arXiv subjects

Noah Fleming

Publications and source records attributed to Noah Fleming.

14 recordsLinked to original sources

arXiv2026

Provable Reductions in TFNP

We introduce a new family of propositional proof systems, denoted , for an arbitrary TFNP search problem $R$. Informally, a refutation of a CNF formula $F$ in is given by a polynomial-time reduction from the false-clause search problem $Search_F$ to $R$, combined with an Extended Frege proof that the reduction is correct. These are motivated in two ways: 1. They are the propositional translations of witnessing theorems in bounded arithmetic, by which proofs of $\forall \Sigma^b_1$ formulas $\phi$ in a theory $T$ imply algorithms solving the search problem for $\phi$ in a TFNP class corresponding to $T$. 2. They are a white-box analogue of the characterizations of proof systems using decision tree reductions to black-box TFNP problems. We consider the proof system , where Iter is a complete problem for PLS. We prove that is polynomially equivalent to the sequent calculus $G_1$, and also to the implicit Resolution proof system [EF, Resolution]. Hence $G_1$ and [EF, Resolution] are equivalent, which is the first characterization of an implicit proof system by a classical proof system beyond the work of Wang. We also consider for general TFNP relations $R$. We observe that if EF can prove that a search problem $R$ is in FP, then is polynomially equivalent to EF. This contrasts to our above result, which shows that Extended-Frege provable reductions to $Iter$, a problem widely believed not to be in FP, yields a proof system ($G_1$) that is believed to be stronger than Extended Frege. Finally, we show that for any proof system $P$ which is sufficiently strong, there is a polynomial-time computable search problem $R_P \in $ FP such that is polynomially equivalent to $P$. Letting $P =$ [EF, Resolution] and combining our two results shows that is polynomially equivalent to .

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arXiv2026

Non-Signaling Locality Lower Bounds for Dominating Set

Minimum dominating set is a basic local covering problem and a core task in distributed computing. Despite extensive study, in the classic LOCAL model there exist significant gaps between known algorithms and lower bounds. Chang and Li prove an $\Omega(\log n)$-locality lower bound for a constant factor approximation, while Kuhn--Moscibroda--Wattenhofer gave an algorithm beating this bound beyond $\log \Delta$-approximation, along with a weaker lower bound for this degree-dependent setting scaling roughly with $\min\{\log \Delta/\log\log \Delta,\sqrt{\log n/\log\log n}\}$. Unfortunately, this latter bound is weak for small $\Delta$, and never recovers the Chang--Li bound, leaving central questions: does $O(\log \Delta)$-approximation require $\Omega(\log n)$ locality, and do such bounds extend beyond LOCAL? In this work, we take a major step toward answering these questions in the non-signaling model, which strictly subsumes the LOCAL, quantum-LOCAL, and bounded-dependence settings. We prove every $O(\log\Delta)$-approximate non-signaling distribution for dominating set requires locality $\Omega(\log n/(\log\Delta \cdot \mathrm{poly}\log\log\Delta))$. Further, we show for some $\beta \in (0,1)$, every $O(\log^\beta \Delta)$-approximate non-signaling distribution requires locality $\Omega(\log n/\log\Delta)$, which combined with the KMW bound yields a degree-independent $\Omega(\sqrt{\log n/\log\log n})$ quantum-LOCAL lower bound for $O(\log^\beta\Delta)$-approximation algorithms. The proof is based on two new low-soundness sensitivity lower bounds for label cover, one via Impagliazzo--Kabanets--Wigderson-style parallel repetition with degree reduction and one from a sensitivity-preserving reworking of the Dinur--Harsha framework, together with the reductions from label cover to set cover to dominating set and the sensitivity-to-locality transfer theorem of Fleming and Yoshida.

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arXiv2026

Separations above TFNP from Sherali-Adams Lower Bounds

Unlike in TFNP, for which there is an abundance of problems capturing natural existence principles which are incomparable (in the black-box setting), Kleinberg et al. [KKMP21] observed that many of the natural problems considered so far in the second level of the total function polynomial hierarchy (TF$\Sigma_2$) reduce to the Strong Avoid problem. In this work, we prove that the Linear Ordering Principle does not reduce to Strong Avoid in the black-box setting, exhibiting the first TF$\Sigma_2$ problem that lies outside of the class of problems reducible to Strong Avoid. The proof of our separation exploits a connection between total search problems in the polynomial hierarchy and proof complexity, recently developed by Fleming, Imrek, and Marciot [FIM25]. In particular, this implies that to show our separation, it suffices to show that there is no small proof of the Linear Ordering Principle in a $\Sigma_2$-variant of the Sherali-Adams proof system. To do so, we extend the classical pseudo-expectation method to the $\Sigma_2$ setting, showing that the existence of a $\Sigma_2$ pseudo-expectation precludes a $\Sigma_2$ Sherali-Adams proof. The main technical challenge is in proving the existence of such a pseudo-expectation, we manage to do so by solving a combinatorial covering problem about permutations. We also show that the extended pseudo-expectation bound implies that the Linear Ordering Principle cannot be reduced to any problem admitting a low-degree Sherali-Adams refutation.

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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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arXiv2024

Truly Supercritical Trade-offs for Resolution, Cutting Planes, Monotone Circuits, and Weisfeiler-Leman

We exhibit supercritical trade-off for monotone circuits, showing that there are functions computable by small circuits for which any circuit must have depth super-linear or even super-polynomial in the number of variables, far exceeding the linear worst-case upper bound. We obtain similar trade-offs in proof complexity, where we establish the first size-depth trade-offs for cutting planes and resolution that are truly supercritical, i.e., in terms of formula size rather than number of variables, and we also show supercritical trade-offs between width and size for treelike resolution. Our results build on a new supercritical width-depth trade-off for resolution, obtained by refining and strengthening the compression scheme for the Cop-Robber game in [Grohe, Lichter, Neuen & Schweitzer 2023]. This yields robust supercritical trade-offs for dimension versus iteration number in the Weisfeiler-Leman algorithm, which also translate into trade-offs between number of variables and quantifier depth in first-order logic. Our other results follow from improved lifting theorems that might be of independent interest.

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arXiv2024

Sensitivity Lower Bounds for Approximaiton Algorithms

Sensitivity measures how much the output of an algorithm changes, in terms of Hamming distance, when part of the input is modified. While approximation algorithms with low sensitivity have been developed for many problems, no sensitivity lower bounds were previously known for approximation algorithms. In this work, we establish the first polynomial lower bound on the sensitivity of (randomized) approximation algorithms for constraint satisfaction problems (CSPs) by adapting the probabilistically checkable proof (PCP) framework to preserve sensitivity lower bounds. From this, we derive polynomial sensitivity lower bounds for approximation algorithms for a variety of problems, including maximum clique, minimum vertex cover, and maximum cut. Leveraging the connection between sensitivity and locality in the non-signaling model, which subsumes the LOCAL, quantum-LOCAL, and bounded dependence models, we establish locality lower bounds for several graph problems in the non-signaling model.

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arXiv2023

Limits of CDCL Learning via Merge Resolution

In their seminal work, Atserias et al. and independently Pipatsrisawat and Darwiche in 2009 showed that CDCL solvers can simulate resolution proofs with polynomial overhead. However, previous work does not address the tightness of the simulation, i.e., the question of how large this overhead needs to be. In this paper, we address this question by focusing on an important property of proofs generated by CDCL solvers that employ standard learning schemes, namely that the derivation of a learned clause has at least one inference where a literal appears in both premises (aka, a merge literal). Specifically, we show that proofs of this kind can simulate resolution proofs with at most a linear overhead, but there also exist formulas where such overhead is necessary or, more precisely, that there exist formulas with resolution proofs of linear length that require quadratic CDCL proofs.

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arXiv2023

Stabbing Planes

We develop a new semi-algebraic proof system called Stabbing Planes which formalizes modern branch-and-cut algorithms for integer programming and is in the style of DPLL-based modern SAT solvers. As with DPLL there is only a single rule: the current polytope can be subdivided by branching on an inequality and its "integer negation." That is, we can (nondeterministically choose) a hyperplane $ax \geq b$ with integer coefficients, which partitions the polytope into three pieces: the points in the polytope satisfying $ax \geq b$, the points satisfying $ax \leq b-1$, and the middle slab $b - 1 < ax < b$. Since the middle slab contains no integer points it can be safely discarded, and the algorithm proceeds recursively on the other two branches. Each path terminates when the current polytope is empty, which is polynomial-time checkable. Among our results, we show that Stabbing Planes can efficiently simulate the Cutting Planes proof system, and is equivalent to a tree-like variant of the RCP system of [Krajicek98]. As well, we show that it possesses short proofs of the canonical family of systems of $\mathbb{F}_2$-linear equations known as the Tseitin formulas. Finally, we prove linear lower bounds on the rank of Stabbing Planes refutations by adapting lower bounds in communication complexity and use these bounds in order to show that Stabbing Planes proofs cannot be balanced. In doing so, we show that real communication protocols cannot be balanced and establish the first lower bound on the real communication complexity of the set disjointness function.

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arXiv2022

Low Degree Testing over the Reals

We study the problem of testing whether a function $f: \mathbb{R}^n \to \mathbb{R}$ is a polynomial of degree at most $d$ in the \emph{distribution-free} testing model. Here, the distance between functions is measured with respect to an unknown distribution $\mathcal{D}$ over $\mathbb{R}^n$ from which we can draw samples. In contrast to previous work, we do not assume that $\mathcal{D}$ has finite support. We design a tester that given query access to $f$, and sample access to $\mathcal{D}$, makes $(d/\varepsilon)^{O(1)}$ many queries to $f$, accepts with probability $1$ if $f$ is a polynomial of degree $d$, and rejects with probability at least $2/3$ if every degree-$d$ polynomial $P$ disagrees with $f$ on a set of mass at least $\varepsilon$ with respect to $\mathcal{D}$. Our result also holds under mild assumptions when we receive only a polynomial number of bits of precision for each query to $f$, or when $f$ can only be queried on rational points representable using a logarithmic number of bits. Along the way, we prove a new stability theorem for multivariate polynomials that may be of independent interest.

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arXiv2021

On the Hierarchical Community Structure of Practical Boolean Formulas

Modern CDCL SAT solvers easily solve industrial instances containing tens of millions of variables and clauses, despite the theoretical intractability of the SAT problem. This gap between practice and theory is a central problem in solver research. It is believed that SAT solvers exploit structure inherent in industrial instances, and hence there have been numerous attempts over the last 25 years at characterizing this structure via parameters. These can be classified as rigorous, i.e., they serve as a basis for complexity-theoretic upper bounds (e.g., backdoors), or correlative, i.e., they correlate well with solver run time and are observed in industrial instances (e.g., community structure). Unfortunately, no parameter proposed to date has been shown to be both strongly correlative and rigorous over a large fraction of industrial instances. Given the sheer difficulty of the problem, we aim for an intermediate goal of proposing a set of parameters that is strongly correlative and has good theoretical properties. Specifically, we propose parameters based on a graph partitioning called Hierarchical Community Structure (HCS), which captures the recursive community structure of a graph of a Boolean formula. We show that HCS parameters are strongly correlative with solver run time using an Empirical Hardness Model, and further build a classifier based on HCS parameters that distinguishes between easy industrial and hard random/crafted instances with very high accuracy. We further strengthen our hypotheses via scaling studies. On the theoretical side, we show that counterexamples which plagued community structure do not apply to HCS, and that there is a subset of HCS parameters such that restricting them limits the size of embeddable expanders.

cs.LO↗

arXiv2021

On the Power and Limitations of Branch and Cut

The Stabbing Planes proof system was introduced to model the reasoning carried out in practical mixed integer programming solvers. As a proof system, it is powerful enough to simulate Cutting Planes and to refute the Tseitin formulas -- certain unsatisfiable systems of linear equations mod 2 -- which are canonical hard examples for many algebraic proof systems. In a recent (and surprising) result, Dadush and Tiwari showed that these short refutations of the Tseitin formulas could be translated into quasi-polynomial size and depth Cutting Planes proofs, refuting a long-standing conjecture. This translation raises several interesting questions. First, whether all Stabbing Planes proofs can be efficiently simulated by Cutting Planes. This would allow for the substantial analysis done on the Cutting Planes system to be lifted to practical mixed integer programming solvers. Second, whether the quasi-polynomial depth of these proofs is inherent to Cutting Planes. In this paper we make progress towards answering both of these questions. First, we show that any Stabbing Planes proof with bounded coefficients SP* can be translated into Cutting Planes. As a consequence of the known lower bounds for Cutting Planes, this establishes the first exponential lower bounds on SP*. Using this translation, we extend the result of Dadush and Tiwari to show that Cutting Planes has short refutations of any unsatisfiable system of linear equations over a finite field. Like the Cutting Planes proofs of Dadush and Tiwari, our refutations also incur a quasi-polynomial blow-up in depth, and we conjecture that this is inherent. As a step towards this conjecture, we develop a new geometric technique for proving lower bounds on the depth of Cutting Planes proofs. This allows us to establish the first lower bounds on the depth of Semantic Cutting Planes proofs of the Tseitin formulas.

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arXiv2020

Towards a Complexity-theoretic Understanding of Restarts in SAT solvers

Restarts are a widely-used class of techniques integral to the efficiency of Conflict-Driven Clause Learning (CDCL) Boolean SAT solvers. While the utility of such policies has been well-established empirically, a theoretical explanation of whether restarts are indeed crucial to the power of CDCL solvers is lacking. In this paper, we prove a series of theoretical results that characterize the power of restarts for various models of SAT solvers. More precisely, we make the following contributions. First, we prove an exponential separation between a {\it drunk} randomized CDCL solver model with restarts and the same model without restarts using a family of satisfiable instances. Second, we show that the configuration of CDCL solver with VSIDS branching and restarts (with activities erased after restarts) is exponentially more powerful than the same configuration without restarts for a family of unsatisfiable instances. To the best of our knowledge, these are the first separation results involving restarts in the context of SAT solvers. Third, we show that restarts do not add any proof complexity-theoretic power vis-a-vis a number of models of CDCL and DPLL solvers with non-deterministic static variable and value selection.

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arXiv2019

Distribution-Free Testing of Linear Functions on R^n

We study the problem of testing whether a function f:R^n->R is linear (i.e., both additive and homogeneous) in the distribution-free property testing model, where the distance between functions is measured with respect to an unknown probability distribution over R. We show that, given query access to f, sampling access to the unknown distribution as well as the standard Gaussian, and eps>0, we can distinguish additive functions from functions that are eps-far from additive functions with O((1/eps)log(1/eps)) queries, independent of n. Furthermore, under the assumption that f is a continuous function, the additivity tester can be extended to a distribution-free tester for linearity using the same number of queries. On the other hand, we show that if we are only allowed to get values of f on sampled points, then any distribution-free tester requires Omega(n) samples, even if the underlying distribution is the standard Gaussian.

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arXiv2017

Random CNFs are Hard for Cutting Planes

The random k-SAT model is the most important and well-studied distribution over k-SAT instances. It is closely connected to statistical physics; it is used as a testbench for satisfiability algorithms, and average-case hardness over this distribution has also been linked to hardness of approximation via Feige's hypothesis. We prove that any Cutting Planes refutation for random k-SAT requires exponential size, for k that is logarithmic in the number of variables, in the (interesting) regime where the number of clauses guarantees that the formula is unsatisfiable with high probability.

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