Faster Algorithms for Sparse ILP and Hypergraph Multi-Packing/Multi-Cover Problems
Dmitry Gribanov, Dmitry Malyshev, Nikolai Zolotykh
TL;DR
This work advances sparse ILP methodology by deriving faster exponential algorithms for feasibility, counting, and optimization over integer points in sparse polyhedra and by unifying a broad class of Edge/Vertex Multi-Packing/Multi-Cover problems on graphs and hypergraphs. The authors develop a generating-function framework centered on Brion's theorem and tangent-cone DP, together with SNF-based group structure and randomized witness vectors, to obtain tight complexity bounds expressed through sparsity/determinant parameters such as $\nu(A)$, $\Delta(A)$, and $\overline{ts}(A)$. They achieve improved counting bounds (e.g., $O(\nu^2 d^4 \Delta^3)$) and provide parameterized algorithms for co-dimension $k$ problems, box-constrained standard forms, and box-free canonical forms, often beating the best-known results for sparse instances. The paper also shows practical impact by applying the theory to hypergraph multi-packing/multi-cover problems, highlighting algorithms parameterized by the number of vertices and offering open questions about polynomial-time bounds for certain multiplicity settings. Overall, the work significantly tightens the frontier for sparse ILP and related combinatorial optimization problems with implications for both theory and applications in graph/hypergraph packing and covering.
Abstract
In our paper, we consider the following general problems: check feasibility, count the number of feasible solutions, find an optimal solution, and count the number of optimal solutions in $P \cap Z^n$, assuming that $P$ is a polyhedron, defined by systems $A x \leq b$ or $Ax = b,\, x \geq 0$ with a sparse matrix $A$. We develop algorithms for these problems that outperform state of the art ILP and counting algorithms on sparse instances with bounded elements. We use known and new methods to develop new exponential algorithms for Edge/Vertex Multi-Packing/Multi-Cover Problems on graphs and hypergraphs. This framework consists of many different problems, such as the Stable Multi-set, Vertex Multi-cover, Dominating Multi-set, Set Multi-cover, Multi-set Multi-cover, and Hypergraph Multi-matching problems, which are natural generalizations of the standard Stable Set, Vertex Cover, Dominating Set, Set Cover, and Maximal Matching problems.