Deterministic algorithms for Mat ching and packing problems based on representative sets

In this work, we study the well-known r-DIMENSIONAL k-MATCHING ((r, k)-DM), and r-SET k-PACKING ((r, k)-SP) problems. Given a universe U := U₁ ⋯ U_r and an r-uniform family F ⊆ U₁ × ⋯ × U_r, the (r, k)-DM problem asks if F admits a collection of k mutually disjoint sets. Given a universe U and an r-uniform family F ⊆ 2^U, the (r,k)SP problem asks if F admits a collection of k mutually disjoint sets. We employ techniques based on dynamic programming and representative families. This leads to a deterministic algorithm with running time O(2.851^(r-1)k. |F|. n log² n-log W) for the weighted version of (r, k)-DM, where W is the maximum weight in the input, and a deterministic algorithm with running time O(2.851(r-0.5501)k · |F| · nlog² n · logW) for the weighted version of (r, k)SP. Thus, we significantly improve the previous best known deterministic running times for (r, k)-DM and (r, k)SP and the previous best known running times for their weighted versions. We rely on structural properties of (r, k)-DM and (r, k)SP to develop algorithms that are faster than those that can be obtained by a standard use of representative sets. Incorporating the principles of iterative expansion, we obtain a better algorithm for (3,k)-DM, running in time O(2.004^3k · |F| · nlog² n). We believe that this algorithm demonstrates an interesting application of representative families in conjunction with more traditional techniques. Furthermore, we present kernels of size O(e^rr(k - 1)^r log W) for the weighted versions of (r, k)-DM and (r, k)-SP, improving the previous best known kernels of size O(r!r(k-1)^r logW) for these problems.