On Approximating Four Covering and Packing Problems

In this paper, we consider approximability issues of the following four problems: triangle packing, full sibling reconstruction, maximum profit coverage and 2-coverage. All of them are generalized or specialized versions of set-cover and have applications in biology ranging from full-sibling reconstructions in wild populations to biomolecular clusterings; however, as this paper shows, their approximability properties differ considerably. Our inapproximability constant for the triangle packing problem improves upon the previous results; this is done by directly transforming the inapproximability gap of Haastad for the problem of maximizing the number of satisfied equations for a set of equations over GF(2) and is interesting in its own right. Our approximability results on the full siblings reconstruction problems answers questions originally posed by Berger-Wolf et al. and our results on the maximum profit coverage problem provides almost matching upper and lower bounds on the approximation ratio, answering a question posed by Hassin and Or.

💡 Research Summary

This paper studies the approximability of four combinatorial optimization problems that arise in computational biology: Triangle Packing (TP), full‑sibling reconstruction (the k‑ALLELEₙ,ℓ problems), Maximum Profit Coverage (MPC), and 2‑Coverage. All four can be expressed as special cases of a general “cover/packing” framework where a universe of elements and a family of subsets are given together with optional weights, cardinality limits, conflict pairs, and other constraints. The goal in each case is to select a sub‑collection of sets satisfying the constraints while optimizing a linear objective in the element and set weights.

Triangle Packing.
Given an undirected graph, the task is to select a maximum number of vertex‑disjoint triangles. Prior work gave an inapproximability factor of 95/94≈1.0106 under P≠NP. The authors improve this bound to 76/75≈1.013 by a direct reduction from Håstad’s 3‑LIN‑2 hardness result. They map each linear equation over GF(2) with three variables to a gadget of triangles such that satisfying a fraction (1−ε) of the equations corresponds to packing (1−ε)·k triangles, while any instance with at most (1/2+ε)·m satisfied equations yields at most (75/76)·k triangles. The reduction preserves the size linearly, yielding the stronger constant under RP≠NP. Additional specialized bounds (e.g., 6725/6724≈1.00014 for a=3, ℓ=2) are also derived.

Full‑Sibling Reconstruction (k‑ALLELEₙ,ℓ).
Each individual is represented by ℓ loci, each locus containing a pair of alleles. A set of individuals is a full‑sibling group if, after possibly reordering each locus, at most k distinct alleles appear in each parental position (k=2 or 4). The objective is to cover all n individuals with the minimum number of sibling groups. Let a denote the size of the largest feasible sibling group. When a is a constant, the problem reduces to a set‑cover instance with at most a‑sized sets, and the classic greedy algorithm yields a (1+ln a)‑approximation. For arbitrary a but a fixed constant c, the authors present a “c‑approximation” scheme: they decompose an optimal solution into disjoint groups, charge each selected set of size b a cost of 1/b per element, and show that the total charge is bounded by c+ln c. This yields a (c+ln c)‑approximation for any constant c, improving over the trivial (a+ln a) bound when a is large. The paper also discusses trivial (a+ln a) approximations via standard set‑cover reductions and shows that the problems are APX‑hard for general a.

Maximum Profit Coverage (MPC).
We are given a family S of subsets of a universe U, each set A having a non‑negative cost q_A and each element i having a profit w_i. For a chosen sub‑family P⊆S, the profit is w(∪{A∈P}A)−q(P). Let a = max{A∈S}|A|. When a≤2 the problem can be solved exactly by reducing to a maximum weight perfect matching in a bipartite graph. For a≥3 the problem becomes NP‑hard; the authors prove APX‑hardness via a reduction from the a‑regular Independent Set problem, obtaining a lower bound of (a+0.5+ε) for any ε>0 under P≠NP. They also give an upper‑bound approximation algorithm: for fixed a, Berman’s local‑improvement method yields a ratio of a+½+ε; for unbounded a, the Berman‑Krysta algorithm achieves a 0.6454·a approximation. Moreover, even under severe restrictions (every set has weight a−1, each element weight 1, each set contains exactly a elements, and sets correspond to equal‑radius balls in a metric space) the same hardness persists.

2‑Coverage.
Given a family S and an integer k, select at most k sets so that the number of elements appearing in at least two of the chosen sets is maximized. Let f be the maximum frequency of any element across S. The authors focus on f=2 and obtain a (1+α)‑approximation, where α is a constant derived from a BPTIME(2^{εn}) hardness result for densest subgraph. They also prove that 2‑Coverage inherits the (1−1/e) lower bound from the classic Maximum Coverage problem, and that for arbitrary f the problem admits an O(√m)‑approximation (m = |S|). The hardness is tied to the densest subgraph problem, which is known to be (1+ε)‑inapproximable unless NP⊆BPTIME(2^{εn}).

Overall Contributions.
Table 1 in the paper summarizes the results:

• TP: (76/75)−ε inapproximability (RP≠NP) and a (7/6)+ε approximation for general a.
• k‑ALLELE: (153/152)−ε inapproximability (RP≠NP) for a=3, ℓ=O(n³); trivial (1+ln a) approximations otherwise.
• MPC: exact for a≤2, APX‑hard for a≥3 with (a+0.5+ε) lower bound, and a 0.6454·a upper bound for arbitrary a.
• 2‑Coverage: (1+α) approximation under BPTIME hardness, O(√m) approximation in general.

The paper thus advances the theoretical understanding of these biologically motivated problems, providing tighter hardness constants (especially for Triangle Packing), new approximation algorithms, and a unified view of how set‑cover‑like structures manifest in diverse applications such as sibling reconstruction, molecular clustering, and seed‑based homology search.