No abstract

Multiobjective combinatorial optimization problems are known to be hard problems for two reasons: their decision versions are often NP-complete, and they are often intractable. Apart from this general observation, are there also variants or cases of multiobjective combinatorial optimization problems that are easy and, if so, what causes them to be easy? This article is a first attempt to provide an answer to these two questions. Thereby, a systematic description of reasons for easiness is envisaged rather than a mere collection of special cases. In particular, the borderline of easy and hard multiobjective optimization problems is explored.

We present a general technique for approximating bicriteria minimization problems with positive-valued, polynomially computable objective functions. Given 0 < ǫ ≤ 1 and a polynomial-time α-approximation algorithm for the corresponding weighted sum problem, we show how to obtain a bicriteria (α · (1 + 2ǫ), α · (1 + 2 ǫ ))-approximation algorithm for the budget-constrained problem whose running time is polynomial in the encoding length of the input and linear in 1 ǫ . Moreover, we show that our method can be extended to compute an (α · (1 + 2ǫ), α · (1 + 2 ǫ ))-approximate Pareto curve under the same assumptions. Our technique applies to many minimization problems to which most previous algorithms for computing approximate Pareto curves cannot be applied because the corresponding gap problem is NP-hard to solve. For maximization problems, however, we show that approximation results similar to the ones presented here for minimization problems are impossible to obtain in polynomial time unless P = NP.

In the last decade, public and industrial research funding has moved quantum computing from the early promises of Shor’s algorithm through experiments to the era of noisy intermediate scale quantum devices (NISQ) for solving real-world problems. It is likely that quantum methods can efficiently solve certain (NP-) hard optimization problems where classical approaches fail. In our perspective, we examine the field of quantum optimization, that is, solving optimization problems using quantum computers. We provide an entry point to quantum optimization for researchers from each topic, optimization or quantum computing, by demonstrating advances and obstacles with a suitable use case. We give an overview on problem formulation, available algorithms, and benchmarking. Although we show a proof-of-concept rather than a full benchmark between classical and quantum methods, this gives an idea of the current quality and capabilities of quantum computers for optimization problems. All observations are incorporated in a discussion on some recent quantum optimization breakthroughs, current status, and future directions.

We provide a comprehensive overview of the literature of algorithmic approaches for multiobjective mixed-integer and integer linear optimization problems. More precisely, we categorize and display exact methods for multiobjective linear problems with integer variables for computing the entire set of nondominated images. Our review lists 108 articles and is intended to serve as a reference for all researchers who are familiar with basic concepts of multiobjective optimization and who have an interest in getting a thorough view on the state-of-the-art in multiobjective mixedinteger programming.