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arxiv: 2604.23334 · v1 · submitted 2026-04-25 · 💻 cs.DS

A Note on Interdiction of Linear Minimization Problems

Yu Cong , Kangyi Tian This is my paper

Pith reviewed 2026-05-08 07:20 UTC · model grok-4.3

classification 💻 cs.DS
keywords interdictionlinear minimizationLagrange multipliertruncated weights2-approximationknapsackfeasible sets
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The pith

The optimal interdiction witness for linear minimization problems is a strict 2-approximate minimizer of the reweighted objective at the optimal Lagrange multiplier.

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

This paper isolates the cut-free part of an argument used in recent connectivity interdiction work. It shows that dualizing the interdiction budget lets deletions be absorbed into truncated weights w_λ(e) = min{w(e), λ c(e)}. At the optimal Lagrange multiplier λ*, any optimal interdiction set is a strict 2-approximate minimizer of the new linear objective over the feasible sets. A reader would care because the reduction yields an exact interdiction algorithm as soon as the reweighted problem can be optimized, its 2-approximate solutions enumerated, and the leftover budget handled as a knapsack; the argument uses only linearity and dualization.

Core claim

After dualizing the interdiction budget, deletion can be absorbed into truncated weights w_λ(e)=min{w(e),λ c(e)}. At an optimal Lagrange multiplier λ*, the unknown optimal interdiction witness is a strict 2-approximate minimizer of the reweighted problem. Thus an exact algorithm can be obtained whenever one can optimize w_λ* over F, enumerate all its 2-approximate minimizers, and solve the remaining knapsack problem.

What carries the argument

Truncated weights w_λ(e) = min{w(e), λ c(e)} that absorb interdiction deletions after dualizing the budget; these weights ensure the optimal witness is a strict 2-approximate minimizer of the objective over the feasible-set family F.

If this is right

  • An exact interdiction algorithm follows whenever the reweighted problem over F can be optimized, all its 2-approximate minimizers enumerated, and the residual budget solved as a knapsack.
  • The reduction applies to any linear minimization problem over a feasible-set family and requires no cut or connectivity structure.
  • The same three-step procedure (optimize reweighted objective, enumerate 2-approximates, solve knapsack) yields exact solutions for any problem where those steps are tractable.

Where Pith is reading between the lines

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

  • The 2-approximation property may hold for interdiction variants in matching or scheduling where cut arguments are unavailable.
  • Designers could search for new feasible-set families F where listing 2-approximate solutions is feasible, producing previously unknown exact interdiction algorithms.
  • The cut-free nature of the argument suggests the reduction can be used to simplify or generalize existing interdiction proofs.

Load-bearing premise

Dualizing the interdiction budget and absorbing deletions into truncated weights preserves the optimality relation without depending on cut structure or other problem-specific properties.

What would settle it

A concrete counterexample with a feasible-set family F, weights w and c, and budget B where at the optimal λ* an optimal interdiction set S satisfies w_λ*(S) strictly greater than twice the minimum of w_λ* over F.

Figures

Figures reproduced from arXiv: 2604.23334 by Kangyi Tian, Yu Cong.

Figure 4.1
Figure 4.1. Figure 4.1: Template for solving interdiction version of linear minimization problem. 4 Algorithmic template The theorem gives the a general template shown in view at source ↗
read the original abstract

Motivated by the FPTAS for connectivity interdiction of Huang et al. (IPCO'24), we isolate the part of the argument that does not use cuts. The setting is a minimization problem over a feasible-set family $\mathcal F$ with a linear objective $w(S)=\sum_{e\in S}w(e)$. After dualizing the interdiction budget, deletion can be absorbed into truncated weights $w_\lambda(e)=\min\{w(e),\lambda c(e)\}$. At an optimal Lagrange multiplier $\lambda^*$, the unknown optimal interdiction witness is a strict $2$-approximate minimizer of the reweighted problem. Thus an exact algorithm can be obtained whenever one can optimize $w_{\lambda^*}$ over $\mathcal F$, enumerate all its $2$-approximate minimizers, and solve the remaining knapsack problem.

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 isolates the duality-based component of the FPTAS for connectivity interdiction from Huang et al. (IPCO'24) and shows that it applies to general linear minimization problems over an arbitrary family F. After dualizing the interdiction budget, deletions are absorbed into truncated weights w_λ(e) = min{w(e), λ c(e)}. The key result is that at the optimal Lagrange multiplier λ*, the unknown optimal interdiction witness is a strict 2-approximate minimizer of the reweighted problem. This yields an exact algorithm whenever one can optimize w_λ* over F, enumerate all its 2-approximate minimizers, and solve the remaining knapsack problem. The argument is presented as relying solely on linearity of the objective.

Significance. If the central claim holds, the note provides a clean modular reduction showing that the Lagrangian duality and 2-approximation enumeration technique apply to arbitrary feasible-set families F without any cut-specific closure properties. This is a strength of the work, as it separates the general technique from the connectivity-specific arguments in the motivating IPCO paper and could enable similar exact algorithms or FPTASes in other interdiction settings where 2-approximates are enumerable. The stress-test concern about hidden reliance on cut structure does not land on the manuscript, which explicitly frames the reduction as holding by linearity alone.

minor comments (3)
  1. The term 'interdiction witness' is introduced in the abstract without a formal definition or reference to its meaning in the interdiction context; adding a precise definition (e.g., as the post-interdiction optimal S* in F) in the introduction or main theorem statement would improve accessibility.
  2. The abstract claims the witness is a 'strict 2-approximate minimizer' but does not spell out the precise inequality or the meaning of 'strict' (e.g., w_λ*(S*) < 2 · OPT or ≤ 2 · OPT − ε); clarifying this in the statement of the main result would aid verification.
  3. A brief illustrative example (even for a trivial F such as all singletons) showing how the truncated weights produce a 2-approximate witness and how the knapsack step recovers the interdiction would help readers follow the reduction.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of the note and the recommendation for minor revision. The manuscript isolates the Lagrangian duality argument from Huang et al. (IPCO'24) and establishes that it applies to arbitrary linear minimization problems over any family F, relying only on linearity of the objective. We appreciate the referee's confirmation that no hidden cut-specific structure is required.

Circularity Check

0 steps flagged

No circularity; derivation uses standard Lagrangian duality on arbitrary linear objectives

full rationale

The paper isolates a general argument for linear minimization over any feasible-set family F: dualizing the interdiction budget yields truncated weights w_λ(e) = min{w(e), λ c(e)}, after which the optimal interdiction witness is shown to be a strict 2-approximate minimizer of the reweighted objective by direct appeal to optimality conditions and linearity. This step does not reduce to any fitted parameter inside the paper, does not invoke a self-citation load-bearing uniqueness theorem, and does not smuggle an ansatz via citation. The reference to Huang et al. is purely motivational; the claimed relation holds by the definition of the dualized problem and the definition of 2-approximate minimizers, making the derivation self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on Lagrangian duality for the interdiction budget and on the property that a 2-approximate minimizer suffices to recover the optimal interdiction; both are standard in combinatorial optimization and not invented here.

axioms (2)
  • domain assumption Lagrangian duality applies to the interdiction budget constraint and produces an optimal multiplier λ* at which the truncated weights capture the optimal deletion set.
    Invoked when the abstract states that deletion can be absorbed into w_λ(e).
  • domain assumption For any linear objective, an optimal interdiction witness is a strict 2-approximate minimizer of the reweighted problem.
    This is the load-bearing step that allows reduction to enumeration of 2-approximates.

pith-pipeline@v0.9.0 · 5434 in / 1502 out tokens · 77867 ms · 2026-05-08T07:20:07.682276+00:00 · methodology

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Reference graph

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