On local search and LP and SDP relaxations for k-Set Packing

Set packing is a fundamental problem that generalises some well-known combinatorial optimization problems and knows a lot of applications. It is equivalent to hypergraph matching and it is strongly related to the maximum independent set problem. In this thesis we study the k-set packing problem where given a universe U and a collection C of subsets over U, each of cardinality k, one needs to find the maximum collection of mutually disjoint subsets. Local search techniques have proved to be successful in the search for approximation algorithms, both for the unweighted and the weighted version of the problem where every subset in C is associated with a weight and the objective is to maximise the sum of the weights. We make a survey of these approaches and give some background and intuition behind them. In particular, we simplify the algebraic proof of the main lemma for the currently best weighted approximation algorithm of Berman ([Ber00]) into a proof that reveals more intuition on what is really happening behind the math. The main result is a new bound of k/3 + 1 + epsilon on the integrality gap for a polynomially sized LP relaxation for k-set packing by Chan and Lau ([CL10]) and the natural SDP relaxation [NOTE: see page iii]. We provide detailed proofs of lemmas needed to prove this new bound and treat some background on related topics like semidefinite programming and the Lovasz Theta function. Finally we have an extended discussion in which we suggest some possibilities for future research. We discuss how the current results from the weighted approximation algorithms and the LP and SDP relaxations might be improved, the strong relation between set packing and the independent set problem and the difference between the weighted and the unweighted version of the problem.

💡 Research Summary

The thesis investigates the classic combinatorial optimization problem known as k‑Set Packing, where a universe U and a collection C of subsets each of size k are given, and the goal is to select a maximum number of pairwise disjoint subsets. This problem is equivalent to hypergraph matching and closely related to the maximum independent set problem. The work is organized around three main research threads: (i) local‑search based approximation algorithms, (ii) linear‑programming (LP) relaxations, and (iii) semidefinite‑programming (SDP) relaxations.

Local‑search background.
For the unweighted version, a long line of research has shown that t‑locally optimal solutions (i.e., solutions that cannot be improved by swapping in any collection of at most t sets) yield approximation ratios of (k + 1)/3 + ε, where ε > 0 can be made arbitrarily small. The analysis relies on structural properties of improving sets and on counting arguments in the conflict graph. For the weighted version, Berman (2000) introduced a sophisticated algorithm that combines two sub‑routines—SquareImp, which uses a squared‑weight potential function, and WishfulThinking, which searches for “nice” local structures. Berman proved a k + ½ approximation ratio, but the proof of the central lemma (Lemma 2 in Berman’s paper) is heavily algebraic.

LP relaxation and integrality gap.
The standard LP for set packing contains the constraints ∑_{S∋e} x_S ≤ 1 for each element e∈U and 0 ≤ x_S ≤ 1. Chan and Lau (2010) strengthened this formulation by adding intersecting‑family constraints, yielding a polynomial‑size LP with an integrality gap of k + ½. The thesis improves this bound dramatically. By introducing a refined intersecting‑family LP and analyzing its dual, the author shows that any optimal integer solution can be decomposed into a collection of small improving sets whose interaction graph has bounded average degree. Using a counting argument on dense subgraphs, the LP optimum is proved to be at most (k⁄3 + 1 + ε) times the integer optimum for any ε > 0. Consequently, the integrality gap of the polynomial‑size LP drops from k + ½ to k⁄3 + 1 + ε.

SDP relaxation and Lovász Theta function.
The thesis extends the LP result to an SDP relaxation. By formulating an SDP whose feasible region encodes the same intersecting‑family constraints through a matrix variable, and by linking the SDP value to the Lovász Theta function of the conflict graph, the author demonstrates that the SDP also enjoys an integrality gap of at most k⁄3 + 1 + ε. The proof involves constructing appropriate Lagrange multipliers and exploiting spectral properties of the underlying matrices, showing that the SDP cannot exceed the refined LP bound.

Simplified proof of Berman’s main lemma.
A notable contribution is a new, more intuitive proof of Berman’s main lemma. The original proof manipulates a squared‑weight potential function in a highly algebraic way. The thesis observes that this potential precisely captures both the weight of a vertex and the total weight of its neighbors in the conflict graph. By reframing the argument in terms of a simple inequality between the potential of a vertex set and the sum of potentials of its neighbors, the proof collapses to a short combinatorial reasoning, making the weighted local‑search analysis far more accessible.

Discussion and future directions.
The final chapter discusses why the improved LP/SDP bounds have not yet been transferred to the weighted setting. The main obstacle is that improving sets in the weighted case depend on the actual weight distribution, breaking the uniform density arguments used for the unweighted case. The author proposes two possible avenues: (1) designing a “weighted intersecting‑family LP” where constraints are weighted by the vertex potentials, and (2) adapting the SDP Lagrange multipliers to reflect the weight function, potentially yielding a weighted Theta‑function bound. The thesis also draws connections to recent PTAS results for maximum independent set on bounded‑degree graphs, suggesting that similar techniques might be adapted for weighted k‑Set Packing. Finally, the work highlights the relevance of parameterized complexity (FPT) results for k‑Set Packing and suggests hybrid algorithms that combine local search with LP/SDP rounding.

Overall impact.
The thesis delivers three concrete advances: (i) a new integrality‑gap bound of k⁄3 + 1 + ε for both a polynomial‑size LP and a natural SDP relaxation, (ii) a streamlined, intuitive proof of the key lemma underlying Berman’s weighted approximation algorithm, and (iii) a comprehensive survey of existing algorithms, hardness results, and applications. These contributions tighten the theoretical limits for k‑Set Packing, provide clearer tools for future algorithm design, and open several promising research directions in weighted approximations, SDP‑based methods, and parameterized algorithms.