Improved deterministic algorithms for weighted matching and packing problems

Abstract

Based on the method of $(n,k)$-universal sets, we present a deterministic parameterized algorithm for the weighted $r\text{<mi mathsize="small" is="true">D-MATCHING</mi>}$ problem with time complexity $O^{\ast }\left(4^{(r-1)k+o\left(k\right)}\right)$, improving the previous best upper bound $O^{\ast }\left(4^{rk+o\left(k\right)}\right)$. In particular, the algorithm applied to the unweighted 3d-matching problem results in a deterministic algorithm with time $O^{\ast }\left(16^{k+o\left(k\right)}\right)$, improving the previous best result $O^{\ast }\left(21.26^k\right)$. For the weighted $r$-set packing problem, we present a deterministic parameterized algorithm with time complexity $O^{\ast }\left(2^{(2r-1)k+o\left(k\right)}\right)$, improving the previous best result $O^{\ast }\left(2^{2rk+o\left(k\right)}\right)$. The algorithm, when applied to the unweighted 3-set packing problem, has running time $O^{\ast }\left(32^{k+o\left(k\right)}\right)$, improving the previous best result $O^{\ast }\left(43.62^{k+o\left(k\right)}\right)$. Moreover, for the weighted $r$-set packing and weighted $r\text{<mi mathsize="small" is="true">D-MATCHING</mi>}$ problems, we give a kernel of size $O\left(k^r\right)$, which is the first kernelization algorithm for the problems on weighted versions.