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Generalized Unrelated Machine Scheduling Problem

We study the generalized load-balancing (GLB) problem, where we are given $n$ jobs, each of which needs to be assigned to one of $m$ unrelated machines with processing times $\{p_{ij}\}$. Under a job assignment $σ$, the load of each machine $i$ is $ψ_i(\mathbf{p}_{i}[σ])$ where $ψ_i:\mathbb{R}^n\rightarrow\mathbb{R}_{\geq0}$ is a symmetric monotone norm and $\mathbf{p}_{i}[σ]$ is the $n$-dimensional vector $\{p_{ij}\cdot \mathbf{1}[σ(j)=i]\}_{j\in [n]}$. Our goal is to minimize the generalized makespan $ϕ(\mathsf{load}(σ))$, where $ϕ:\mathbb{R}^m\rightarrow\mathbb{R}_{\geq0}$ is another symmetric monotone norm and $\mathsf{load}(σ)$ is the $m$-dimensional machine load vector. This problem significantly generalizes many classic optimization problems, e.g., makespan minimization, set cover, minimum-norm load-balancing, etc. We obtain a polynomial time randomized algorithm that achieves an approximation factor of $O(\log n)$, matching the lower bound of set cover up to constant factor. We achieve this by rounding a novel configuration LP relaxation with exponential number of variables. To approximately solve the configuration LP, we design an approximate separation oracle for its dual program. In particular, the separation oracle can be reduced to the norm minimization with a linear constraint (NormLin) problem and we devise a polynomial time approximation scheme (PTAS) for it, which may be of independent interest.