On‐line machine covering

Online Bounded Analysis

“…For two related machines with speed ratio s, we present an algorithm with an online-bounded ratio better than 1 s and show that no online algorithm has a higher online-bounded ratio. For this problem, it is known that the best possible competitive ratio for identical machines is 1 m , and the best possible competitive ratio for two related machines is 1 s+1 [35,5,22]. For classic bin packing, we show that any Any-Fit algorithm has an online-bounded ratio of at least 3 2 .…”

“…For First-Fit, the corollary follows from Theorem 11, since Corollary 1 and Theorem 4 in [13] imply that First-Fit's competitive ratio on accommodating sequences is 5 8 . For Worst-Fit, the corollary follows from Theorem 11, since Theorems 1 and 5 in [13] imply that Worst-Fit's competitive ratio on accommodating sequences is 1 2 .…”

“…Corollary 2 Both Best-Fit and First-Fit have online-bounded ratios of 5 8 . Worst-Fit has an online-bounded ratio of 1 2 .…”

Online Bounded Analysis

“…For two related machines with speed ratio s, we present an algorithm with an online-bounded ratio better than 1 s and show that no online algorithm has a higher online-bounded ratio. For this problem, it is known that the best possible competitive ratio for identical machines is 1 m , and the best possible competitive ratio for two related machines is 1 s+1 [35,5,22]. For classic bin packing, we show that any Any-Fit algorithm has an online-bounded ratio of at least 3 2 .…”

“…For First-Fit, the corollary follows from Theorem 11, since Corollary 1 and Theorem 4 in [13] imply that First-Fit's competitive ratio on accommodating sequences is 5 8 . For Worst-Fit, the corollary follows from Theorem 11, since Theorems 1 and 5 in [13] imply that Worst-Fit's competitive ratio on accommodating sequences is 1 2 .…”

“…Corollary 2 Both Best-Fit and First-Fit have online-bounded ratios of 5 8 . Worst-Fit has an online-bounded ratio of 1 2 .…”

Fair Division of Indivisible Goods: Recent Progress and Open Questions

“…It remains an open question whether there exist competitive online algorithms for the computation of (approximately) MMS or Prop1/PropX allocations. For the case of identical valuation functions, approximate MMS allocations correspond to maximizing the minimum load on the job scheduling problem, for which optimal competitive ratios have been proved by Azar and Epstein [1997] and Tan and Wu [2007].…”

Fair Allocation of Indivisible Items with Conflict Graphs

“…For k ≥ 3, one can simply add jobs of length one half of the sum of weights in the instance of Partition. If k is not fixed, but part of the input, the same scheduling problem is strongly NP-hard as mentioned in [8] (a PTAS was derived in [64]). In fact, an instance of the strongly NP-complete 3-Partition problem with 3m elements and target bound B could be decided by any algorithm for the scheduling problem with n = 3m jobs, k = m machines and a desired minimal completion time equal to B.…”

Online Bounded Analysis

“…For two related machines with speed ratio s, we present an algorithm with an online-bounded ratio better than 1 s and show that no online algorithm has a higher online-bounded ratio. For this problem, it is known that the best possible competitive ratio for identical machines is 1 m , and the best possible competitive ratio for two related machines is 1 s+1 [35,5,22]. For classic bin packing, we show that any Any-Fit algorithm has an online-bounded ratio of at least 3 2 .…”

“…For First-Fit, the corollary follows from Theorem 11, since Corollary 1 and Theorem 4 in [13] imply that First-Fit's competitive ratio on accommodating sequences is 5 8 . For Worst-Fit, the corollary follows from Theorem 11, since Theorems 1 and 5 in [13] imply that Worst-Fit's competitive ratio on accommodating sequences is 1 2 .…”

“…Corollary 2 Both Best-Fit and First-Fit have online-bounded ratios of 5 8 . Worst-Fit has an online-bounded ratio of 1 2 .…”

Fair Division of Indivisible Goods: Recent Progress and Open Questions

“…It remains an open question whether there exist competitive online algorithms for the computation of (approximately) MMS or Prop1/PropX allocations. For the case of identical valuation functions, approximate MMS allocations correspond to maximizing the minimum load on the job scheduling problem, for which optimal competitive ratios have been proved by Azar and Epstein [1997] and Tan and Wu [2007].…”

Fair Allocation of Indivisible Items with Conflict Graphs

“…For k ≥ 3, one can simply add jobs of length one half of the sum of weights in the instance of Partition. If k is not fixed, but part of the input, the same scheduling problem is strongly NP-hard as mentioned in [8] (a PTAS was derived in [64]). In fact, an instance of the strongly NP-complete 3-Partition problem with 3m elements and target bound B could be decided by any algorithm for the scheduling problem with n = 3m jobs, k = m machines and a desired minimal completion time equal to B.…”

Online Bounded Analysis

“…For two related machines with speed ratio s, we present an algorithm with an online-bounded ratio better than 1 s and show that no online algorithm has a higher online-bounded ratio. For this problem, it is known that the best possible competitive ratio for identical machines is 1 m , and the best possible competitive ratio for two related machines is 1 s+1 [35,5,22]. For classic bin packing, we show that any Any-Fit algorithm has an online-bounded ratio of at least 3 2 .…”

“…For First-Fit, the corollary follows from Theorem 11, since Corollary 1 and Theorem 4 in [13] imply that First-Fit's competitive ratio on accommodating sequences is 5 8 . For Worst-Fit, the corollary follows from Theorem 11, since Theorems 1 and 5 in [13] imply that Worst-Fit's competitive ratio on accommodating sequences is 1 2 .…”

“…Corollary 2 Both Best-Fit and First-Fit have online-bounded ratios of 5 8 . Worst-Fit has an online-bounded ratio of 1 2 .…”

Fair Division of Indivisible Goods: Recent Progress and Open Questions

“…It remains an open question whether there exist competitive online algorithms for the computation of (approximately) MMS or Prop1/PropX allocations. For the case of identical valuation functions, approximate MMS allocations correspond to maximizing the minimum load on the job scheduling problem, for which optimal competitive ratios have been proved by Azar and Epstein [1997] and Tan and Wu [2007].…”

Fair Allocation of Indivisible Items with Conflict Graphs

“…For k ≥ 3, one can simply add jobs of length one half of the sum of weights in the instance of Partition. If k is not fixed, but part of the input, the same scheduling problem is strongly NP-hard as mentioned in [8] (a PTAS was derived in [64]). In fact, an instance of the strongly NP-complete 3-Partition problem with 3m elements and target bound B could be decided by any algorithm for the scheduling problem with n = 3m jobs, k = m machines and a desired minimal completion time equal to B.…”