A greedy approximation algorithm for minimum-gap scheduling

Marek Chrobak, Uriel Feige, Mohammad Taghi Hajiaghayi, Sanjeev Khanna, Fei Li, Seffi Naor

Abstract: We consider scheduling of unit-length jobs with release times and deadlines, where the objective is to minimize the number of gaps in the schedule. Polynomial-time algorithms for this problem are known, yet they are rather inefficient, with the best algorithm running in time $$O(n^4)$$ O ( n 4 ) and requiring $$O(n^3)$$ O ( n 3 ) memory. We present a greedy algorithm that approximates the optimum solution within a factor of 2 and show that our analysis is tight. Our algorithm runs in time $$O(n^2 \log n)$$ O ( n 2 log n ) and needs only O(n) memory. In fact, the running time is $$O(n (g^*+1)\log n)$$ O ( n ( g ∗ + 1 ) log n ) , where $$g^*$$ g ∗ is the minimum number of gaps.