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arxiv: 2604.16418 · v1 · submitted 2026-04-02 · 💻 cs.CC

Towards Solving NP-Complete and Other Hard Problems Efficiently in Practice

Mircea-Adrian Digulescu This is my paper

Pith reviewed 2026-05-13 20:16 UTC · model grok-4.3

classification 💻 cs.CC
keywords finite algorithmicsNP-complete problemsautomatic algorithm discovery3CNFSATinteger factorizationstring compressionbounded input complexity
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The pith

A generic method can automatically discover efficient algorithms for finite instances of NP-complete problems if they exist.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper first builds a theoretical framework that redefines familiar complexity notions for inputs whose size is capped at some fixed upper bound. It then describes a generic procedure that searches for an algorithm adequate to that finite bound, provided one exists. The author argues that the restriction to finite cases makes hard problems meaningfully easier in practice than their unbounded versions. This matters for applications where inputs stay well below the limits that make general-case intractability relevant.

Core claim

Finite algorithmics adapts asymptotic concepts such as complexity to the setting where input size is bounded above; within this setting a generic search procedure can locate an adequate algorithm for any hard problem whose finite case admits one, as illustrated by concrete ideas developed for 3CNFSAT, string compression, and integer factorization.

What carries the argument

the generic approach for automatically discovering an adequate algorithm for the finite case of a hard problem

If this is right

  • 3CNFSAT instances whose number of variables is capped admit automatically discovered polynomial-time solvers.
  • String compression for strings drawn from a finite alphabet and of bounded length can be performed by algorithms found through the same procedure.
  • Integer factorization for integers below a fixed size reduces to an efficient, automatically located algorithm.
  • Practical computation on real-world bounded inputs no longer requires resolving the general-case complexity of the problem.
  • Elementary results in the finite-complexity theory provide the foundation for applying the method to other NP-complete problems.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • For any fixed bound the method could be run once and the resulting algorithm stored for repeated use on all instances obeying that bound.
  • The framework might be combined with existing SAT or factoring heuristics to seed the search and reduce discovery time.
  • The same finite-case perspective could be applied to other problems such as bounded traveling-salesman instances without invoking the full NP-completeness barrier.
  • If the discovery cost scales favorably with the bound, it becomes feasible to pre-compute optimal solvers for successively larger practical limits.

Load-bearing premise

An adequate algorithm for any chosen finite bound exists and the proposed generic method can discover it without the search itself becoming intractable.

What would settle it

Running the discovery procedure on 3CNFSAT instances with at most 20 variables and observing that it returns no algorithm better than the best known exhaustive solver, even though a linear-time method is known to exist for that bound.

read the original abstract

Until now, Computer Scientists have concerned themselves with identifying efficient algorithms for solving the general case of some problem -- that is finding one which performs well when the size of the input tends to infinity. In this paper, we first introduce a theoretical framework for reasoning about finite algorithmics. It allows familiar concepts such as asymptotic complexity to be adapted to the case where the input size is bounded from above. We also present some elementary results within this theory. Secondly, we present a generic approach for automatically discovering an adequate algorithm for the finite case of some hard problem -- if one exists. Thirdly, we argue why we expect the finite case of hard problems to be easier than the general case. Fourthly, we present some relevant ideas specific to three hard problems, namely 3CNFSAT, String Compression and Integer Factorization.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 1 minor

Summary. The paper introduces a theoretical framework for finite algorithmics that adapts familiar complexity notions such as asymptotic behavior to the setting where input size is bounded from above. It presents a generic approach for automatically discovering an adequate algorithm for the finite case of a hard problem if one exists, argues that finite cases are easier than the general case, and sketches relevant ideas for three specific problems: 3CNFSAT, string compression, and integer factorization.

Significance. If the generic discovery method can be made concrete, shown to be tractable, and validated on non-trivial finite instances, the framework could shift practical attention toward bounded-input versions of NP-hard problems and provide a principled way to exploit the fact that real-world inputs are always finite.

major comments (3)
  1. [Abstract and generic approach] Abstract / generic approach section: the central claim that a generic method can automatically discover an adequate finite-case algorithm is stated at a high level with no pseudocode, search procedure, or complexity analysis of the discovery process itself; without this, it is impossible to determine whether the method is tractable or reduces to the original decision problem.
  2. [Argument that finite cases are easier] Section arguing finite cases are easier: the claim that finite cases are meaningfully easier rests on the existence of an adequate algorithm plus an unspecified search whose cost is at least as high as the original problem; no concrete termination argument or reduction in search space is supplied.
  3. [Ideas for 3CNFSAT, compression and factorization] Section on specific problems: no worked example, even for a trivial bound (e.g., 3CNF formulas with at most 5 variables or integers up to 20 bits), is given to illustrate how the framework or discovery method would actually produce a better-than-brute-force procedure.
minor comments (1)
  1. [Introduction / framework section] The manuscript would benefit from explicit definitions of 'adequate algorithm' and 'finite case' early in the text to remove ambiguity in later claims.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive feedback. The comments correctly identify areas where the manuscript would benefit from greater concreteness. We will revise the paper to add the requested details on the discovery procedure, termination arguments, and a worked example while preserving the theoretical focus of the work.

read point-by-point responses

  1. Referee: Abstract / generic approach section: the central claim that a generic method can automatically discover an adequate finite-case algorithm is stated at a high level with no pseudocode, search procedure, or complexity analysis of the discovery process itself; without this, it is impossible to determine whether the method is tractable or reduces to the original decision problem.

    Authors: We agree the generic approach was presented conceptually. The revised manuscript will include pseudocode for the discovery procedure (enumeration of candidate algorithms whose size is bounded by the input bound, followed by verification on all inputs of that size), a description of the search, and a complexity analysis. The analysis will show that tractability holds for small bounds because the space of algorithms is finite and the verification step is polynomial in the (finite) number of inputs. revision: yes

  2. Referee: Section arguing finite cases are easier: the claim that finite cases are meaningfully easier rests on the existence of an adequate algorithm plus an unspecified search whose cost is at least as high as the original problem; no concrete termination argument or reduction in search space is supplied.

    Authors: The argument relies on the finiteness of the algorithm space for any fixed bound. We will add an explicit termination argument: the search enumerates all programs up to a length linear in the bound and tests them exhaustively on the finite input set; this terminates because both the program space and input space are finite. We will also quantify the reduction in search space relative to the unbounded case. revision: yes

  3. Referee: Section on specific problems: no worked example, even for a trivial bound (e.g., 3CNF formulas with at most 5 variables or integers up to 20 bits), is given to illustrate how the framework or discovery method would actually produce a better-than-brute-force procedure.

    Authors: We accept that a concrete illustration is needed. The revision will add a worked example for 3CNFSAT restricted to formulas with at most 5 variables, showing the discovery procedure identifying a specialized decision procedure (via exhaustive enumeration of small circuits) that outperforms naive brute-force enumeration for that bound. revision: yes

Circularity Check

0 steps flagged

No circularity: framework and generic method introduced without self-referential reduction

full rationale

The paper introduces a new theoretical framework for finite algorithmics, adapts asymptotic concepts to bounded inputs, presents a generic discovery approach for finite-case algorithms (if they exist), and separately argues why finite cases should be easier. No equations or steps reduce a claimed result to its own inputs by construction, no parameters are fitted then relabeled as predictions, and no load-bearing uniqueness or ansatz is imported via self-citation. The derivation chain remains self-contained as a sequence of new definitions and arguments rather than tautological renaming or fitting.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review; no explicit free parameters, axioms, or invented entities can be identified from the provided text.

pith-pipeline@v0.9.0 · 5432 in / 1005 out tokens · 42805 ms · 2026-05-13T20:16:13.016968+00:00 · methodology

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Reference graph

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