Optimal algorithms and proofs (Dagstuhl Seminar 14421)

This report documents the programme and the outcomes of the Dagstuhl Seminar 14421 "Optimal algorithms and proofs". The seminar brought together researchers working in computational and proof complexity, logic, and the theory of approximations. Each of these areas has its own, but connected notion of optimality; and the main aim of the seminar was to bring together researchers from these different areas, for an exchange of ideas, techniques, and open questions, thereby triggering new research collaborations across established research boundaries

Upward Translation of Optimal and P-Optimal Proof Systems in the Boolean Hierarchy over NP

We study the existence of optimal and p-optimal proof systems for classes in the Boolean hierarchy over $\mathrm{NP}$. Our main results concern $\mathrm{DP}$, i.e., the second level of this hierarchy: If all sets in $\mathrm{DP}$ have p-optimal proof systems, then all sets in $\mathrm{coDP}$ have p-optimal proof systems. The analogous implication for optimal proof systems fails relative to an oracle. As a consequence, we clarify such implications for all classes $\mathcal{C}$ and $\mathcal{D}$ in the Boolean hierarchy over $\mathrm{NP}$: either we can prove the implication or show that it fails relative to an oracle. Furthermore, we show that the sets $\mathrm{SAT}$ and $\mathrm{TAUT}$ have p-optimal proof systems, if and only if all sets in the Boolean hierarchy over $\mathrm{NP}$ have p-optimal proof systems which is a new characterization of a conjecture studied by Pudl\'ak

Do there exist complete sets for promise classes?

In this paper we investigate the following two questions: Q1: Do there exist optimal proof systems for a given language L? Q2: Do there exist complete problems for a given promise class ? For concrete languages L (such as TAUT or SAT) and concrete promise classes (such as , , , disjoint -pairs etc.), these questions have been intensively studied during the last years, and a number of characterizations have been obtained. Here we provide new characterizations for Q1 and Q2 that apply to almost all promise classes and languages L, thus creating a unifying framework for the study of these practically relevant questions. While questions Q1 and Q2 are left open by our results, we show that they receive affirmative answers when a small amount of advice is available in the underlying machine model. For promise classes with promise condition in, the advice can be replaced by a tally -oracle