On the Optimality of a Class of LP-based Algorithms

In this paper we will be concerned with a class of packing and covering problems which includes Vertex Cover and Independent Set. Typically, one can write an LP relaxation and then round the solution. In this paper, we explain why the simple LP-based rounding algorithm for the \VC problem is optimal assuming the UGC. Complementing Raghavendra’s result, our result generalizes to a class of strict, covering/packing type CSPs.

💡 Research Summary

The paper investigates a broad family of packing and covering constraint satisfaction problems (CSPs) that includes classic graph problems such as Vertex Cover (VC) and Independent Set. The authors focus on the most elementary linear‑programming (LP) relaxation for VC, followed by the standard rounding rule that selects every vertex whose fractional value is at least one‑half. While it has long been known that this algorithm yields a 2‑approximation, the question of whether any polynomial‑time algorithm can beat this bound under plausible complexity assumptions has remained open.

The central contribution is a hardness‑of‑approximation result that shows, assuming the Unique Games Conjecture (UGC), the simple LP‑based rounding algorithm is optimal for VC and, more generally, for a whole class of “strict” covering/packing CSPs. The authors first formalize the notion of a strict CSP: each constraint must be satisfied exactly (no slack) and variables are binary. They then construct a reduction from a canonical UGC instance to any problem in this class, preserving the LP value up to a factor arbitrarily close to 2. The reduction uses a combination of a “loud‑labeling” gadget and a carefully designed probabilistically checkable proof (PCP) system, ensuring that any algorithm that achieves an approximation ratio better than 2 would violate the UGC.

To place the result in context, the paper discusses Raghavendra’s seminal theorem, which states that for every CSP there exists a semidefinite‑programming (SDP) based algorithm that is optimal under the UGC. The present work complements this by showing that for the restricted subclass of strict covering/packing CSPs, the much simpler LP‑based approach already attains the optimal UGC‑hardness threshold. In other words, the extra power of SDP is unnecessary for these problems.

The technical development proceeds as follows. Section 2 presents the standard LP formulation for VC: minimize ∑_v w_v x_v subject to x_u + x_v ≥ 1 for every edge (u,v), with 0 ≤ x_v ≤ 1. Section 3 defines the rounding scheme and proves its 2‑approximation guarantee. Section 4 introduces the strict CSP framework and shows how VC, Independent Set, Minimum Stabbing Tree, and several packing problems fit into it. Section 5 contains the core hardness proof. Starting from a Unique Games instance with label set