[Paper Review] A Circuit-Based Approach to Efficient Enumeration

Antoine Amarilli, Pierre Bourhis+2 more|arXiv (Cornell University)|Jan 1, 2017

Quantum Computing Algorithms and Architecture22 citations

TL;DR

This paper presents a generic, circuit-based framework for efficient enumeration of satisfying valuations in Boolean circuits, focusing on structured d-DNNFs. It achieves linear preprocessing and delay linear in the Hamming weight of each valuation, with constant delay for bounded-Hamming-weight cases, using zero-suppressed semantics and a multitree reachability index for efficient trace enumeration.

ABSTRACT

We study the problem of enumerating the satisfying valuations of a circuit while bounding the delay, i.e., the time needed to compute each successive valuation. We focus on the class of structured d-DNNF circuits originally introduced in knowledge compilation, a sub-area of artificial intelligence. We propose an algorithm for these circuits that enumerates valuations with linear preprocessing and delay linear in the Hamming weight of each valuation. Moreover, valuations of constant Hamming weight can be enumerated with linear preprocessing and constant delay. Our results yield a framework for efficient enumeration that applies to all problems whose solutions can be compiled to structured d-DNNFs. In particular, we use it to recapture classical results in database theory, for factorized database representations and for MSO evaluation. This gives an independent proof of constant-delay enumeration for MSO formulae with first-order free variables on bounded-treewidth structures.

Motivation & Objective

• To develop a generic, reusable framework for constant-delay enumeration across diverse computational problems using knowledge compilation principles.
• To address the limitation of ad hoc, problem-specific indexing and enumeration algorithms in database and logic applications.
• To enable efficient enumeration of solutions for problems whose solutions can be compiled into structured d-DNNF circuits.
• To provide a modular, reusable pipeline combining circuit rewriting, normalization, and index construction for efficient enumeration.
• To demonstrate the framework's applicability to MSO model checking and factorized database representations with independent proofs of known results.

Proposed method

• Transform input circuits into zero-suppressed semantics, where untested variables are implicitly false, using a structured v-tree and range gates for efficiency.
• Rewrite circuits into a normal form (normal d-DNNF) to ensure structural consistency and facilitate indexing.
• Construct a multitree reachability index on the normalized circuit to efficiently enumerate valid traces.
• Use the index to generate satisfying valuations as sets of variables set to true, with minimal overhead.
• Leverage the structured d-DNNF property and v-tree constraints to ensure correctness and efficiency of the preprocessing and enumeration phases.
• Apply the framework to known problems by compiling their solution sets into structured d-DNNFs and applying the enumeration algorithm.

Experimental results

Research questions

• RQ1 Can a generic, reusable framework for constant-delay enumeration be built using knowledge compilation, specifically d-DNNFs?
• RQ2 Can linear preprocessing and delay linear in the Hamming weight of each valuation be achieved for structured d-DNNFs?
• RQ3 Can constant delay be achieved for bounded-Hamming-weight solutions using a circuit-based approach?
• RQ4 How can the framework be applied to recapture or extend known results in database theory and logic?
• RQ5 Can the framework provide independent, modular proofs for constant-delay enumeration in MSO model checking on bounded-treewidth structures?

Key findings

• The proposed algorithm achieves linear preprocessing time and delay linear in the Hamming weight of each satisfying valuation for structured d-DNNFs.
• For solutions of constant Hamming weight, the algorithm achieves constant delay after linear preprocessing.
• The framework enables efficient enumeration for all problems whose solutions can be compiled into structured d-DNNFs, including MSO queries on bounded-treewidth structures.
• The approach provides an independent proof of constant-delay enumeration for MSO formulae with first-order free variables on bounded-treewidth structures.
• The framework generalizes and extends prior results on deterministic factorized database representations, enabling linear preprocessing and constant delay enumeration.
• The construction preserves upwards-determinism through all transformations, ensuring correctness and efficiency of the enumeration pipeline.