Hello Stack Overflow community,

I’m tackling a homework question for an algorithms course that involves a bin packing problem where item weights are rounded up, and I’m struggling with the second part of the assignment.

Problem Overview:

• Each item u from the original set U has a weight w_u >= δ.
• We round up each weight to the nearest multiple of δ/k, resulting in a new weight w_u' for each item.
• This rounding can potentially push the total weight in a bin over its capacity (each bin has a capacity of 1 unit).

Challenge:
The first question was straightforward, but the second question is proving difficult. I need to derive the relationship between OPT(U') and OPT(U), where:

• OPT(U) is the minimal number of bins needed without rounding.
• OPT(U') is the minimal number of bins needed with rounding.

Current Understanding:
Given the worst-case scenario where every item w_u is just below a rounding threshold, most items are rounded up nearly by δ/k. Moreover, each bin is almost full, so adding δ/k weight often necessitates opening a new bin.

Formulation Attempt:
∑(u’ ∈ U’) w_u’ ≤ ∑(u ∈ U) w_u + n * (δ/k) = |OPT(U)| + n * (δ/k) ≤ |OPT(U)| + |OPT(U)|/k

I hypothesize that:
|OPT(U’)| ≤ |OPT(U)| + |OPT(U)|/k

We know that the total rounded weight of all items in the new set U' is:
∑(u’ ∈ U’) w_u’ ≤ OPT(U’)

We aim to establish the relationship:
|OPT(U’)| ≤ |OPT(U)| + Additional bins needed

This represents the increase in the number of bins required due to the increase in item weights after rounding.

Questions:

1. Can someone guide me on how to validate or refine my hypothesis that:
   |OPT(U’)| ≤ |OPT(U)| + |OPT(U)|/k

Are there specific methodologies or considerations I should follow to ensure the accuracy of this formulation?

2. What strategies or analytical approaches are commonly used to estimate the additional bins required due to weight rounding in bin packing problems?

Any insights or references to academic papers or resources that discuss similar problems would be greatly appreciated!