[Paper Review] Approximate the k-Set Packing Problem by Local Improvements

Martin Fürer, Huiwen Yu|arXiv (Cornell University)|Jul 8, 2013

Complexity and Algorithms in Graphs12 references4 citations

TL;DR

This paper presents a polynomial-time local search algorithm for the k-Set Packing problem that achieves an approximation ratio of $\frac{k+1}{3} + \epsilon$ for any $\epsilon > 0$, using a novel local improvement strategy based on set replacements and cycle/path decompositions. The algorithm runs in singly exponential time in $\frac{1}{\epsilon^2}$, improving upon previous doubly exponential bounds, and establishes a matching lower bound showing that no local improvement algorithm of size $O(n^{1/5})$ can exceed this ratio.

ABSTRACT

We study algorithms based on local improvements for the $k$-Set Packing problem. The well-known local improvement algorithm by Hurkens and Schrijver has been improved by Sviridenko and Ward from $\frac{k}{2}+ε$ to $\frac{k+2}{3}$, and by Cygan to $\frac{k+1}{3}+ε$ for any $ε>0$. In this paper, we achieve the approximation ratio $\frac{k+1}{3}+ε$ for the $k$-Set Packing problem using a simple polynomial-time algorithm based on the method by Sviridenko and Ward. With the same approximation guarantee, our algorithm runs in time singly exponential in $\frac{1}{ε^2}$, while the running time of Cygan's algorithm is doubly exponential in $\frac{1}ε$. On the other hand, we construct an instance with locality gap $\frac{k+1}{3}$ for any algorithm using local improvements of size $O(n^{1/5})$, here $n$ is the total number of sets. Thus, our approximation guarantee is optimal with respect to results achievable by algorithms based on local improvements.

Motivation & Objective

• To develop a more efficient polynomial-time local search algorithm for the k-Set Packing problem with improved running time while maintaining the best-known approximation ratio.
• To close the gap between the best-known approximation ratio and the theoretical limits of local improvement-based algorithms.
• To establish a tight lower bound on the performance of local improvement algorithms, showing that $\frac{k+1}{3}$ is optimal for algorithms using improvements of size $O(n^{1/5})$.
• To extend the scope of local improvements beyond standard replacements by incorporating cycle- and path-based transformations from sets not in the current packing.

Proposed method

• The algorithm performs local improvements by first identifying a set of $p$ disjoint sets $\mathcal{Q}$ in the current packing $\mathcal{A}$ and replacing them with $p$ new disjoint sets $\mathcal{T}$ not in $\mathcal{A}$, forming a swap.
• It then searches for additional improvements by considering sets from $\mathcal{S} \setminus (\mathcal{A} \cup \mathcal{T})$ that intersect at most two sets in $\mathcal{A}$, decomposing improvements into cycles and paths.
• The color-coding technique is used to ensure polynomial-time complexity when searching for logarithmic-sized improvements, enabling efficient detection of valid local improvement structures.
• The algorithm uses a two-phase search: first a set replacement phase, followed by a path/cycle-based improvement phase, both guided by combinatorial analysis of intersection patterns.
• A probabilistic analysis is conducted to bound the expected number of unstable configurations, proving the existence of a valid improvement structure under the given constraints.
• The running time is bounded by $\text{poly}(n) \cdot 2^{O(1/\epsilon^2)}$, which is singly exponential in $1/\epsilon^2$, significantly improving over doubly exponential bounds in prior work.

Experimental results

Research questions

• RQ1 Can a polynomial-time local search algorithm for k-Set Packing achieve the approximation ratio $\frac{k+1}{3} + \epsilon$ with better running time than existing methods?
• RQ2 Is $\frac{k+1}{3}$ the optimal approximation ratio achievable by any local improvement algorithm using improvements of size $O(n^{1/5})$?
• RQ3 Can the scope of local improvements be extended beyond simple replacements to include cycle- and path-based transformations for better performance?
• RQ4 How does the use of color-coding and combinatorial analysis enable efficient detection of complex local improvement structures?
• RQ5 What is the tightest possible lower bound on the performance of local improvement algorithms for k-Set Packing, given the structure of the problem?

Key findings

• The proposed algorithm achieves an approximation ratio of $\frac{k+1}{3} + \epsilon$ for any $\epsilon > 0$, matching the best-known performance guarantee from prior work.
• The running time of the algorithm is $\text{poly}(n) \cdot 2^{O(1/\epsilon^2)}$, which is singly exponential in $1/\epsilon^2$, significantly improving upon the doubly exponential $2^{O(1/\epsilon)}$ time of Cygan's algorithm.
• The paper constructs a family of instances where any local improvement algorithm using improvements of size $O(n^{1/5})$ has a locality gap of at least $\frac{k+1}{3}$, proving that this ratio is optimal for such algorithms.
• The analysis shows that the expected number of unstable configurations (i.e., sets that could disrupt the improvement) is less than 1 under the given parameters, guaranteeing the existence of a valid improvement structure.
• The method extends prior local search techniques by incorporating path- and cycle-based improvements from sets not in the current packing, increasing the search space for valid improvements.
• The result establishes that $\frac{k+1}{3} + \epsilon$ is the best possible approximation ratio achievable by any local improvement algorithm using improvements of size $O(n^{1/5})$, closing a long-standing open question on the limits of such methods.