[Paper Review] On the Optimality of Pseudo-polynomial Algorithms for Integer Programming

Fedor V. Fomin, Fahad Panolan+2 more|arXiv (Cornell University)|Jan 1, 2018

Formal Methods in Verification5 citations

TL;DR

This paper establishes tight conditional lower bounds for integer programming (IP) under the Exponential Time Hypothesis (ETH), showing that recent pseudo-polynomial algorithms—particularly those by Jansen and Rohwedder—are nearly optimal. It proves that solving IP with a constant number of constraints cannot be done in time $ n^{o(m / \log m)} \cdot \|b\|_\infty^{o(m)} $, even for non-negative matrices, and provides matching upper and lower bounds when the path-width of the column-matroid is bounded, resolving a long-standing complexity question in parameterized IP.

ABSTRACT

In the classic Integer Programming (IP) problem, the objective is to decide whether, for a given m x n matrix A and an m-vector b=(b_1,..., b_m), there is a non-negative integer n-vector x such that Ax=b. Solving (IP) is an important step in numerous algorithms and it is important to obtain an understanding of the precise complexity of this problem as a function of natural parameters of the input. The classic pseudo-polynomial time algorithm of Papadimitriou [J. ACM 1981] for instances of (IP) with a constant number of constraints was only recently improved upon by Eisenbrand and Weismantel [SODA 2018] and Jansen and Rohwedder [ArXiv 2018]. We continue this line of work and show that under the Exponential Time Hypothesis (ETH), the algorithm of Jansen and Rohwedder is nearly optimal. We also show that when the matrix A is assumed to be non-negative, a component of Papadimitriou's original algorithm is already nearly optimal under ETH. This motivates us to pick up the line of research initiated by Cunningham and Geelen [IPCO 2007] who studied the complexity of solving (IP) with non-negative matrices in which the number of constraints may be unbounded, but the branch-width of the column-matroid corresponding to the constraint matrix is a constant. We prove a lower bound on the complexity of solving (IP) for such instances and obtain optimal results with respect to a closely related parameter, path-width. Specifically, we prove matching upper and lower bounds for (IP) when the path-width of the corresponding column-matroid is a constant.

Motivation & Objective

• To establish conditional lower bounds on the complexity of integer programming (IP) under the Exponential Time Hypothesis (ETH).
• To analyze the optimality of pseudo-polynomial algorithms for IP with a constant number of constraints.
• To resolve the complexity of IP when the path-width of the column-matroid of the constraint matrix is bounded.
• To extend prior work by Cunningham and Geelen on branch-width and IP to path-width, providing tight bounds.

Proposed method

• Uses the Exponential Time Hypothesis (ETH) and Strong Exponential Time Hypothesis (SETH) to derive conditional lower bounds on IP algorithms.
• Reduces known hard problems (e.g., 3-SUM) to IP instances to establish lower bounds via problem equivalence.
• Applies the Steinitz lemma and proximity arguments to bound solution sizes and guide algorithmic design.
• Introduces a constructive algorithm for computing path-decompositions of matroids with bounded path-width.
• Employs a reduction from 3-SUM′ to show that improvements in IP runtime would imply breakthroughs in 3-SUM.
• Proves matching upper and lower bounds for IP when path-width is constant, using dynamic programming on path-decompositions.

Experimental results

Research questions

• RQ1 Can the running time of pseudo-polynomial IP algorithms be significantly improved under ETH?
• RQ2 Is the algorithm of Jansen and Rohwedder optimal up to sub-exponential factors?
• RQ3 What is the precise complexity of IP when the path-width of the column-matroid is bounded by a constant?
• RQ4 Can the dependence on $ \|b\|_\infty $ in IP algorithms be reduced below $ \|b\|_\infty^{o(m)} $ under ETH?
• RQ5 Is there a super-polynomial gap between the complexity of IP with bounded branch-width and bounded path-width?

Key findings

• Under ETH, no algorithm can solve IP with m constraints in time $ n^{o(m / \log m)} \cdot \|b\|_\infty^{o(m)} $, even for non-negative matrices.
• The algorithm of Jansen and Rohwedder is nearly optimal, as any improvement would violate ETH.
• For non-negative matrices, the component of Papadimitriou’s algorithm that bounds solution size is already nearly optimal under ETH.
• When the path-width of the column-matroid is bounded by a constant, IP admits matching upper and lower bounds of the form $ (\|b\|_\infty + 1)^{O(\text{path-width})} \cdot n^{O(1)} $.
• The result implies that further improvements in IP runtime for bounded path-width instances would require a fundamentally new algorithmic approach.
• The paper shows that a super-polynomial improvement in 3-SUM would imply a similar improvement in IP algorithms, indicating strong complexity barriers.