Total Search Problems in $\mathsf{ZPP}$

TL;DR

This work formalizes ${\mathsf{TFZPP}}$, the randomized-time subset of ${\mathsf{TFNP}}$, and establishes a powerful black-box separation framework showing ${\mathsf{TFZPP}^{dt}}$ cannot be contained in any uniformly generated ${\mathsf{TFNP}^{dt}}$ class unless ${\mathsf{NP\subseteq QP}}$. Central to their program is a randomized-proof-complexity correspondence: total search problems are characterized by randomized reductions that correspond to randomized tree-like proofs, with ${\mathsf{Lossy-Code}}$ emerging as a robust, complete problem for the class ${\mathsf{LOSSY}}$. They introduce the Nephew problem as a natural candidate in ${\mathsf{TFZPP}}$, develop a rich taxonomy around ${\mathsf{Lossy-Code}}$, and prove numerous reductions placing many natural total problems within LOSSY or related classes, including dense linear ordering. A suite of reductions grounded in ${\mathsf{APC}_1}$ and reconstructive pseudorandom generators (notably the Nisan–Wigderson construction) demonstrate that several problems, such as AMGM-LC and Inclusion-Exclusion, are lossily reducible to ${\mathsf{Lossy-Code}}$, culminating in LOSSY-completeness results. The paper further connects these ideas to open questions about derandomization frontiers and the separations among TFNP subclasses, and it lays out explicit open problems, including a potential natural ${\mathsf{TFZPP}}^{dt}$ problem beyond Lossy-Code and bounds for Bertrand–Chebyshev and Razborov–Smolensky type tasks. The results collectively offer a compelling framework for a black-box program in complexity theory, tying randomized total-search to proof complexity and providing a roadmap for explicit separations and future derandomization insights.

Abstract

We initiate a systematic study of ${\sf TFZPP}$, the class of total ${\sf NP}$ search problems solvable by polynomial time randomized algorithms. ${\sf TFZPP}$ contains a variety of important search problems such as $\text{Bertrand-Chebyshev}$ (finding a prime between $N$ and $2N$), refuter problems for many circuit lower bounds, and $\text{Lossy-Code}$. The $\text{Lossy-Code}$ problem has found prominence due to its fundamental connections to derandomization, catalytic computing, and the metamathematics of complexity theory, among other areas. While ${\sf TFZPP}$ collapses to ${\sf FP}$ under standard derandomization assumptions in the white-box setting, we are able to separate ${\sf TFZPP}$ from the major ${\sf TFNP}$ subclasses in the black-box setting. In fact, we are able to separate it from every uniform ${\sf TFNP}$ class assuming that ${\sf NP}$ is not in quasi-polynomial time. To do so, we extend the connection between proof complexity and black-box ${\sf TFNP}$ to randomized proof systems and randomized reductions. Next, we turn to developing a taxonomy of ${\sf TFZPP}$ problems. We highlight a problem called $\text{Nephew}$, originating from an infinity axiom in set theory. We show that $\text{Nephew}$ is in $\mathsf{PWPP}\cap \mathsf{TFZPP}$ and conjecture that it is not reducible to $\text{Lossy-Code}$. Intriguingly, except for some artificial examples, most other black-box ${\sf TFZPP}$ problems that we are aware of reduce to $\text{Lossy-Code}$.

Figures (10)

• Figure 1: The $\textup{TFZPP}\xspace$ zoo.
• Figure 2: An example $G_f$ with levels marked.
• Figure 3: The procedure performed by ${\text{\upshape\ttfamily Find-Children}}\xspace_{f, g}(v)$. Solid arrows represent $f$ and dashed arrows represent $g$. Parentheses are omitted in labels. The dotted boxes indicate that vertices $g(v)$ and $h(v)$ will be returned as the children of $v$. The procedure will check if the shaded vertices are ${\text{\upshape\scshape Nephew}}\xspace$ solutions, and in doing so will visit the unshaded vertices (but will not detect if these are solutions). Note that $f(h(v)) = v$ is possible, but $f(h(v)) = f(g(v))$ is not.
• Figure 4: The argument behind \ref{['claim: different children']}. Solid arrows represent $f$ and dashed arrows represent $g$. Parentheses are omitted in labels. The shaded vertices are possible locations of the $w \in V^*$ such that either $L(w) = u$ or $R(w) = u$.
• Figure 5: Illustration of cases I and II of \ref{['lem: pair of vertices']}. Solid arrows represent $f$ and dashed arrows represent $g$. Zigzag $f$ arrows are between vertices of level 0. Parentheses are omitted in labels.

Theorems & Definitions (88)

• Example 1.1
• Example 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Theorem 1.10