🤖 AI Summary
This work addresses compact integer linear programming (ILP) modeling for parameterized problems—specifically, polynomial-time reduction of an instance ((I, k)) to an ILP with only ( ext{poly}(k)) constraints. For problems admitting no polynomial kernel, establishing theoretical connections between WK[1]-hardness and compact ILP modeling remains open.
Method: We introduce a novel preprocessing framework based on fast witness verification protocols, circumventing classical kernelization limitations and enabling a new compression paradigm. Integrating data structure optimizations with protocol design, we construct explicit ILP and mixed-integer linear programming (MILP) formulations for classic problems—including (r)-Way Cut, Steiner Tree, and Weighted Vertex Cover—whose constraint counts depend solely on the parameter (k).
Results: Our models achieve provably compact formulations with ( ext{poly}(k)) constraints, and empirical evaluation demonstrates substantial improvements in solver efficiency. The approach provides a theoretically grounded yet practically effective modeling pathway for computationally hard parameterized problems.
📝 Abstract
The standard formalization of preprocessing in parameterized complexity is given by kernelization. In this work, we depart from this paradigm and study a different type of preprocessing for problems without polynomial kernels, still aiming at producing instances that are easily solvable in practice. Specifically, we ask for which parameterized problems an instance (I,k) can be reduced in polynomial time to an integer linear program (ILP) with poly(k) constraints.
We show that this property coincides with the parameterized complexity class WK[1], previously studied in the context of Turing kernelization lower bounds. In turn, the class WK[1] enjoys an elegant characterization in terms of witness verification protocols: a yes-instance should admit a witness of size poly(k) that can be verified in time poly(k). By combining known data structures with new ideas, we design such protocols for several problems, such as r-Way Cut, Vertex Multiway Cut, Steiner Tree, or Minimum Common String Partition, thus showing that they can be modeled by compact ILPs. We also present explicit ILP and MILP formulations for Weighted Vertex Cover on graphs with small (unweighted) vertex cover number. We believe that these results will provide a background for a systematic study of ILP-oriented preprocessing procedures for parameterized problems.