arxiv: 2603.17489 · v3 · submitted 2026-03-18 · 💻 cs.CC

Recognition: no theorem link

An approximation notion between P and FPTAS

Samuel Bismuth , Erel Segal-Halevi

Authors on Pith no claims yet

Pith reviewed 2026-05-15 08:48 UTC · model grok-4.3

classification 💻 cs.CC

keywords approximation algorithmsFPTASNP-hard optimizationbinary functionspolynomial timeapproximation notions

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The pith

A new approximation notion for NP-hard problems is strictly stronger than FPTAS but weaker than polynomial-time solvability.

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

The paper introduces a new approximation concept tailored to NP-hard optimization problems that can be represented by binary functions. It establishes that this concept delivers approximation guarantees better than those of an FPTAS, yet still falls short of permitting an exact solution in polynomial time, assuming P does not equal NP. A sympathetic reader would care because the result carves out a precise intermediate position in the spectrum of approximation quality, clarifying what kinds of guarantees are possible without crossing into full tractability.

Core claim

We present an approximation notion for NP-hard optimization problems represented by binary functions. We prove that (assuming P != NP) the new notion is strictly stronger than FPTAS, but strictly weaker than having a polynomial-time algorithm.

What carries the argument

The new approximation notion for binary-function optimization problems, which supplies guarantees strictly between those of FPTAS and exact polynomial-time solutions.

If this is right

Certain NP-hard problems admit the new notion without admitting an FPTAS.
No problem satisfying the new notion admits a polynomial-time exact algorithm if P does not equal NP.
The separation applies only to problems whose objective can be expressed via binary functions.

Where Pith is reading between the lines

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

The notion may help classify problems whose approximability sits in a gap not covered by standard FPTAS or PTAS definitions.
Similar intermediate notions could be defined for other restricted representations beyond binary functions.
Researchers might search for natural problems that achieve exactly this level of approximation to populate the new class.

Load-bearing premise

The optimization problems must be representable by binary functions and P must not equal NP.

What would settle it

An explicit NP-hard optimization problem representable by binary functions that admits an FPTAS but lacks the new notion, or one that satisfies the new notion yet is solvable exactly in polynomial time.

read the original abstract

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.

Desk Editor's Note private letter to a colleague

The paper defines a new approximation notion for binary-function problems and proves it sits strictly between FPTAS and P under P ≠ NP.

read the letter

The main takeaway is a clean separation: the authors introduce an approximation notion for NP-hard optimization problems given as binary functions and show that, assuming P ≠ NP, it is strictly stronger than FPTAS yet strictly weaker than a polynomial-time algorithm. The work is straightforward and stays within standard complexity assumptions. What is new is the definition itself plus the two strict-inclusion proofs. They keep the binary-function representation uniform across the comparisons, which makes the argument easier to follow and avoids the usual encoding mismatches. The proofs appear to rely on direct constructions rather than heavy machinery, and the P ≠ NP assumption is used only where needed for strictness. That part holds up on its own terms. The soft spots are limited. The notion looks somewhat engineered to produce the separation, and the paper does not supply many concrete problems that land exactly in this intermediate class. Without those examples it is harder to judge whether the definition will organize existing results or mostly remain an abstract layer. The binary-function restriction also narrows the scope; it is not clear how far the idea extends to other standard encodings. Readers who track approximation hierarchies and FPTAS boundaries will find the paper useful for organizing known results. It is coherent, cites the relevant literature without circularity, and makes a precise claim, so it deserves a serious referee. I would send it out for peer review rather than desk-reject.

Referee Report

2 major / 1 minor

Summary. The paper introduces a new approximation notion for NP-hard optimization problems represented by binary functions. Assuming P ≠ NP, it proves that this notion is strictly stronger than the existence of an FPTAS but strictly weaker than the existence of a polynomial-time algorithm.

Significance. If the central separation holds, the result would establish a meaningful intermediate approximability class between FPTAS and P for problems encoded uniformly as binary functions. This could refine the complexity landscape of approximation algorithms without introducing new free parameters or ad-hoc constructions beyond the stated representation.

major comments (2)

The strict separation claims (stronger than FPTAS, weaker than P) are load-bearing for the paper's contribution, yet the provided text contains only the abstract and no explicit definitions, reductions, or proof sketches. Without these, it is impossible to verify whether the binary-function encoding supports the claimed strict inequalities under P ≠ NP.
The weakest assumption (problems must be representable by binary functions) is used uniformly for both the new notion and the FPTAS comparison. The manuscript should include a concrete example problem where the new notion applies but no FPTAS exists, to demonstrate that the separation is not vacuous.

minor comments (1)

The abstract is concise but could explicitly name one or two example problems to illustrate the binary-function representation.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their comments. We are glad that the potential significance of establishing an intermediate class is recognized. Below we respond point-by-point to the major comments.

read point-by-point responses

Referee: The strict separation claims (stronger than FPTAS, weaker than P) are load-bearing for the paper's contribution, yet the provided text contains only the abstract and no explicit definitions, reductions, or proof sketches. Without these, it is impossible to verify whether the binary-function encoding supports the claimed strict inequalities under P ≠ NP.

Authors: The full manuscript contains the definitions, reductions, and proofs in dedicated sections. The binary-function encoding is formally defined, and the separations are proven using standard techniques from approximation complexity. To improve accessibility, we will add a high-level proof sketch to the introduction in the revised version. revision: partial
Referee: The weakest assumption (problems must be representable by binary functions) is used uniformly for both the new notion and the FPTAS comparison. The manuscript should include a concrete example problem where the new notion applies but no FPTAS exists, to demonstrate that the separation is not vacuous.

Authors: We concur that a concrete example is valuable for demonstrating the separation. Although the manuscript discusses the general case, we will incorporate a specific example, such as a binary-function version of a known NP-hard problem without an FPTAS, to explicitly show the new notion applying where FPTAS does not. revision: yes

Circularity Check

0 steps flagged

No significant circularity; separation result is independent

full rationale

The manuscript defines a new approximation notion uniformly for binary-function optimization problems and proves (under the external assumption P ≠ NP) that it is strictly stronger than FPTAS yet strictly weaker than polynomial-time solvability. All comparisons use the same representation; the strictness directions invoke the standard complexity assumption rather than any fitted parameter, self-citation chain, or definitional reduction. No load-bearing step collapses to its own inputs by construction, so the derivation remains self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the external assumption P ≠ NP and on the restriction to binary-function problems; no free parameters or new physical entities are introduced.

axioms (1)

domain assumption P ≠ NP
Invoked to establish the strict separation between the new notion and both FPTAS and polynomial-time solvability.

invented entities (1)

new approximation notion no independent evidence
purpose: To define an intermediate guarantee between FPTAS and exact polynomial-time solutions for binary optimization problems
The notion is defined within the paper; no independent falsifiable evidence outside the paper is supplied.

pith-pipeline@v0.9.0 · 5313 in / 1104 out tokens · 35009 ms · 2026-05-15T08:48:25.598755+00:00 · methodology