On the extension complexity of scheduling

Abstract

Linear programming is a powerful method in combinatorial optimization with many applications in theory and practice. For solving a linear program quickly it is desirable to have a formulation of small size for the given problem. A useful approach for this is the construction of an extended formulation, which is a linear program in a higher dimensional space whose projection yields the original linear program. For many problems it is known that a small extended formulation cannot exist. However, most of these problems are either $\mathsf{NP}$-hard (like TSP), or only quite complicated polynomial time algorithms are known for them (like for the matching problem). In this work we study the minimum makespan problem on identical machines in which we want to assign a set of $n$ given jobs to $m$ machines in order to minimize the maximum load over the machines. We prove that the canonical formulation for this problem has extension complexity $2^{\Omega(n/\log n)}$, even if each job has size 1 or 2 and the optimal makespan is 2. This is a case that a trivial greedy algorithm can solve optimally! For the more powerful configuration integer program, we even prove a lower bound of $2^{\Omega(n)}$. On the other hand, we show that there is an abstraction of the configuration integer program admitting an extended formulation of size $f(\text{opt})\cdot \text{poly}(n,m)$. In addition, we give an $O(\log n)$-approximate integral formulation of polynomial size, even for arbitrary processing times and for the far more general setting of unrelated machines.