QUBO reformulations of the M$k$CS problem by performing numerical tests on D-Wave's quantum annealing devices. These tests also illustrate the numerical power gained by using the latest D-Wave's quantum annealing device.