Equivalent Instances for Scheduling and Packing Problems

2025-12-11

AI Summary

This work addresses the construction of equivalent instances for parameterized scheduling and bin-packing problems—specifically integer linear programming (ILP), knapsack, and load balancing—aiming to achieve polynomial-size static equivalent instances that enable parameterized tractability via compression. Leveraging FPT algorithms, the ConfILP framework, the Eisenbrand–Weismantel proximity theorem, and a balanced partitioning lemma, we derive the first $O(d^2 log p_{max})$-size equivalent instance for load balancing. We improve the $ell_1$-norm bound on equivalent ILP vectors, yielding a tight $O(n^2 log n)$-size equivalent instance for knapsack and an $O(MN^2 log(NU))$-size equivalent instance for feasibility ILPs; additionally, we obtain an $O(M^2 N log(NMDelta))$ kernel for general ILPs. These results collectively advance kernelization efficiency and compression accuracy across multiple fundamental combinatorial optimization problems.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Constraint Programming
• Data Mining & Knowledge Management: Data Compression

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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 $O(MN^2log(NU))$ 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 $O(n^2log(n))$ static equivalent instance for Knapsack where $n$ is the number of items. Moreover, we give an $O(M^2Nlog(NMΔ))$ 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 $O(d^2log(p_{max}))$ 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.

Problem

• Construct equivalent instances for scheduling and packing problems
• Improve bounds for equivalent vectors in integer linear programs
• Provide kernels for feasibility ILPs and load balancing generalizations

Innovation

• Construct equivalent instances for scheduling and packing problems
• Improve bound for equivalent vector norm in integer linear programs
• Provide kernel for feasibility ILPs using balancing and proximity results