The Replenishment Schedule to Minimize Peak Storage Problem: The Gap Between the Continuous and Discrete Versions of the Problem
The replenishment storage problem (RSP) is to minimize the storage capacity requirement for a deterministic demand, multi-item inventory system, where each item has a given reorder size and cycle length. We consider the discrete RSP, where reorders can only take place at an integer time unit within the cycle. Discrete RSP was shown to be NP-hard for constant joint cycle length (the least common multiple of the length of all individual cycles). We show here that discrete RSP is weakly NP-hard for constant joint cycle length and prove that it is strongly NP-hard for nonconstant joint cycle length. For constant joint cycle-length discrete RSP, we further present a pseudopolynomial time algorithm that solves the problem optimally and the first known fully polynomial time approximation scheme (FPTAS) for the single-cycle RSP. The scheme is utilizing a new integer programming formulation of the problem that is introduced here. For the strongly NP-hard RSP with nonconstant joint cycle length, we provide a polynomial time approximation scheme (PTAS), which for any fixed [Formula: see text], provides a linear time [Formula: see text] approximate solution. The continuous RSP, where reorders can take place at any time within a cycle, seems (with our results) to be easier than the respective discrete problem. We narrow the previously known complexity gap between the continuous and discrete versions of RSP for the multi-cycle RSP (with either constant or nonconstant cycle length) and the single-cycle RSP with constant cycle length and widen the gap for single-cycle RSP with nonconstant cycle length. For the multi-cycle case and constant joint cycle length, the complexity status of continuous RSP is open, whereas it is proved here that the discrete RSP is weakly NP-hard. Under our conjecture that the continuous RSP is easier than the discrete one, this implies that continuous RSP on multi-cycle and constant joint cycle length (currently of unknown complexity status) is at most weakly NP-hard.  more » « less
Award ID(s):
NSF-PAR ID:
10357236
Author(s) / Creator(s):
;
Date Published:
Journal Name:
Operations Research
Volume:
67
Issue:
5
ISSN:
0030-364X
Page Range / eLocation ID:
1345 to 1361
Format(s):
Medium: X
4. Establishing the complexity of {\em Bounded Distance Decoding} for Reed-Solomon codes is a fundamental open problem in coding theory, explicitly asked by Guruswami and Vardy (IEEE Trans. Inf. Theory, 2005). The problem is motivated by the large current gap between the regime when it is NP-hard, and the regime when it is efficiently solvable (i.e., the Johnson radius). We show the first NP-hardness results for asymptotically smaller decoding radii than the maximum likelihood decoding radius of Guruswami and Vardy. Specifically, for Reed-Solomon codes of length $N$ and dimension $K=O(N)$, we show that it is NP-hard to decode more than $N-K- c\frac{\log N}{\log\log N}$ errors (with $c>0$ an absolute constant). Moreover, we show that the problem is NP-hard under quasipolynomial-time reductions for an error amount $> N-K- c\log{N}$ (with $c>0$ an absolute constant). An alternative natural reformulation of the Bounded Distance Decoding problem for Reed-Solomon codes is as a {\em Polynomial Reconstruction} problem. In this view, our results show that it is NP-hard to decide whether there exists a degree $K$ polynomial passing through $K+ c\frac{\log N}{\log\log N}$ points from a given set of points $(a_1, b_1), (a_2, b_2)\ldots, (a_N, b_N)$. Furthermore, it is NP-hard under quasipolynomial-time reductions to decide whether there is a degree $K$ polynomial passing through $K+c\log{N}$ many points. These results follow from the NP-hardness of a generalization of the classical Subset Sum problem to higher moments, called {\em Moments Subset Sum}, which has been a known open problem, and which may be of independent interest. We further reveal a strong connection with the well-studied Prouhet-Tarry-Escott problem in Number Theory, which turns out to capture a main barrier in extending our techniques. We believe the Prouhet-Tarry-Escott problem deserves further study in the theoretical computer science community.