Succinct approximate convex pareto curves

We study the succinct approximation of convex Pareto curves of multiobjective optimization problems. We propose the concept of ε-convex Pareto (ε-CP) set as the appropriate one for the convex setting, and observe that it can offer arbitrarily more compact representations than ε-Pareto sets in this context. We characterize when an ε-CP can be constructed in polynomial time in terms of an efficient routine Comb for optimizing (exactly or approximately) monotone linear combinations of the objectives. We investigate the problem of computing minimum size ε-convex Pareto sets, both for discrete (combinatorial) and continuous (convex) problems, and present general algorithms using a Comb routine. For bi-objective problems, we show that if we have an exact Comb optimization routine, then we can compute the minimum ε-CP for continuous problems (this applies for example to bi-objective Linear Programming and Markov Decision Processes), and factor 2 approximation to the minimum ε-CP for discrete problems (this applies for example to bi-objective versions of polynomial-time solvable combinatorial problems such as Shortest Paths, Spanning Tree, etc.). If we have an approximate Comb routine, then we can compute factor 3 and 6 approximations respectively to the minimum ε-CP for continuous and discrete bi-objective problems. We consider also the case of three and more objectives and present some upper and lower bounds.