arXiv subjects

Hanlin Ren

Publications and source records attributed to Hanlin Ren.

12 recordsLinked to original sources

arXiv2026

The Weak Rank Principle: Lower Bounds and Applications

Given two symbolic matrices $X$ and $Y$ of dimensions $m\times n$ and $n\times m$, the *weak rank principle* (WRank) states the equation $XY = A$ is unsatisfiable when $m>n$ and rank of $A$ exceeds $n$. We study this principle as an algebraic generalisation of the weak pigeonhole principle (WPHP). As a strengthening of WPHP, it admits proof complexity lower bounds in settings where none are known for WPHP, while still supporting analogous applications. *Generators for PCR$_{F_2}$*: We prove exponential size lower bounds for algebraic, perfect matching, and bamboo-tree encodings of WRank in PCR$_{F_2}$. The latter encoding is the most relevant for applications to circuit lower-bound formulas, as considered by Alekhnovich, Ben-Sasson, Razborov, and Wigderson (SIAM J. Comput., 2004) and Razborov (Ann. Math., 2015). Using a standard iteration technique we amplify the stretch to exponential. This resolves the open problem concerning the construction of proof complexity generators with good stretch for PCR$_{F_2}$. *Generators for Sherali--Adams:* We develop a new size lower-bound technique showing that WRank, encoded as a bamboo-tree CNF, serves as a proof complexity generator for SA. Our method introduces a pseudoexpectation tailored specifically to the rank principle (and incompatible with WPHP). *Circuit lower bound formulas:* We show that PCR$_{F_2}$ does not admit short proofs of lower-bound statements against Boolean circuits, nor against weak models of algebraic circuits. This settles the open problem raised by Razborov (Ann. Math., 2015) concerning the provability of such lower bounds in PCR$_{F_2}$. *Strength of the weak rank principle:* Finally, we show that WRank is *necessary* for proving NC$^2$ circuit lower bounds and, for odd primes $p$, *sufficient* within the theory corresponding to AC$^{0}[p]$ for deriving AC$^{0}[p]$ lower bounds.

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arXiv2026

Near-Maximum Circuit Lower Bounds for Exponential Time with Merlin-Arthur Queries

We prove a near-maximum ($2^n / n$) circuit lower bound for the complexity class $\mathsf{E}^{\mathrm{pr}\mathsf{MA}}/_1$, corresponding to exponential time with access to a promise-$\mathsf{MA}$ oracle and one bit of advice. Our proof incorporates the iterative win-win paradigm (Chen--Lu--Oliveira--Ren--Santhanam, FOCS'23), the reduction from the Range Avoidance problem to circuit lower bounds (Je\v{r}\'abek, Ann. Pure Appl. Log. '04; Korten, FOCS'21), and the PCP theorem. Crucial to our proof is the analysis of the complexity class $\mathsf{P}^\mathsf{NP}[{\textsf{#rounds}}=r, {\textsf{length}}=s]$, which is $\mathsf{P}^\mathsf{NP}$ with $r(n)$ adaptive rounds of $\mathsf{NP}$ queries, where each $\mathsf{NP}$ query has witness length $s(n)$.

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arXiv2025

Total Search Problems in $\mathsf{ZPP}$

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 $\text{Lossy-Code}$.

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arXiv2025

New Algebrization Barriers to Circuit Lower Bounds via Communication Complexity of Missing-String

The *algebrization barrier*, proposed by Aaronson and Wigderson (STOC '08, ToCT '09), captures the limitations of many complexity-theoretic techniques based on arithmetization. Notably, several circuit lower bounds that overcome the relativization barrier (Buhrman--Fortnow--Thierauf, CCC '98; Vinodchandran, TCS '05; Santhanam, STOC '07, SICOMP '09) remain subject to the algebrization barrier. In this work, we establish several new algebrization barriers to circuit lower bounds by studying the communication complexity of the following problem, called XOR-Missing-String: For $m < 2^{n/2}$, Alice gets a list of $m$ strings $x_1, \dots, x_m\in\{0, 1\}^n$, Bob gets a list of $m$ strings $y_1, \dots, y_m\in\{0, 1\}^n$, and the goal is to output a string $s\in\{0, 1\}^n$ that is not equal to $x_i\oplus y_j$ for any $i, j\in [m]$. 1. We construct an oracle $A_1$ and its multilinear extension $\widetilde{A_1}$ such that ${\sf PostBPE}^{\widetilde{A_1}}$ has linear-size $A_1$-oracle circuits on infinitely many input lengths. 2. We construct an oracle $A_2$ and its multilinear extension $\widetilde{A_2}$ such that ${\sf BPE}^{\widetilde{A_2}}$ has linear-size $A_2$-oracle circuits on all input lengths. 3. Finally, we study algebrization barriers to circuit lower bounds for $\sf MA_E$. Buhrman, Fortnow, and Thierauf proved a *sub-half-exponential* circuit lower bound for $\sf MA_E$ via algebrizing techniques. Toward understanding whether the half-exponential bound can be improved, we define a natural subclass of $\sf MA_E$ that includes their hard $\sf MA_E$ language, and prove the following result: For every *super-half-exponential* function $h(n)$, we construct an oracle $A_3$ and its multilinear extension $\widetilde{A_3}$ such that this natural subclass of ${\sf MA}_{\sf E}^{\widetilde{A_3}}$ has $h(n)$-size $A_3$-oracle circuits on all input lengths.

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arXiv2025

Hardness of Range Avoidance and Proof Complexity Generators from Demi-Bits

Given a circuit $G: \{0, 1\}^n \to \{0, 1\}^m$ with $m > n$, the *range avoidance* problem ($\text{Avoid}$) asks to output a string $y\in \{0, 1\}^m$ that is not in the range of $G$. Besides its profound connection to circuit complexity and explicit construction problems, this problem is also related to the existence of *proof complexity generators* -- circuits $G: \{0, 1\}^n \to \{0, 1\}^m$ where $m > n$ but for every $y\in \{0, 1\}^m$, it is infeasible to prove the statement "$y\not\in\mathrm{Range}(G)$" in a given propositional proof system. This paper connects these two problems with the existence of *demi-bits generators*, a fundamental cryptographic primitive against nondeterministic adversaries introduced by Rudich (RANDOM '97). $\bullet$ We show that the existence of demi-bits generators implies $\text{Avoid}$ is hard for nondeterministic algorithms. This resolves an open problem raised by Chen and Li (STOC '24). Furthermore, assuming the demi-hardness of certain LPN-style generators or Goldreich' PRG, we prove the hardness of $\text{Avoid}$ even when the instances are constant-degree polynomials over $\mathbb{F}_2$. $\bullet$ We show that the dual weak pigeonhole principle is unprovable in Cook's theory $\mathsf{PV}_1$ under the existence of demi-bits generators secure against $\mathbf{AM}$, thereby separating Jerabek's theory $\mathsf{APC}_1$ from $\mathsf{PV}_1$. $\bullet$ We transform demi-bits generators to proof complexity generators that are *pseudo-surjective* with nearly optimal parameters. Our constructions build on the recent breakthroughs on the hardness of $\text{Avoid}$ by Ilango, Li, and Williams (STOC '23) and Chen and Li (STOC '24). We use *randomness extractors* to significantly simplify the construction and the proof.

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arXiv2024

Finding Bugs in Short Proofs: The Metamathematics of Resolution Lower Bounds

We study the *refuter* problems for proof complexity lower bounds. Suppose $\varphi$ is a hard tautology that does not admit any length-$s$ proof in some proof system $P$. In the corresponding refuter problem, we are given (query access to) a purported length-$s$ proof $\pi$ in $P$ that claims to have proved $\varphi$, and our goal is to find an invalid derivation step within $\pi$. As suggested by witnessing theorems in bounded arithmetic, the *computational complexity* of these refuter problems is closely tied to the *metamathematics* of the underlying lower bounds. We focus on refuter problems corresponding to lower bounds for *resolution*, which is arguably the single most studied system in proof complexity. To capture the complexity of refuter problems for resolution *size* lower bounds, we introduce a new class $\mathrm{rwPHP}(\mathsf{PLS})$ in decision-tree $\mathsf{TFNP}$, which can be seen as a randomized version of $\mathsf{PLS}$. Interpreted in bounded arithmetic, our results show that the theory $\mathsf{T}^1_2(\alpha) + \mathrm{dwPHP}(\mathsf{PV}(\alpha))$ characterizes the "reasoning power" required to prove (the "easiest") resolution size lower bounds. As a corollary, we obtain surprisingly efficient proofs of resolution lower bounds. In particular, we show that many resolution size lower bounds can be proved in low-width *random resolution* [Pudl\'ak--Thapen, CCC'17].

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arXiv2023

Symmetric Exponential Time Requires Near-Maximum Circuit Size

We show that there is a language in $\mathsf{S}_2\mathsf{E}/_1$ (symmetric exponential time with one bit of advice) with circuit complexity at least $2^n/n$. In particular, the above also implies the same near-maximum circuit lower bounds for the classes $\Sigma_2\mathsf{E}$, $(\Sigma_2\mathsf{E}\cap\Pi_2\mathsf{E})/_1$, and $\mathsf{ZPE}^{\mathsf{NP}}/_1$. Previously, only "half-exponential" circuit lower bounds for these complexity classes were known, and the smallest complexity class known to require exponential circuit complexity was $\Delta_3\mathsf{E} = \mathsf{E}^{\Sigma_2\mathsf{P}}$ (Miltersen, Vinodchandran, and Watanabe COCOON'99). Our circuit lower bounds are corollaries of an unconditional zero-error pseudodeterministic algorithm with an $\mathsf{NP}$ oracle and one bit of advice ($\mathsf{FZPP}^{\mathsf{NP}}/_1$) that solves the range avoidance problem infinitely often. This algorithm also implies unconditional infinitely-often pseudodeterministic $\mathsf{FZPP}^{\mathsf{NP}}/_1$ constructions for Ramsey graphs, rigid matrices, two-source extractors, linear codes, and $\mathrm{K}^{\mathrm{poly}}$-random strings with nearly optimal parameters. Our proofs relativize. The two main technical ingredients are (1) Korten's $\mathsf{P}^{\mathsf{NP}}$ reduction from the range avoidance problem to constructing hard truth tables (FOCS'21), which was in turn inspired by a result of Je\v{r}\'abek on provability in Bounded Arithmetic (Ann. Pure Appl. Log. 2004); and (2) the recent iterative win-win paradigm of Chen, Lu, Oliveira, Ren, and Santhanam (FOCS'23).

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arXiv2023

Polynomial-Time Pseudodeterministic Construction of Primes

A randomized algorithm for a search problem is *pseudodeterministic* if it produces a fixed canonical solution to the search problem with high probability. In their seminal work on the topic, Gat and Goldwasser posed as their main open problem whether prime numbers can be pseudodeterministically constructed in polynomial time. We provide a positive solution to this question in the infinitely-often regime. In more detail, we give an *unconditional* polynomial-time randomized algorithm $B$ such that, for infinitely many values of $n$, $B(1^n)$ outputs a canonical $n$-bit prime $p_n$ with high probability. More generally, we prove that for every dense property $Q$ of strings that can be decided in polynomial time, there is an infinitely-often pseudodeterministic polynomial-time construction of strings satisfying $Q$. This improves upon a subexponential-time construction of Oliveira and Santhanam. Our construction uses several new ideas, including a novel bootstrapping technique for pseudodeterministic constructions, and a quantitative optimization of the uniform hardness-randomness framework of Chen and Tell, using a variant of the Shaltiel--Umans generator.

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arXiv2021

Maintaining Exact Distances under Multiple Edge Failures

We present the first compact distance oracle that tolerates multiple failures and maintains exact distances. Given an undirected weighted graph $G = (V, E)$ and an arbitrarily large constant $d$, we construct an oracle that given vertices $u, v \in V$ and a set of $d$ edge failures $D$, outputs the exact distance between $u$ and $v$ in $G - D$ (that is, $G$ with edges in $D$ removed). Our oracle has space complexity $O(d n^4)$ and query time $d^{O(d)}$. Previously, there were compact approximate distance oracles under multiple failures [Chechik, Cohen, Fiat, and Kaplan, SODA'17; Duan, Gu, and Ren, SODA'21], but the best exact distance oracles under $d$ failures require essentially $\Omega(n^d)$ space [Duan and Pettie, SODA'09]. Our distance oracle seems to require $n^{\Omega(d)}$ time to preprocess; we leave it as an open question to improve this preprocessing time.

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arXiv2021

Constructing a Distance Sensitivity Oracle in $O(n^{2.5794}M)$ Time

We continue the study of distance sensitivity oracles (DSOs). Given a directed graph $G$ with $n$ vertices and edge weights in $\{1, 2, \dots, M\}$, we want to build a data structure such that given any source vertex $u$, any target vertex $v$, and any failure $f$ (which is either a vertex or an edge), it outputs the length of the shortest path from $u$ to $v$ not going through $f$. Our main result is a DSO with preprocessing time $O(n^{2.5794}M)$ and constant query time. Previously, the best preprocessing time of DSOs for directed graphs is $O(n^{2.7233}M)$, and even in the easier case of undirected graphs, the best preprocessing time is $O(n^{2.6865}M)$ [Ren, ESA 2020]. One drawback of our DSOs, though, is that it only supports distance queries but not path queries. Our main technical ingredient is an algorithm that computes the inverse of a degree-$d$ polynomial matrix (i.e. a matrix whose entries are degree-$d$ univariate polynomials) modulo $x^r$. The algorithm is adapted from [Zhou, Labahn, and Storjohann, Journal of Complexity, 2015], and we replace some of its intermediate steps with faster rectangular matrix multiplication algorithms. We also show how to compute unique shortest paths in a directed graph with edge weights in $\{1, 2, \dots, M\}$, in $O(n^{2.5286}M)$ time. This algorithm is crucial in the preprocessing algorithm of our DSO. Our solution improves the $O(n^{2.6865}M)$ time bound in [Ren, ESA 2020], and matches the current best time bound for computing all-pairs shortest paths.

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arXiv2020

Improved Distance Sensitivity Oracles with Subcubic Preprocessing Time

We consider the problem of building Distance Sensitivity Oracles (DSOs). Given a directed graph $G=(V, E)$ with edge weights in $\{1, 2, \dots, M\}$, we need to preprocess it into a data structure, and answer the following queries: given vertices $u,v\in V$ and a failed vertex or edge $f\in (V\cup E)$, output the length of the shortest path from $u$ to $v$ that does not go through $f$. Our main result is a simple DSO with $\tilde{O}(n^{2.7233}M)$ preprocessing time and $O(1)$ query time. Moreover, if the input graph is undirected, the preprocessing time can be improved to $\tilde{O}(n^{2.6865}M)$. The preprocessing algorithm is randomized with correct probability $\ge 1-1/n^C$, for a constant $C$ that can be made arbitrarily large. Previously, there is a DSO with $\tilde{O}(n^{2.8729}M)$ preprocessing time and $\operatorname{polylog}(n)$ query time [Chechik and Cohen, STOC'20]. At the core of our DSO is the following observation from [Bernstein and Karger, STOC'09]: if there is a DSO with preprocessing time $P$ and query time $Q$, then we can construct a DSO with preprocessing time $P+\tilde{O}(n^2)\cdot Q$ and query time $O(1)$. (Here $\tilde{O}(\cdot)$ hides $\operatorname{polylog}(n)$ factors.)

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arXiv2020

Approximate Distance Oracles Subject to Multiple Vertex Failures

Given an undirected graph $G=(V,E)$ of $n$ vertices and $m$ edges with weights in $[1,W]$, we construct vertex sensitive distance oracles (VSDO), which are data structures that preprocess the graph, and answer the following kind of queries: Given a source vertex $u$, a target vertex $v$, and a batch of $d$ failed vertices $D$, output (an approximation of) the distance between $u$ and $v$ in $G-D$ (that is, the graph $G$ with vertices in $D$ removed). An oracle has stretch $\alpha$ if it always holds that $\delta_{G-D}(u,v)\le\tilde{\delta}(u,v)\le\alpha\cdot\delta_{G-D}(u,v)$, where $\delta_{G-D}(u,v)$ is the actual distance between $u$ and $v$ in $G-D$, and $\tilde{\delta}(u,v)$ is the distance reported by the oracle. In this paper we construct efficient VSDOs for any number $d$ of failures. For any constant $c\geq 1$, we propose two oracles: $\bullet$ The first oracle has size $n^{2+1/c}(\log n/\epsilon)^{O(d)}\cdot \log W$, answers a query in ${\rm poly}(\log n,d^c,\log\log W,\epsilon^{-1})$ time, and has stretch $1+\epsilon$, for any constant $\epsilon>0$. $\bullet$ The second oracle has size $n^{2+1/c}{\rm poly}(\log (nW),d)$, answers a query in ${\rm poly}(\log n,d^c,\log\log W)$ time, and has stretch ${\rm poly}(\log n,d)$. Both of these oracles can be preprocessed in time polynomial in their space complexity. These results are the first approximate distance oracles of poly-logarithmic query time for any constant number of vertex failures in general undirected graphs. Previously there are $(1+\epsilon)$-approximate $d$-edge sensitive distance oracles [Chechik et al. 2017] answering distance queries when $d$ edges fail, which have size $O(n^2(\log n/\epsilon)^d\cdot d\log W)$ and query time ${\rm poly}(\log n, d, \log\log W)$.

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