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A Combinatorial Approximation Algorithm for Graph Balancing with Light Hyper Edges

Makespan minimization in restricted assignment $(R|p_{ij}\in \{p_j, \infty\}|C_{\max})$ is a classical problem in the field of machine scheduling. In a landmark paper in 1990 [8], Lenstra, Shmoys, and Tardos gave a 2-approximation algorithm and proved that the problem cannot be approximated within 1.5 unless P=NP. The upper and lower bounds of the problem have been essentially unimproved in the intervening 25 years, despite several remarkable successful attempts in some special cases of the problem [2,4,12] recently. In this paper, we consider a special case called graph-balancing with light hyper edges, where heavy jobs can be assigned to at most two machines while light jobs can be assigned to any number of machines. For this case, we present algorithms with approximation ratios strictly better than 2. Specifically, Two job sizes: Suppose that light jobs have weight $w$ and heavy jobs have weight $W$, and $w < W$. We give a $1.5$-approximation algorithm (note that the current 1.5 lower bound is established in an even more restrictive setting [1,3]). Indeed, depending on the specific values of $w$ and $W$, sometimes our algorithm guarantees sub-1.5 approximation ratios. Arbitrary job sizes: Suppose that $W$ is the largest given weight, heavy jobs have weights in the range of $(βW, W]$, where $4/7\leq β< 1$, and light jobs have weights in the range of $(0,βW]$. We present a $(5/3+β/3)$-approximation algorithm. Our algorithms are purely combinatorial, without the need of solving a linear program as required in most other known approaches.