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arxiv: 2604.18661 · v1 · submitted 2026-04-20 · 💻 cs.DS · cs.CC

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Coordinatewise Balanced Covering for Linear Gain Graphs, with an Application to Coset-List Min-2-Lin over Powers of Two

Faruk Alpay , Levent Sarioglu
Authors on Pith no claims yet

Pith reviewed 2026-05-10 03:38 UTC · model grok-4.3

classification 💻 cs.DS cs.CC
keywords linear gain graphsbalanced coveringcoset-list min-2-lindyadic cosetscycle spacerandomized algorithmsrank compression
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The pith

A coordinatewise balanced covering theorem for linear gain graphs labeled by vectors in F_2^r isolates all low-cost balanced subgraphs with probability 2^{-O(k^2 r)} via tensoring and rank compression.

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

The paper establishes a randomized covering procedure for linear gain graphs with vector labels from F_2^r. Given one balanced subgraph of cost at most k, the procedure produces a vertex set S and deletion set F such that the induced subgraph on S minus F remains balanced, and with the stated probability it contains every other balanced subgraph of cost at most k while recording all relevant edge deletions. This covering is obtained by tensoring a one-coordinate theorem across the r coordinates and compressing the ambient dimension down to the intrinsic cycle-label rank rho. The result is then applied to obtain a randomized algorithm for Coset-List Min-2-Lin over powers of two that runs in time 2^{O(k^2 rho + k log(k rho + 2))} n^{O(1)} plus near-linear terms in m, d, and rho^omega, and among all deletion sets of size at most k it returns one of minimum weight.

Core claim

The central claim is that any linear gain graph labeled by vectors in F_2^r admits a coordinatewise balanced covering: given a balanced subgraph of cost at most k, a randomized procedure outputs S and F such that (G-F)[S] is balanced and, with probability 2^{-O(k^2 r)}, every hidden balanced subgraph of cost at most k is contained in S while all its incident deletions are captured by F. The proof proceeds by tensoring the one-coordinate balanced-covering theorem across coordinates, followed by a rank-compression step that replaces the lifted dimension by the cycle-label rank rho. The same framework yields an explicit bit-lifting analysis and a cycle-space formulation of balancedness for dyad

What carries the argument

The tensor product of one-coordinate balanced coverings across the r coordinates of the F_2^r labels, reduced by cycle-label rank compression to intrinsic dimension rho, which produces the vertex set S and deletion set F that simultaneously cover all low-cost balanced subgraphs.

If this is right

  • The covering theorem directly supplies a one-sided-error randomized algorithm that solves Coset-List Min-2-Lin^pm over Z/2^d Z and returns a minimum-weight deletion set of size at most k.
  • The same construction produces a cycle-space formulation and a cut-space/potential characterization of balancedness for the labeled graphs.
  • Rank compression reduces the exponential dependence from the lifted dimension to the intrinsic cycle-label rank rho.
  • An explicit bit-lifting analysis is obtained for systems restricted to dyadic cosets a + 2^ell (Z/2^d Z).

Where Pith is reading between the lines

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

  • The tensoring technique may extend to gain graphs labeled by vectors over other finite fields or to higher-modulus constraints that admit a similar coordinate decomposition.
  • Because the covering probability depends only on k and r (or rho), the method could be combined with kernelization or other preprocessing to handle larger instances where rho remains small.
  • The minimal-dimension statement for equivalent labelings suggests that the framework might apply to other deletion problems whose feasibility is characterized by linear dependencies in a vector space.

Load-bearing premise

The input must be a linear gain graph whose edges are labeled by vectors in F_2^r and at least one balanced subgraph of cost at most k must exist.

What would settle it

A concrete counterexample would be a linear gain graph over F_2^r together with a balanced subgraph of cost k=1 such that every run of the randomized procedure either fails to produce a balanced induced subgraph on S or misses at least one other balanced subgraph of cost 1 with probability substantially larger than 2^{-O(r)}.

read the original abstract

We study a list-constrained extension of modular equation deletion over powers of two, called Coset-List Min-2-Lin$^{\pm}$ over $\mathbb{Z}/2^d\mathbb{Z}$. Each variable is restricted to a dyadic coset $a+2^{\ell}(\mathbb{Z}/2^d\mathbb{Z})$, each binary constraint is of the form $x_u=x_v$, $x_u=-x_v$, or $x_u=2x_v$, and the goal is to delete a minimum number of constraints so that the remaining system is satisfiable. This problem lies between the no-list case and the poorly understood fully conservative list setting. Our main technical result is a coordinatewise balanced covering theorem for linear gain graphs labeled by vectors in $\mathbb{F}_2^r$. Given any balanced subgraph of cost at most $k$, a randomized procedure outputs a vertex set $S$ and an edge set $F$ such that $(G-F)[S]$ is balanced and, with probability $2^{-O(k^2r)}$, every hidden balanced subgraph of cost at most $k$ is contained in $S$ while all incident deletions are captured by $F$. The proof tensors the one-coordinate balanced-covering theorem of Dabrowski, Jonsson, Ordyniak, Osipov, and Wahlstr\"om across coordinates, and is combined with a rank-compression theorem replacing the ambient lifted dimension by the intrinsic cycle-label rank $\rho$. We also develop a cycle-space formulation, a cut-space/potential characterization of balancedness, a minimal-dimension statement for equivalent labelings, and an explicit bit-lifting analysis for dyadic coset systems. These yield a randomized one-sided-error algorithm running in \[ 2^{O(k^2\rho+k\log(k\rho+2))}\cdot n^{O(1)}+\widetilde{O}(md+\rho^\omega), \] and the same framework returns a minimum-weight feasible deletion set among all solutions of size at most $k$.

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

0 major / 3 minor

Summary. The paper claims a coordinatewise balanced covering theorem for linear gain graphs labeled by vectors in F_2^r. Given any balanced subgraph of cost at most k, a randomized procedure outputs a vertex set S and an edge set F such that (G-F)[S] is balanced and, with probability 2^{-O(k^2 r)}, every hidden balanced subgraph of cost at most k is contained in S while all incident deletions are captured by F. The proof tensors the one-coordinate balanced-covering theorem across the r coordinates and combines it with a rank-compression theorem replacing ambient dimension by the intrinsic cycle-label rank ρ. These tools, together with cycle-space and cut-space characterizations plus bit-lifting analysis, yield a randomized one-sided-error algorithm for Coset-List Min-2-Lin^± over Z/2^d Z with runtime 2^{O(k^2 ρ + k log(kρ+2))} · n^{O(1)} + ~O(md + ρ^ω) that also returns a minimum-weight feasible deletion set among all solutions of size at most k.

Significance. If the central theorem holds, the result is significant because it supplies a parameterized covering tool that bridges the well-understood no-list case and the poorly-understood fully conservative list setting for modular-equation deletion. The tensoring construction together with rank compression to intrinsic cycle-label rank ρ is technically clean, and the explicit runtime dependence on ρ rather than ambient dimension is a concrete algorithmic improvement. The supporting cycle-space formulation, potential characterization, and minimal-dimension statement for equivalent labelings strengthen the foundational toolkit for linear gain graphs.

minor comments (3)
  1. [Abstract and §4] Abstract and §4: the success probability is stated as 2^{-O(k^2 r)} while the final runtime uses ρ; the text should explicitly state whether rank compression is applied before or after the covering procedure and how the probability bound transforms (or remains unchanged) under the compression.
  2. The abstract promises an 'explicit bit-lifting analysis for dyadic coset systems'; a dedicated lemma or subsection in the application section should isolate the lifting argument so that the reduction from the gain-graph covering theorem to the Min-2-Lin instance is self-contained.
  3. Notation: the symbol ρ is introduced as the cycle-label rank, but its precise definition (as the rank of the cycle space over F_2 after labeling) should be restated once in the main theorem statement for readers who skip the preliminary sections.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful reading and positive evaluation of the manuscript. We are grateful for the recommendation of minor revision and for highlighting the significance of the coordinatewise balanced covering theorem, the tensoring construction, rank compression to the intrinsic cycle-label rank ρ, and the resulting algorithmic improvements for Coset-List Min-2-Lin^±.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The central derivation tensors an external one-coordinate balanced-covering theorem (Dabrowski et al.) across F_2^r coordinates and augments it with newly developed cycle-space formulation, cut-space/potential characterization, minimal-dimension statement for equivalent labelings, and bit-lifting analysis. These support the rank-compression step replacing ambient dimension by intrinsic cycle-label rank ρ without reducing to self-definition, fitted parameters renamed as predictions, or load-bearing self-citations. The success probability 2^{-O(k^2 r)} is the direct product of per-coordinate probabilities, and the algorithm runtime follows from these independent components. No quoted step equates a claimed prediction or uniqueness result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper relies on standard linear algebra over F_2, graph theory notions of balanced subgraphs, and cycle-space properties. No free parameters or invented entities are evident from the abstract. Axioms are standard_math such as properties of vector spaces and graph balance.

axioms (2)
  • standard math Linear algebra over the field F_2 and properties of cycle spaces in graphs
    Invoked in the rank-compression theorem and cycle-space formulation described in the abstract.
  • domain assumption Existence of a balanced subgraph of cost at most k in the input gain graph
    Used to invoke the randomized covering procedure.

pith-pipeline@v0.9.0 · 5685 in / 1564 out tokens · 26477 ms · 2026-05-10T03:38:22.379719+00:00 · methodology

discussion (0)

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Reference graph

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