On Approximating Partial Set Cover and Generalizations
Pith reviewed 2026-05-24 23:47 UTC · model grok-4.3
The pith
Partial Set Cover can be approximated within (1-1/e)(β + 1) when Set Cover admits a β-approximation via the natural LP.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The authors show that the coverage function for Partial Set Cover remains submodular under the deletion-closed family assumption from earlier work, so the standard greedy algorithm for submodular maximization yields an improved approximation of (1-1/e)(β + 1) when the underlying Set Cover problem has a β-approximation via the natural LP; the same technique handles generalizations with several partial constraints and sparse instances.
What carries the argument
The submodular coverage function arising from the natural LP relaxation, which permits a greedy selection step that tightens the reduction from Partial Set Cover to Set Cover.
If this is right
- PSC on deletion-closed families now has approximation ratio (1-1/e)(β + 1) instead of 2(β + 1).
- The submodularity argument extends the reduction to handle multiple simultaneous partial-coverage constraints.
- The improved bounds and simplifications apply in the sparse setting as well.
- Prior results on PSC and its generalizations are recovered and tightened as direct corollaries of the submodular view.
Where Pith is reading between the lines
- The same submodularity observation could be tested on other partial covering problems that use LP-based reductions.
- For geometric set systems, the tighter factor may translate into measurable improvements in covering algorithms used in practice.
- If stronger submodular maximization routines become available, they could be plugged in to further improve the PSC guarantee without new reductions.
Load-bearing premise
The coverage function remains submodular when the set system is restricted to a deletion-closed family.
What would settle it
An explicit deletion-closed family with a β-approximation for Set Cover via LP but where no algorithm achieves better than (1-1/e)(β + 1) + ε for Partial Set Cover on some instance would falsify the improved ratio.
read the original abstract
Partial Set Cover (PSC) is a generalization of the well-studied Set Cover problem (SC). In PSC the input consists of an integer $k$ and a set system $(U,S)$ where $U$ is a finite set, and $S \subseteq 2^U$ is a collection of subsets of $U$. The goal is to find a subcollection $S' \subseteq S$ of smallest cardinality such that sets in $S'$ cover at least $k$ elements of $U$; that is $|\cup_{A \in S'} A| \ge k$. SC is a special case of PSC when $k = |U|$. In the weighted version each set $X \in S$ has a non-negative weight $w(X)$ and the goal is to find a minimum weight subcollection to cover $k$ elements. Approximation algorithms for SC have been adapted to obtain comparable algorithms for PSC in various interesting cases. In recent work Inamdar and Varadarajan, motivated by geometric set systems, obtained a simple and elegant approach to reduce PSC to SC via the natural LP relaxation. They showed that if a deletion-closed family of SC admits a $\beta$-approximation via the natural LP relaxation, then one can obtain a $2(\beta + 1)$-approximation for PSC on the same family. In a subsequent paper, they also considered a generalization of PSC that has multiple partial covering constraints which is partly inspired by and generalizes previous work of Bera et al on the Vertex Cover problem. Our main goal in this paper is to demonstrate some useful connections between the results in previous work and submodularity. This allows us to simplify, and in some cases improve their results. We improve the approximation for PSC to $(1-1/e)(\beta + 1)$. We extend the previous work to the sparse setting.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper claims that by establishing connections between the deletion-closed reductions of Inamdar-Varadarajan and the submodularity of the coverage function, the approximation ratio for Partial Set Cover (PSC) on families admitting a β-approximation via the natural LP can be improved from 2(β + 1) to (1 - 1/e)(β + 1). It further extends the approach to a sparse setting and to a generalization with multiple partial covering constraints.
Significance. If the submodularity connection holds, the result strengthens prior work by replacing the factor-2 loss with the standard (1 - 1/e) greedy factor for submodular maximization; this is a clean improvement for geometric and other LP-based set systems. The manuscript also supplies a simplification of the earlier reduction technique.
major comments (1)
- [Abstract / introduction paragraph on Inamdar-Varadarajan] The central improvement rests on the claim that the coverage function remains submodular after the deletion-closed reduction (abstract). A concrete verification that the marginal-gain property is preserved exactly under the same family assumption used for the 2(β + 1) result would strengthen the argument; without it the improvement factor could be viewed as conditional on an unstated property.
minor comments (2)
- The abstract states that the work 'extends the previous work to the sparse setting' but does not define the sparse model or cite the precise prior result being extended; adding a one-sentence definition or reference would improve readability.
- Notation for the multiple-constraint generalization is introduced only at the end of the abstract; a brief forward reference to the section containing the formal statement would help readers.
Simulated Author's Rebuttal
We thank the referee for the positive recommendation of minor revision and for the constructive comment on strengthening the submodularity argument. We address the point below.
read point-by-point responses
-
Referee: [Abstract / introduction paragraph on Inamdar-Varadarajan] The central improvement rests on the claim that the coverage function remains submodular after the deletion-closed reduction (abstract). A concrete verification that the marginal-gain property is preserved exactly under the same family assumption used for the 2(β + 1) result would strengthen the argument; without it the improvement factor could be viewed as conditional on an unstated property.
Authors: We agree that an explicit verification would improve clarity. The coverage function f(S) = |∪_{A∈S} A| is monotone submodular on any set system by definition, with marginal gains satisfying f(A ∪ {e}) − f(A) ≥ f(B ∪ {e}) − f(B) whenever A ⊆ B. The deletion-closed reduction of Inamdar-Varadarajan constructs a new instance whose objective remains exactly this coverage function (elements deleted from consideration are those already accounted for in the partial-cover threshold), so the marginal-gain inequality is inherited verbatim. The β-approximability assumption on the natural LP is used only to bound the cost of the selected sets and does not alter the objective function itself; hence the same family assumption that yields 2(β + 1) also yields the submodular (1 − 1/e) factor. We will add a short lemma (or a dedicated paragraph in Section 3) that records this preservation explicitly, citing the original reduction construction. revision: yes
Circularity Check
No significant circularity identified
full rationale
The paper's core improvement from 2(β + 1) to (1-1/e)(β + 1) for PSC rests on connecting the deletion-closed family assumption from Inamdar-Varadarajan (distinct authors) to the independent fact that coverage functions remain submodular. This is a standard external property of set coverage, not derived from or equivalent to the paper's own inputs, equations, or self-citations. No step reduces a prediction to a fitted parameter by construction, renames a known result, or loads the central claim on a self-citation chain. The derivation is self-contained against external benchmarks.
Axiom & Free-Parameter Ledger
axioms (1)
- domain assumption The coverage function is submodular
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.