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Curvature batching gives single-exponential integer quadratic programming

Cinar Ari, Robert Hildebrand

Abstract

Integer Quadratic Programming (IQP), $\min\{x^T Q x + c^T x : Ax \le b,\, x\in\Z^n\}$, is a fundamental problem in combinatorial optimization. While the convex and concave special cases admit polynomial-time algorithms for fixed~$n$, the general indefinite case is considerably harder: it was only recently shown to lie in NP, and the FPT algorithm, due to Lokshtanov, establishes fixed-parameter tractability parameterized by $n$ and the largest coefficient~$L$ without giving an explicit running time. We give the first single-exponential algorithm for IQP, solving it in time $ \bigl(n\,L^n_A\,Δ(A)\,L_Q\bigr)^{O(n)}\cdot\mathrm{poly}(\varphi), $ which is $(nL)^{O(n^2)}\cdot\mathrm{poly}(\varphi)$ in general using the same parameterization. We achieve improvements for structured cases like total unimodularity and further state explicit complexity results for a number of FPT algorithms and optimization problems. The single-exponential bound is achieved via curvature batching: we classify kernel directions by the sign of their quadratic curvature and observe that when no negative-curvature direction exists, all gradient constraints can be imposed simultaneously in a single batch. This replaces the chain of determinant squarings inherent in sequential branching with a single polynomial inflation, after which the remaining problem is an ILP. As a secondary contribution, we give an explicit bound for concave integer minimization over a polytope $\{Ax \le b\} \cap \Z^n$ whose parametric complexity depends only on the constraint matrix~$A$ and is independent of the right-hand side~$b$.

Curvature batching gives single-exponential integer quadratic programming

Abstract

Integer Quadratic Programming (IQP), , is a fundamental problem in combinatorial optimization. While the convex and concave special cases admit polynomial-time algorithms for fixed~, the general indefinite case is considerably harder: it was only recently shown to lie in NP, and the FPT algorithm, due to Lokshtanov, establishes fixed-parameter tractability parameterized by and the largest coefficient~ without giving an explicit running time. We give the first single-exponential algorithm for IQP, solving it in time which is in general using the same parameterization. We achieve improvements for structured cases like total unimodularity and further state explicit complexity results for a number of FPT algorithms and optimization problems. The single-exponential bound is achieved via curvature batching: we classify kernel directions by the sign of their quadratic curvature and observe that when no negative-curvature direction exists, all gradient constraints can be imposed simultaneously in a single batch. This replaces the chain of determinant squarings inherent in sequential branching with a single polynomial inflation, after which the remaining problem is an ILP. As a secondary contribution, we give an explicit bound for concave integer minimization over a polytope whose parametric complexity depends only on the constraint matrix~ and is independent of the right-hand side~.

Paper Structure

This paper contains 23 sections, 19 theorems, 34 equations, 2 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Literature review
  3. Approach
  4. Preliminaries
  5. Lokshtanov's algorithm
  6. Further notation
  7. Sequential bound
  8. Adjugate basis construction
  9. Positive eigenvalue bound on gradient steps
  10. Subdeterminant growth
  11. Running time analysis
  12. Curvature batch: proof of Theorem \ref{['thm:main']}
  13. The curvature-batch algorithm
  14. Correctness and post-batch structure
  15. Running time
  16. ...and 8 more sections

Key Result

Theorem 1.1

IQP eq:iqp can be solved in time where $L_A = \|A\|_{max}$, $L_Q = \|Q\|_{max}$, and $\Delta(A)$ denote the maximum absolute value of any subdeterminant of $A$. Since $\Delta(A)\le (nL_A)^n$, this is $(nL)^{O(n^2)}\cdot\mathrm{poly}(\varphi)$ where $L=\max(L_A,L_Q)$. $\blacktriangleleft$$\blacktriangleleft$

Figures (2)

  • Figure 1: Visual representation of branches
  • Figure 2: Batch branching.

Theorems & Definitions (37)

  • Theorem 1.1: Curvature batch bound
  • Corollary 1.2: TU constraints
  • Lemma 2.1: Structural branching rule Lokshtanov17
  • Theorem 3.1: Sequential bound
  • Lemma 3.2: Adjugate basis construction
  • proof
  • Lemma 3.3: Deep optimality forces non-negative curvature
  • proof
  • Lemma 3.4: Gradient branching characterization
  • proof
  • ...and 27 more