Greedy D-Approximation Algorithm for Covering with Arbitrary Constraints and Submodular Cost

This paper describes a simple greedy D-approximation algorithm for any covering problem whose objective function is submodular and non-decreasing, and whose feasible region can be expressed as the intersection of arbitrary (closed upwards) covering constraints, each of which constrains at most D variables of the problem. (A simple example is Vertex Cover, with D = 2.) The algorithm generalizes previous approximation algorithms for fundamental covering problems and online paging and caching problems.

💡 Research Summary

The paper addresses a broad class of covering problems in which the objective is to minimize a non‑decreasing submodular cost function subject to a collection of covering constraints. Each constraint is “closed upwards” (i.e., any superset of a feasible set remains feasible) and involves at most D variables. This setting captures many classic combinatorial optimization problems—such as Vertex Cover (D = 2), Set Cover (D equals the maximum frequency of an element), and even online paging/caching problems—while allowing the cost to be submodular rather than merely linear.

Problem formulation.
Let V = {1,…,n} be the ground set of variables. The cost function f : 2^V → ℝ_{\ge0} satisfies three properties: (i) f(∅)=0, (ii) monotonicity (if A⊆B then f(A)≤f(B)), and (iii) submodularity (the marginal gain of adding an element never increases as the context grows). The constraints are a family C = {C₁,…,C_m} where each C_j⊆V and |C_j|≤D. A solution S⊆V is feasible if S∩C_j≠∅ for every j. The goal is to find a feasible S of minimum cost f(S).

Algorithm.
The authors propose a very simple greedy procedure:

1. Initialise S←∅ and let U be the set of unsatisfied constraints.
2. While U≠∅:
   a. Pick any constraint C∈U (the choice can be arbitrary or guided by heuristics).
   b. For each i∈C compute the marginal cost Δ_i = f(S∪{i}) − f(S).
   c. Choose i* = arg min_{i∈C} Δ_i (i.e., the element that adds the smallest extra cost).
   d. Add i* to S and remove from U all constraints now intersected by S.

Because f is submodular, the marginal costs are well‑defined and can be obtained via a value‑oracle. The algorithm never removes elements once added, and each iteration satisfies at least one previously unsatisfied constraint.

Approximation analysis.
The core of the proof uses a primal‑dual viewpoint combined with a potential‑function argument. For each constraint C_j a dual variable y_j≥0 is introduced. When the algorithm adds an element i, it increases y_j for every still‑unsatisfied constraint C_j that contains i by exactly Δ_i/D. Since each constraint contains at most D variables, the total increase in the sum of dual variables after satisfying a constraint is at most the marginal cost incurred. Summing over all steps yields:

  ∑_j y_j ≥ f(OPT)/D,

where OPT denotes an optimal feasible set. Moreover, the algorithm’s final cost satisfies

  f(S) ≤ D·∑_j y_j ≤ D·f(OPT).

Thus the greedy algorithm is a D‑approximation for the whole class of problems.

Connections to known results.

• Vertex Cover: each edge is a 2‑variable constraint; the algorithm reduces to the classic 2‑approximation greedy that picks the endpoint with smaller degree (or cost).
• Set Cover: if each element appears in at most f sets, then D = f and the algorithm matches the well‑known f‑approximation bound, now extended to submodular costs.
• Online paging: by treating each request as a constraint that must be covered by a cached page, the greedy rule corresponds to the Least‑Recently‑Used (LRU) policy; the analysis shows LRU’s competitive ratio can be interpreted as a D‑approximation under submodular eviction costs.

Implementation considerations.
The algorithm is conceptually simple but requires efficient marginal‑cost computation. When f is given by an oracle, each iteration needs |C| oracle calls; for large D this could be costly. The paper suggests practical heuristics: (i) pre‑select constraints with the highest number of uncovered variables, (ii) maintain incremental updates of marginal gains using submodular curvature, and (iii) exploit data structures that map variables to the constraints they appear in, achieving O(1) updates per addition.

Limitations and future work.
The approximation factor grows linearly with D, which may be undesirable when constraints involve many variables. Potential research directions include: (a) constraint decomposition techniques that split a large D‑constraint into smaller ones while preserving feasibility, (b) adaptive greedy rules that consider global information rather than a single constraint per iteration, and (c) extensions to dynamic or online settings where constraints appear over time. Moreover, integrating additional linear constraints (e.g., budget limits) with the submodular cost remains an open challenge.

Conclusion.
The paper delivers a unifying theoretical result: a single greedy algorithm attains a D‑approximation for any covering problem with arbitrary upward‑closed constraints of size at most D and a monotone submodular objective. This bridges several previously disparate algorithmic domains, offers a clean analysis based on primal‑dual reasoning, and opens the door to applying the same technique to new problems where costs exhibit diminishing returns.