[Paper Review] On local search and LP and SDP relaxations for k-Set Packing

TS Tim Oosterwijk|arXiv (Cornell University)|Jul 27, 2015

Complexity and Algorithms in Graphs84 references3 citations

TL;DR

This thesis investigates local search, linear programming (LP), and semidefinite programming (SDP) relaxations for the k-set packing problem, a fundamental NP-hard combinatorial optimization problem. It simplifies Berman's weighted approximation algorithm proof, establishes a new integrality gap bound of $\frac{k}{3}+1+\varepsilon$ for a polynomial-sized LP and the natural SDP relaxation, and discusses open problems in improving approximation guarantees.

ABSTRACT

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.

Motivation & Objective

• To simplify the algebraic proof of the main lemma in Berman's weighted approximation algorithm for k-set packing, revealing deeper intuition behind the mathematical structure.
• To establish a new integrality gap bound of $\frac{k}{3}+1+\varepsilon$ for a polynomially sized LP relaxation and the natural SDP relaxation for k-set packing.
• To explore the relationship between k-set packing and the maximum independent set problem, particularly in the context of weighted and unweighted variants.
• To identify and discuss open problems and future research directions, including improving approximation guarantees via larger improving sets in local search.

Proposed method

• Simplifies the proof of the main lemma in Berman's weighted approximation algorithm using intuitive combinatorial reasoning instead of complex algebra.
• Analyzes the structure of improving sets in local search by modeling them as subgraphs with more edges than vertices in the conflict hypergraph.
• Applies techniques from semidefinite programming and the Lovász theta function to derive bounds on the SDP relaxation for k-set packing.
• Uses a polynomial-sized LP relaxation, building on Chan and Lau's work, to achieve a tighter integrality gap bound.
• Employs a reduction from weighted k-set packing to the independent set problem on $k+1$-claw-free graphs as a foundation for algorithmic design.
• Conducts a detailed discussion on the differences between weighted and unweighted k-set packing and the limitations of current approximation techniques.

Experimental results

Research questions

• RQ1 Can the proof of Berman's weighted approximation algorithm for k-set packing be simplified to reveal more intuitive insights into the underlying combinatorial structure?
• RQ2 What is the tightest possible integrality gap for a polynomial-sized LP relaxation of the k-set packing problem?
• RQ3 Does the natural SDP relaxation for k-set packing achieve the same integrality gap bound as the polynomial-sized LP relaxation?
• RQ4 Can local search algorithms that consider improving sets of size larger than 2 (e.g., $t$-locally optimal solutions for $t > 2$) yield better approximation guarantees for the weighted k-set packing problem?
• RQ5 Is it possible to achieve an approximation guarantee of $\frac{k+c}{3}+\varepsilon$ for some fixed $c \geq 0$ in the weighted k-set packing problem?

Key findings

• The proof of the main lemma in Berman's weighted approximation algorithm is simplified, revealing clearer combinatorial intuition behind the weight function and improving set selection.
• A new integrality gap bound of $\frac{k}{3}+1+\varepsilon$ is established for a polynomially sized LP relaxation of the k-set packing problem.
• The same integrality gap bound of $\frac{k}{3}+1+\varepsilon$ is shown to hold for the natural SDP relaxation of k-set packing.
• The proofs of Theorem 17 and its corollaries (Theorems 19, 23, 24) are found to be invalid due to an error in the definition of improving sets, leaving the truth of these results open.
• The thesis conjectures that a better approximation guarantee of $\frac{k+c}{3}+\varepsilon$ for some fixed $c \geq 0$ may be achievable by extending local search to consider larger improving sets.
• The paper identifies that current weighted approximation algorithms rely heavily on reducing k-set packing to the independent set problem on $k+1$-claw-free graphs, suggesting this reduction as a promising avenue for future improvement.