[Paper Review] The Computational Complexity of Integer Programming with Alternations

Danny Nguyen, Igor Pak|arXiv (Cornell University)|Jan 1, 2017

Complexity and Algorithms in Graphs7 references7 citations

TL;DR

This paper establishes that integer programming with three alternating quantifiers (∃x∀y∃z) is NP-complete even with a fixed number of variables, resolving a long-standing open question. The authors prove this via a reduction from the NP-complete Good Simultaneous Approximation (GSA) problem, and further show that counting integer points in the projection of the difference of two 3D polytopes (Q\P) is #P-complete, contrasting with Barvinok and Woods' polynomial-time result for projections of convex sets.

ABSTRACT

We prove that integer programming with three alternating quantifiers is NP-complete, even for a fixed number of variables. This complements earlier results by Lenstra and Kannan, which together say that integer programming with at most two alternating quantifiers can be done in polynomial time for a fixed number of variables. As a byproduct of the proof, we show that for two polytopes P, Q in R^4, counting the projection of integer points in Q\P is #P-complete. This contrasts the 2003 result by Barvinok and Woods, which allows counting in polynomial time the projection of integer points in P and Q separately.

Motivation & Objective

• To resolve Kannan's 1992 question on whether integer programming with three alternating quantifiers is decidable in polynomial time.
• To establish the computational complexity of integer programming with three alternating quantifiers in fixed dimension.
• To analyze the complexity of counting integer points in projections of non-convex sets, particularly Q\P for 3D polytopes.
• To contrast the tractability of counting projections of convex polytopes (Barvinok-Woods) with the intractability of projections of their complements.

Proposed method

• Reduction from the NP-complete Good Simultaneous Approximation (GSA) problem to prove NP-completeness of 3-alternation integer programming.
• Use of geometric constructions based on Fibonacci numbers to encode number-theoretic constraints into polyhedral systems.
• Application of the Doignon–Bell–Scarf theorem to reduce systems with many inequalities to finitely many subproblems in the two-quantifier case.
• Proof that the conjunction of subproblems commutes with universal quantifiers in the two-quantifier case, but fails in the three-quantifier case, breaking the reduction.
• Construction of polytopes in R^3 whose projection difference Q\P encodes hard counting problems.
• Use of the pigeonhole principle and convex position arguments to prove tightness of dimension bounds in the projection complexity result.

Experimental results

Research questions

• RQ1 Is integer programming with three alternating quantifiers in fixed dimension decidable in polynomial time?
• RQ2 Can the projection of the difference of two 3D polytopes Q\P be counted in polynomial time?
• RQ3 Does the presence of disjunctions in Presburger formulas provide a computational advantage over pure inequality systems in integer programming?
• RQ4 Is the complexity of projecting integer points preserved under set complementation in fixed dimension?
• RQ5 Can short generating functions efficiently represent projections of non-convex sets like Q\P?

Key findings

• Integer programming with three alternating quantifiers (∃x∀y∃z) is NP-complete, even when the number of variables is fixed.
• Counting the number of integer points in the vertical projection of Q\P for 3D polytopes Q and P is #P-complete.
• The NP-completeness result holds even when P is an interval and Q is a rectangle, indicating hardness persists in low-dimensional settings.
• The problem remains NP-complete even when the number of inequalities is unbounded, but the number of variables is fixed.
• The complement of a convex polytope (Q\P) can encode hard counting problems, making its projection computationally intractable despite convexity of P and Q.
• The result shows that the Barvinok-Woods polynomial-time algorithm for projecting convex polytopes cannot be extended to their complements.