Approximations and Hardness of Packing Partially Ordered Items
arxiv(2024)
摘要
Motivated by applications in production planning and storage allocation in
hierarchical databases, we initiate the study of covering partially ordered
items (CPO). Given a capacity k ∈ℤ^+, and a directed graph
G=(V,E) where each vertex has a size in {0,1, …,k}, we seek a
collection of subsets of vertices S_1, …, S_m that cover all the
vertices, such that for any 1 ≤ j ≤ m, the total size of vertices in
S_j is bounded by k, and there are no edges from V ∖ S_j to
S_j. The objective is to minimize the number of subsets m. CPO is closely
related to the rule caching problem (RCP) that is of wide interest in the
networking area. The input for RCP is a directed graph G=(V,E), a profit
function p:V →ℤ_0^+, and k ∈ℤ^+. The
output is a subset S ⊆ V of maximum profit such that |S| ≤ k and
there are no edges from V ∖ S to S.
Our main result is a 2-approximation algorithm for CPO on out-trees,
complemented by an asymptotic 1.5-hardness of approximation result. We also
give a two-way reduction between RCP and the densest k-subhypergraph problem,
surprisingly showing that the problems are equivalent w.r.t. polynomial-time
approximation within any factor ρ≥ 1. This implies that RCP cannot be
approximated within factor |V|^1- for any fixed >0, under
standard complexity assumptions. Prior to this work, RCP was just known to be
strongly NP-hard. We further show that there is no EPTAS for the special case
of RCP where the profits are uniform, assuming Gap-ETH. Since this variant
admits a PTAS, we essentially resolve the complexity status of this problem.
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