Polynomial time approximation algorithms for machine scheduling: ten open problems

Approximating Precedence-Constrained Single Machine Scheduling by Coloring

“…For the general version of 1|prec| j w j C j , closing the approximability gap is considered a longstanding open problem in scheduling theory (see e.g. [21]). …”

On the Approximability of Single-Machine Scheduling with Precedence Constraints

“…More precisely, in §9, we establish a connection between 1 prec w j C j and the maximum edge biclique problem, which proves that this problem does not admit a PTAS unless all problems in the complexity class NP can be solved by probabilistic algorithms of subexponential running time. This result makes a first step toward closing the approximability gap for this scheduling problem, a prominent problem in scheduling theory (see Schuurman and Woeginger [36]). Subsequent to our work, Bansal and Khot [6] showed that the gap indeed closes, assuming a variant of the unique games conjecture (Khot [21]), by providing a 2-inapproximability result based on that assumption.…”

“…For the general version of 1 prec w j C j , closing the approximability gap has been listed as one of ten outstanding open problems in scheduling theory (see, e.g., Schuurman and Woeginger [36]). On the positive side, several polynomial-time 2-approximation algorithms are known.…”

“…Subsequent to our work, Bansal and Khot [6] showed that the gap indeed closes, assuming a variant of the unique games conjecture (Khot [21]), by providing a 2-inapproximability result based on that assumption. Furthermore, this year, the fourth author (Svensson [40]) established an interesting relationship between 1 prec w j C j and the problem P prec C max by showing that a 2 − -hardness for 1 prec w j C j would imply essentially the same hardness result for P prec C max and would thus settle another prominent open question listed in Schuurman and Woeginger [36]. This underscores the importance of understanding the approximability of 1 prec w j C j .…”

Approximability of Average Completion Time Scheduling on Unrelated Machines

“…Such an algorithm is called a (randomized) α-approximation algorithm. A paper by Schuurman and Woeginger [18] of 1999 gives an interesting (and still accurate) overview of scheduling problems for which the approximability is unknown. Arguably, the most interesting are the ones with precedence constraints.…”