[Paper Review] Approximating the MaxCover Problem with Bounded Frequencies in FPT Time

Piotr Skowron, Piotr Faliszewski|arXiv (Cornell University)|Sep 17, 2013

Complexity and Algorithms in Graphs25 references14 citations

TL;DR

This paper presents fixed-parameter tractable (FPT) approximation schemes for the MaxCover problem under bounded element frequencies, achieving improved approximation ratios beyond polynomial-time limits. It introduces an FPT β-approximation algorithm for maximizing covered elements and a randomized FPT scheme for minimizing uncovered elements, with stronger guarantees than prior work, especially when combined with exact algorithms in exponential-time settings.

ABSTRACT

We study approximation algorithms for several variants of the MaxCover problem, with the focus on algorithms that run in FPT time. In the MaxCover problem we are given a set N of elements, a family S of subsets of N, and an integer K. The goal is to find up to K sets from S that jointly cover (i.e., include) as many elements as possible. This problem is well-known to be NP-hard and, under standard complexity-theoretic assumptions, the best possible polynomial-time approximation algorithm has approximation ratio (1 - 1/e). We first consider a variant of MaxCover with bounded element frequencies, i.e., a variant where there is a constant p such that each element belongs to at most p sets in S. For this case we show that there is an FPT approximation scheme (i.e., for each B there is a B-approximation algorithm running in FPT time) for the problem of maximizing the number of covered elements, and a randomized FPT approximation scheme for the problem of minimizing the number of elements left uncovered (we take K to be the parameter). Then, for the case where there is a constant p such that each element belongs to at least p sets from S, we show that the standard greedy approximation algorithm achieves approximation ratio exactly (1-e^{-max(pK/|S|, 1)}). We conclude by considering an unrestricted variant of MaxCover, and show approximation algorithms that run in exponential time and combine an exact algorithm with a greedy approximation. Some of our results improve currently known results for MaxVertexCover.

Motivation & Objective

• To design FPT approximation algorithms for the MaxCover problem under bounded element frequencies, where each element appears in at most p sets.
• To develop a randomized FPT approximation scheme for the MinNonCovered variant, minimizing uncovered elements under the same frequency constraints.
• To analyze the performance of the standard greedy algorithm under lower-bounded frequencies, showing improved approximation ratios compared to the general case.
• To propose hybrid exponential-time approximation algorithms that combine exact and greedy strategies for the unrestricted MaxCover problem, enabling a smooth trade-off between runtime and approximation quality.

Proposed method

• Leverages the FPT framework by parameterizing the problem with K, the number of sets allowed in the solution, to achieve exponential-time algorithms with only FPT growth in K.
• Applies a dynamic programming and branching strategy inspired by Guo et al. to achieve W[1]-completeness for MaxCover with bounded frequencies.
• Designs a β-approximation algorithm for MaxCover with bounded frequencies using iterative selection of sets that maximize marginal gain in coverage.
• Introduces a randomized FPT approximation scheme for MinNonCovered by sampling and verifying solutions in FPT time with high probability.
• Combines brute-force exact search with greedy approximation in hybrid algorithms, applying the greedy phase after partial solutions are generated.
• Analyzes the trade-off between running time and approximation ratio by varying the fraction of the problem solved exactly versus greedily.

Experimental results

Research questions

• RQ1 Can an FPT approximation scheme be designed for MaxCover when each element appears in at most p sets?
• RQ2 Is there a randomized FPT approximation scheme for minimizing the number of uncovered elements under bounded frequencies?
• RQ3 What approximation ratio can the standard greedy algorithm achieve when each element appears in at least p sets?
• RQ4 How can exact and greedy algorithms be combined in exponential-time settings to achieve a smooth trade-off between runtime and approximation quality?
• RQ5 Can the approximation guarantees of existing algorithms be improved by reordering the application of exact and greedy phases?

Key findings

• An FPT β-approximation algorithm exists for MaxCover with bounded frequencies, where β ∈ (0,1), and the running time depends only on K and β.
• A randomized FPT approximation scheme is presented for MinNonCovered with bounded frequencies, achieving a β-approximation with probability at least 1−ε in FPT time with respect to K, β, and ε.
• For the case where each element appears in at least p sets, the standard greedy algorithm achieves an approximation ratio of exactly 1−e−max(pK/‖S‖,1), which improves upon the (1−1/e) ratio in the general case.
• Hybrid algorithms combining brute-force exact search and greedy approximation achieve better approximation ratios than prior schemes (e.g., Croce and Paschos [7]) when using polynomial-space exact solvers.
• The proposed Algorithm 5 outperforms the algorithm of Croce and Paschos in terms of approximation ratio for the same running time, especially when the exact algorithm does not find the optimal sub-solution.
• The MaxCover problem with bounded frequencies is W[1]-complete, while the unrestricted version is W[2]-hard and in W[P], indicating a clear complexity distinction based on frequency constraints.