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.

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In this article, we introduce the rectangular knapsack problem as a special case of the quadratic knapsack problem consisting in the maximization of the product of two separate knapsack profits subject to a cardinality constraint. We propose a polynomial time algorithm for this problem that provides a constant approximation ratio of 4.5. Our experimental results on a large number of artificially generated problem instances show that the average ratio is far from theoretical guarantee. In addition, we suggest refined versions of this approximation algorithm with the same time complexity and approximation ratio that lead to even better experimental results.

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