Maximilian Witek

Approximability and hardness in multi-objective optimization

We study the approximability and the hardness of combinatorial multi-objective NP optimization problems (multi-objective problems, for short). Our contributions are: - We define and compare several solution notions that capture reasonable algorithmic tasks for computing optimal solutions. - These solution notions induce corresponding NP-hardness notions for which we prove implication and separation results. - We define approximative solution notions and investigate in which cases polynomial-time solvability translates from one to another notion. Moreover, for problems where all objectives have to be minimized, approximability results translate from single-objective to multi-objective optimization such that the relative error degrades only by a constant factor. Such translations are not possible for problems where all objectives have to be maximized (unless P = NP). As a consequence we see that in contrast to single-objective problems (where the solution notions coincide), the situation is more subtle for multiple objectives. So it is important to exactly specify the NP-hardness notion when discussing the complexity of multi-objective problems.

Autoreducibility of complete sets for log-space and polynomial-time reductions

We investigate the autoreducibility and mitoticity of complete sets for several classes with respect to different polynomial-time and logarithmic-space reducibility notions. Previous work in this area focused on polynomial-time reducibility notions. Here we obtain new mitoticity and autoreducibility results for the classes EXP and NEXP with respect to some restricted truth-table reductions (e.g.,

Applications of Discrepancy Theory in Multiobjective Approximation

\leq^{p}_{2-tt},\leq^{p}_{ctt},\leq^{p}_{dtt}

Autoreducibility and Mitoticity of Logspace-Complete Sets for NP and Other Classes

). Moreover, we start a systematic study of logarithmic-space autoreducibility and mitoticity which enables us to also consider P and smaller classes. Among others, we obtain the following results: · Regarding

Introduction to Autoreducibility and Mitoticity

\leq^{log}_{m}, \leq^{log}_{2-tt}, \leq^{log}_{dtt}

Structural complexity of multiobjective NP search problems

and

Autoreducibility and mitoticity of logspace-complete sets for NP and other classes

\leq^{log}_{ctt}

The Complexity of Solving Multiobjective Optimization Problems and its Relation to Multivalued Functions.

, complete sets for PSPACE and EXP are mitotic, and complete sets for NEXP are autoreducible. · All

Improved and Derandomized Approximations for Two-Criteria Metric Traveling Salesman

\leq^{log}_{1-tt}

Hardness and Approximability in Multi-Objective Optimization.

-complete sets for NL and P are