Equivalent Instances for Scheduling and Packing Problems
Klaus Jansen, Kai Kahler, Corinna Wambsganz
TL;DR
The paper studies static equivalent instances for scheduling and packing problems, formalizing equivalence notions that preserve solution sets under efficient transformations. It sharpens the coefficient-reduction toolkit, notably improving the $\\ell_1$-norm bound for equivalent vectors, and derives small static equivalents for Knapsack, SubsetSum, and various ILP variants. It further provides a kernel for feasibility ILPs and, via balancing techniques, an compact equivalent for LoadBalancing on identical machines. These results advance kernelization and FPT-style preprocessing by enabling substantial instance compression while preserving solvability, albeit with exponential-time constructibility in general.
Abstract
Two instances $(I,k)$ and $(I',k')$ of a parameterized problem $P$ are equivalent if they have the same set of solutions (static equivalent) or if the set of solutions of $(I,k)$ can be constructed by the set of solutions for $(I',k')$ and some computable pre-solutions. If the algorithm constructing such a (static) equivalent instance whose size is polynomial bounded runs in fixed-parameter tractable (FPT) time, we say that there exists a (static) equivalent instance for this problem. In this paper we present (static) equivalent instances for Scheduling and Knapsack problems. We improve the bound for the $\ell_1$-norm of an equivalent vector given by Eisenbrand, Hunkenschröder, Klein, Koutecký, Levin, and Onn and show how this yields equivalent instances for integer linear programs (ILPs) and related problems. We obtain an $O(MN^2\log(NU))$ static equivalent instance for feasibility ILPs where $M$ is the number of constraints, $N$ is the number of variables and $U$ is an upper bound for the $\ell_\infty$-norm of the smallest feasible solution. With this, we get an $O(n^2\log(n))$ static equivalent instance for Knapsack where $n$ is the number of items. Moreover, we give an $O(M^2N\log(NMΔ))$ kernel for feasibility ILPs where $Δ$ is an upper bound for the $\ell_\infty$-norm of the given constraint matrix. Using balancing results by Knop et al., the ConfILP and a proximity result by Eisenbrand and Weismantel we give an $O(d^2\log(p_{\max}))$ equivalent instance for LoadBalancing, a generalization of scheduling problems. Here $d$ is the number of different processing times and $p_{\max}$ is the largest processing time.