Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2003.02323

Towards a Complexity-theoretic Understanding of Restarts in SAT solvers

Chunxiao Li·Noah Fleming·Marc Vinyals·Toniann Pitassi·Vijay Ganesh
View original source ↗Read source document ↗Export metadata ↓

Abstract

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.

Explore related subjects

cs.CC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Chunxiao Li, Noah Fleming, Marc Vinyals, Toniann Pitassi, Vijay Ganesh. 2020-05-11. Towards a Complexity-theoretic Understanding of Restarts in SAT solvers. https://arxiv.org/abs/2003.02323

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

CVP Is NP-Complete for Principal Cyclotomic Ideals

We prove that exact Euclidean decision-CVP is $\mathsf{NP}$-complete on the coefficient lattices of nonzero principal ideals in the power-of-two cyclotomic rings $R_d:=\mathbb{Z}[y]/(y^d+1)$. Our deterministic reduction from Exact Cover by 3-Sets (X3C) produces a target and a squared threshold $Δ$ such that the closest squared distance is exactly $Δ$ in YES instances and at least $Δ+4$ in NO instances. This also implies $\mathsf{NP}$-hardness of exact search-CVP under polynomial-time Turing reductions. We also transfer the resulting principal-ideal CVP instances to full-rank principal ideals of the cyclic quotient ring $\mathbb{Z}[X]/(X^D-1)$, where $D:=2d$. Their coefficient lattices are invariant under cyclic coordinate shifts. The lift preserves principality and multiplies corresponding squared distances by eight. Thus, on principal cyclic ideal lattices, exact decision-CVP is $\mathsf{NP}$-complete and exact search-CVP is $\mathsf{NP}$-hard. We also obtain uniformly computable fixed cyclotomic and cyclic families in which only the target and threshold depend on the X3C collection. Consequently, a polynomial-time solution to exact decision-CVPP on either family would imply $\mathsf{NP}\subseteq\mathsf{P}/\mathrm{poly}$ and collapse the polynomial hierarchy to $Σ_2^{\mathsf{P}}$. To our knowledge, the cyclic results answer Micciancio's questions of whether exact decision-CVP is $\mathsf{NP}$-hard on cyclic lattices and on a fixed family of cyclic lattices, even when restricted to full-rank principal cyclic ideals. Finally, under the coefficient embedding, we prove that exact decision-module-SIVP is $\mathsf{NP}$-complete on free rank-two modules over the same cyclotomic rings.

cs.CC↗
arXiv2026

Fooling Thresholds of Halfspaces

We initiate the study of constructing explicit pseudorandom generators for thresholds of halfspaces with seed length polylogarithmic in the number of halfspaces. This class of functions lies at the frontier of circuit complexity [CTW26]. We show that the generator designed by O'Donnell, Servedio, and Tan for polytopes [OST22] also fools this broader class. To analyze the generator, we develop a threshold-specific smooth approximation framework based on a Bentkus-type mollifier. We prove derivative bounds for this mollifier and also establish a Boolean anticoncentration theorem for thresholds of halfspaces via a random thinning argument. These ingredients imply that the generator $δ$-fools every $k$-out-of-$m$ threshold of $m$ halfspaces over $\{-1,1\}^n$ with seed length $\widetilde{O}(κ^{6+2\varepsilon}\log^{6+2\varepsilon}\!m\cdotδ^{-(2+2\varepsilon)}\log n)$, for any arbitrarily small constant $\varepsilon>0$, where $κ=\min\{k,m-k+1\}$. The random thinning argument also yields bounds on the noise sensitivity and Gaussian surface area for thresholds of halfspaces, leading to learning algorithms under both the uniform and Gaussian distributions.

cs.CC↗
arXiv2026

FPT=PTIME for Homomorphism Problems on Sparse-Incidence and Bounded-Independence Patterns

Assuming the Exponential Time Hypothesis (ETH), fixed-parameter tractability and polynomial-time solvability coincide for homomorphism problems specified by classes of pattern hypergraphs of bounded incidence degeneracy or bounded primal independence number. In both cases, tractability is characterised by bounded fractional hypertree width. Grohe (JACM 2007) established the corresponding FPT-PTIME equivalence under bounded arity. Our result allows unbounded arity and covers important cases such as bounded-degree patterns and patterns whose incidence graphs exclude a fixed minor. Building on the recent fractional balanced-separator framework and rounding theorem of Korchemna et al. (FOCS 2024), we prove a near-linear bound on fractional hypertree width ($\mathsf{fhw}$) in terms of adaptive width ($\mathsf{adw}$). For every hypergraph $H$ with $\mathsf{adw}(H)\geq 2$, \[ \mathsf{fhw}(H)=O\bigl(λ(H)\mathsf{adw}(H)\log\mathsf{adw}(H)\bigr), \] where $λ(H)=\min\{μ(H),\max\{1,\logα(H)\}\}$, with $μ(H)$ denoting incidence degeneracy and $α(H)$ the independence number of the primal graph. As a further consequence, we obtain a corresponding FPT-PTIME collapse for exact homomorphism counting on every bounded-$λ$ class. More generally, for every recursively enumerable class of pattern hypergraphs, fixed-parameter tractability of the parameterised homomorphism problem implies quasipolynomial-time solvability of the corresponding unparameterised problem, assuming ETH.

cs.CC↗