FPT-Algorithms for the \ell-Matchoid Problem with a Coverage Objective

TL;DR

This paper develops fixed-parameter and streaming algorithms for the ll-matchoid coverage problem, demonstrating new tractability results and separations from prior work in matroid and submodular optimization.

Contribution

It introduces the first fixed-parameter algorithms for the ll-matchoid coverage problem in the general oracle model, including deterministic and randomized methods, and shows how to bypass recent space lower bounds.

Findings

Exact solutions for linear coverage functions in the oracle model.

Streaming algorithms with space depending only on ll and z, independent of total size n.

Separation between space and time complexity for coverage versus submodular functions.

Abstract

We consider the problem of optimizing a coverage function under a $\ell$-matchoid of rank $k$. We design fixed-parameter algorithms as well as streaming algorithms to compute an exact solution. Unlike previous work that presumes linear representativity of matroids, we consider the general oracle model. For the special case where the coverage function is linear, we give a deterministic fixed-parameter algorithm parameterized by $\ell$ and $k$. This result, combined with the lower bounds of Lovasz, and Jensen and Korte demonstrates a separation between the $\ell$-matchoid and the matroid $\ell$-parity problems in the setting of fixed-parameter tractability. For a general coverage function, we give both deterministic and randomized fixed-parameter algorithms, parameterized by $\ell$ and $z$, where $z$ is the number of points covered in an optimal solution. The resulting algorithms can be directly translated into streaming algorithms. For unweighted coverage functions, we show that we can find an exact solution even when the function is given in the form of a value oracle (and so we do not have access to an explicit representation of the set system). Our result can be implemented in the streaming setting and stores a number of elements depending only on $\ell$ and $z$, but completely indpendent of the total size $n$ of the ground set. This shows that it is possible to circumvent the recent space lower bound of Feldman et al, by parameterizing the solution value. This result, combined with existing lower bounds, also provides a new separation between the space and time complexity of maximizing an arbitrary submodular function and a coverage function in the value oracle model.