# Frontiers of Extended Formulations

## Final Report Summary - FOREFRONT (Frontiers of Extended Formulations)

Convex polytopes generalize to higher dimensions the familiar 2-dimensional polygons (triangles, quadrilaterals, pentagons, ...) and 3-dimensional polyhedra (tetrahedra, pyramids, cubes, octahedra, dodecahedra, ...). Surprisingly, they play a key role in the modern theory of discrete optimization, a research area that studies problems such as the shortest path problem. Nowadays, instances of the shortest path problem are quickly and reliably solved by route planning systems. Research in discrete optimization made that possible!

Discrete optimization problems have so many solutions that going through all solutions to select the best one is prohibitive. In order to find good solutions faster, we need to find shortcuts. For some problems, this can be done, and efficient algorithms can be found. For instance, this is the case of the shortest path problem. For the other problems, those for which no efficient algorithm is in sight, we believe that any algorithm has to perform a huge amount of work on some instances. Computational complexity was invented to classify the problems into "easy" and "hard".

For solving instances of hard problems, it is necessary to avoid complete enumeration. Sometimes, good bounds on the optimal value can be obtained and an optimal solution be found through partial enumeration. This is precisely where high-dimensional convex polytopes come into play. Using convex polytopes in combination with linear programming, huge instances of hard discrete optimization problems such as the famous traveling salesman problem are solved to optimality.

In order to formulate a discrete optimization problem as a linear program, one has to choose variables for which the constraints translate as linear constraints and the objective function as a linear function. In order to truly capture the problem, it is usually necessary to add the requirement that certain variables take only integer values. By allowing these variables to vary continuously, one obtains a relaxation of the problem. Solving the (continuous) linear program provides a bound on the optimal value of the problem.

The main goal of ERC project FOREFRONT was to study the tradeoff between the size of linear programming relaxations for a given problem and the quality of the bound it computes. For which problems can we write down a small linear program that solves the problem exactly? For which problems can't we get good bounds at all, even with enormous linear programs? What if the more powerful semidefinite programming is used instead of linear programming?

During the course of the project, the state of the art evolved tremendously, and ERC project FOREFRONT played a big role in this. We have a far much clearer picture of the field of linear programming relaxations (also known as extended formulations). We obtained precise tradeoffs for various key problems, invented novel ways to construct extended formulations, and explored the boundary between linear programming and semidefinite programming. The results obtained are unconditional, that is, do not depend on open conjectures in complexity theory such as P versus NP.