Applying Practice to Theory

How can complexity theory and algorithms benefit from practical advances in computing? We give a short overview of some prior work using practical computing to attack problems in computational complexity and algorithms, informally describe how linear program solvers may be used to help prove new lower bounds for satisfiability, and suggest a research program for developing new understanding in circuit complexity.

💡 Research Summary

The paper “Applying Practice to Theory” by Ryan Williams argues that the ever‑increasing computational power available today should be deliberately harnessed to advance fundamental questions in theoretical computer science. After a brief motivational introduction that cites distributed‑computing successes such as Folding@Home, ClimatePrediction.net, and SETI@Home, the author proposes allocating “spare cycles” to the systematic study of complexity theory, algorithm design, and circuit lower bounds.

The first substantive section surveys several concrete areas where practical computing has already made a difference. In the realm of moderately exponential algorithms, researchers have used computers to analyse intricate recurrence relations that arise from branching‑type backtracking procedures. Traditional hand‑derived analyses often rely on a single progress measure (e.g., the number of vertices) and quickly become intractable when multiple measures (vertices, edges, degree distributions) interact. Eppstein’s quasi‑convex programming technique converts multivariate recurrences into a linear‑programming problem, automatically selecting optimal weight parameters (αi) for each measure. This method has yielded improved exponential constants for problems such as Dominating Set (≈1.52ⁿ), Maximum Independent Set (≈1.23ⁿ), and Minimum Vertex Cover, surpassing earlier hand‑crafted analyses.

A second line of work replaces exhaustive case‑by‑case reasoning with automated enumeration. Robson’s unpublished program, later extended by Fomin‑Kulikov and others, systematically checks every possible local configuration of a backtracking algorithm up to a bounded size, thereby certifying that the algorithm respects a desired time bound (e.g., O(2ⁿ/⁴) for a Max‑Independent‑Set routine). The automation also discovers new simplification rules (degree‑0, degree‑1 reductions, etc.) that human designers might overlook, leading to faster algorithms for SAT and MAX‑SAT.

The paper then turns to approximation and in‑approximation results, focusing on gadget constructions. By encoding a 3‑SAT clause into a set of 2‑SAT clauses with an auxiliary variable, one obtains a reduction that preserves a precise satisfaction ratio (7 out of 10 clauses versus 6 out of 10). This gadget yields a quantitative transfer: any (1‑ε)‑approximation for MAX‑2‑SAT would imply a (1‑7ε)‑approximation for MAX‑3‑SAT, which is known to be impossible for ε>0 unless P=NP. The author highlights the formal gadget framework of Trevisan‑Sorkin‑Sudan‑Williamson, which treats gadget design as a family of linear programs over auxiliary variables and weights. By fixing the number of auxiliary variables, the space of feasible gadgets can be explored algorithmically, suggesting a path toward automatically generating stronger in‑approximation reductions.

The most speculative contribution is a proposal to use linear‑programming solvers to prove new lower bounds for SAT. The idea is to model a SAT instance as a weighted collection of constraints and let an LP solver search for a weight assignment that demonstrates infeasibility for any circuit of a given size. In effect, the LP would certify that no small circuit can satisfy the instance, providing a numeric lower bound on circuit size or depth. This approach differs from traditional combinatorial or proof‑complexity arguments and could be especially powerful for small, concrete circuits (e.g., a 7×7 matrix‑multiplication circuit) where exhaustive search is still feasible but human insight is limited.

Finally, the author sketches a long‑term research program aimed at circuit complexity. The vision is a pipeline that (1) harnesses large‑scale distributed computing to generate and test thousands of small circuits, (2) integrates automated LP and SAT solvers to evaluate each circuit’s capability, and (3) iteratively refines the search space based on discovered lower bounds. Such a system would automate much of the “gadget‑design” and “simplification‑rule” discovery that currently relies on expert intuition, potentially creating a self‑reinforcing loop where theoretical advances yield more computational resources, which in turn enable deeper theoretical insights.

Overall, the paper makes a compelling case that practical computation is no longer merely a tool for verifying proofs but can be an active engine for generating new theorems, tighter exponential bounds, and stronger hardness results. By systematically integrating automated recurrence analysis, case‑enumeration, gadget‑generation via linear programming, and large‑scale circuit testing, the community can move toward a more experimental, data‑driven style of complexity research.