Chien-Chung Huang

Chien-Chung Huang, Naonori Kakimura, Yusuke Kobayashi et al.

This paper studies randomized polynomial kernelization for the weighted $d$-matroid intersection problem. While the problem is known to have a kernel of size $O(d^{(k - 1)d})$ where $k$ is the solution size, the existence of a polynomial kernel is not known, except for the cases when either all the given matroids are partition matroids~(i.e., the $d$-dimensional matching problem) or all the given matroids are linearly representable. The main contribution of this paper is to develop a new kernelization technique for handling general matroids. We first show that the weighted $d$-matroid intersection problem admits a polynomial kernel when one matroid is arbitrary and the other $d-1$ matroids are partition matroids. Interestingly, the obtained kernel has size $\tilde{O}(k^d)$, which matches the optimal bound~(up to logarithmic factors) for the $d$-dimensional matching problem. This approach can be adapted to the case when $d-1$ matroids in the input belong to a more general class of matroids, including graphic, cographic, and transversal matroids. We also show that the problem has a kernel of pseudo-polynomial size when given $d-1$ matroids are laminar. Our technique finds a kernel such that any feasible solution of a given instance can reach a better solution in the kernel, which is sufficiently versatile to allow us to design parameterized streaming algorithms and faster EPTASs.

Chien-Chung Huang, Naonori Kakimura, Simon Mauras et al.

Maximizing a monotone submodular function under various constraints is a classical and intensively studied problem. However, in the single-pass streaming model, where the elements arrive one by one and an algorithm can store only a small fraction of input elements, there is much gap in our knowledge, even though several approximation algorithms have been proposed in the literature. In this work, we present the first lower bound on the approximation ratios for cardinality and matroid constraints that beat $1-\frac{1}{e}$ in the single-pass streaming model. Let $n$ be the number of elements in the stream. Then, we prove that any (randomized) streaming algorithm for a cardinality constraint with approximation ratio $\frac{2}{2+\sqrt{2}}+\varepsilon$ requires $Ω\left(\frac{n}{K^2}\right)$ space for any $\varepsilon>0$, where $K$ is the size limit of the output set. We also prove that any (randomized) streaming algorithm for a (partition) matroid constraint with approximation ratio $\frac{K}{2K-1}+\varepsilon$ requires $Ω\left(\frac{n}{K}\right)$ space for any $\varepsilon>0$, where $K$ is the rank of the given matroid. In addition, we give streaming algorithms when we only have a weak oracle with which we can only evaluate function values on feasible sets. Specifically, we show weak-oracle streaming algorithms for cardinality and matroid constraints with approximation ratios $\frac{K}{2K-1}$ and $\frac{1}{2}$, respectively, whose space complexity is exponential in $K$ but is independent of $n$. The former one exactly matches the known inapproximability result for a cardinality constraint in the weak oracle model. The latter one almost matches our lower bound of $\frac{K}{2K-1}$ for a matroid constraint, which almost settles the approximation ratio for a matroid constraint that can be obtained by a streaming algorithm whose space complexity is independent of $n$.