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Towards a Quantitative Theory of Integer Programming

Periodic Reporting for period 1 - QIP (Towards a Quantitative Theory of Integer Programming)

Reporting period: 2019-01-01 to 2020-06-30

Integer programming (IP), i.e. linear optimization with integrality constraints
on variables, is one of the most successful methods for solving large scale
optimization problems in practice. While many of the base IP problems such as
the traveling salesman problem (TSP) or satisfiability (SAT) are NP-Complete,
IPs with tens of thousands of variables are routinely solved in just a few hours
by current state of the art IP solvers.

The main goal of this proposal is to develop a quantitative theory capable of
explaining when and how well different IP solver techniques will work on a wide
range of instances. Here we will study many of the principal tools used to solve
IPs including branch \& bound, the simplex method, cutting planes and rounding
heuristics. Our first direction of study will be to develop parametrized classes
of instances, inspired by the structure of realistic models, on which branch &
bound and the simplex method are provably efficient. The second research
direction will be to develop alternatives to ad hoc rounding heuristics and
cutting plane selection strategies with provable guarantees and provide their
applications to important classes of IPs. Lastly, we will explore the power and
limitations of IP techniques in the context of algorithm design by comparing
them to powerful techniques in theoretical computer science and analyzing their
worst-case performance for solving general integer programs. While the main
thrust of this proposal is theoretical, it will be complimented by an
experimental component performed in collaboration with well-known experts in
computational IP, both to gain valuable insights on the structure of real-world
instances and to validate the effectiveness newly suggested approaches. The
proposed research is designed to make breakthroughs in our quantitative
understanding of IP techniques, many of which have long resisted theoretical
analysis.
"In this first period, my team has pursued work on three separate topics.

The first topic has been to more deeply understand fundamental systems for proving infeasibility of integer programs. This is a topic with deep connections to the field of proof complexity, which seeks to understand how difficult it is to demonstrate the infeasibility of unsatisfiable logical formulae using different systems of logical deduction. In this regard, we showed that a powerful integer programming strategy, known as general branching and cutting plane proofs, can be used to provide much shorter proofs of infeasibility for so-called Tseitin formulas than was expected to be possible. Our work refuted a conjecture due to Cook, Coullard and Turan from the 1987 and was recently awarded the best paper award at the 2020 conference on computational complexity (CCC 2020).

The second topic was to explore the power of interior point methods (IPM) for solving linear programs exactly. Interior points methods are the most theoretical efficient methods for solving linear programs, though has mostly been theoretically analyzed from the perspective of obtaining approximate instead of exact solutions. Exact solutions to linear programming approximations of integer programs (IP) are crucially needed by modern IP solvers, and thus understanding how to efficiently extract exact solutions from an IPM is of great importance. Our contribution in this line has been to improve the ""numerical complexity"" of an IPM solver. More precisely, we developed a novel IPM whose running time for producing exact solutions can be controlled by very refined measure of an LPs numerical complexity, which was conjectured to be sufficient by Monteiro and Tsuchiya in 2003.

The final research topic was to develop a new theoretical framework for an integer programming technique known as cutting planes. Cutting planes provide a way of automatically strengthening linear programming relaxations of integer programs and are crucial for the efficiency of modern IP solvers. From the theoretical perspective however, even proving basic convergence results for cutting plane methods has proven very challenging. Our work is addressing this difficulty by providing a novel framework for selecting cutting planes which automatically ensures convergence."
The results we have achieved so far in this project have already pushed us passed the state of the art in terms of our theoretical understanding of basic IP techniques such as branching and cutting planes. They has also provided provided new avenues for improvement for interior point methods for solving linear programs, which serve as crucial relaxations in the context of integer programming. In the future, we expect the theoretical progress to continue and to use the developed methods to help improve modern IP solvers.
Integer Program