# A General Approximation Method for Bicriteria Minimization Problems

## TL;DR

This paper introduces a versatile approximation technique for bicriteria minimization problems, enabling near-optimal solutions and Pareto curves efficiently, even when previous methods are infeasible due to NP-hardness.

## Contribution

The authors develop a general polynomial-time method to approximate bicriteria minimization problems and Pareto curves using existing weighted sum algorithms, applicable to many problems with NP-hard gap issues.

## Key findings

- Provides a polynomial-time approximation algorithm for bicriteria minimization.
- Extends the method to compute approximate Pareto curves.
- Shows similar results are unlikely for maximization problems unless P=NP.

## Abstract

We present a general technique for approximating bicriteria minimization problems with positive-valued, polynomially computable objective functions. Given $0<\epsilon\leq1$ and a polynomial-time $\alpha$-approximation algorithm for the corresponding weighted sum problem, we show how to obtain a bicriteria $(\alpha\cdot(1+2\epsilon),\alpha\cdot(1+\frac{2}{\epsilon}))$-approximation algorithm for the budget-constrained problem whose running time is polynomial in the encoding length of the input and linear in $\frac{1}{\epsilon}$. Moreover, we show that our method can be extended to compute an $(\alpha\cdot(1+2\epsilon),\alpha\cdot(1+\frac{2}{\epsilon}))$-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 $\textsf{NP}$-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 $\textsf{P}=\textsf{NP}$.