Question

Here’s the problem:

A user has a shopping basket containing items, where each item has a set of available delivery types which is a subset selected from some defined overall set (e.g. [“UK 2nd class”, “UK 1st class”, “UK Recorded Delivery”], but don’t get too hung up on the exact names).

When going through the checkout process, the user should be presented with the option of either separate or combined delivery.

Separate is easy – a table is displayed where each item is on its own row, and the set of columns matches the union of available delivery types among the items. Each row contains a radio button set, one button for each column that is of a delivery type available to that item.

It’s combined I’m unsure how to approach. Items can be combined into a group only when all items in that group share a subset of available delivery types. Items with mutually exclusive delivery types can never be in the same group. The basket can contain items from multiple vendors; when this is the case, products from different vendors may not be grouped together, even if they share a delivery type.

The client requires that the minimum number of groups is calculated, irrelevant of the cost of each type of postage. Yes, that means that if separate deliveries for four items would cost £1+£2+£3+£4 (four different delivery types), but the items all share a fifth delivery type that costs £15, then the user would be presented with that single more-expensive group for the “combined” option.

Here is an HTML example of separate/combined options.

The available delivery types for an item are retrieved from the database using a stored procedure, and have a unique identifier created from the type ID and vendor ID, so that it’s easy to compare equality between types. However, I can’t seem to conjure up an efficient algorithm for doing the comparisons and generating groups.

The end result I’m hoping for in terms of data structures is (pseudo-code):

Basket { List<Item> Items }
=>
GroupsTable {
    List<SelectableDeliveryType> Columns,
    List<Group { List<Item> }> Rows
}

Where each Item contains its own List<AvailableDeliveryType> which can easily then be used to work out which SelectableDeliveryType column to put its radio buttons in.

Any thoughts, pointers to general algorithmic concepts which cover this (I don’t think it’s a set cover problem for example) etc. greatly appreciated.

Answer 1

This problem is NP-hard by a reduction from graph coloring. Given a graph consisting of a set of nodes connected by edges, a k-coloring in that graph is a way of coloring each node in the graph so that no two connected nodes have the same color. The chromatic number problem is the following – what is the minimum number of colors with which it is possible to color a graph?

We can reduce any instance of the chromatic number problem, which is NP-hard, to your problem as follows: for each node, create a new product. For each edge, mark that those two products cannot be grouped into the same cluster. Then, any clustering of these products corresponds to a coloring – just color all the nodes in each cluster the same color. Consequently, solving your problem optimally is equivalent to solving graph coloring, and therefore (under the assumption that P ≠ NP) has no polynomial-time algorithm.

Unfortunately, graph coloring is known to be very hard to approximate; there is no known polynomial-time algorithm that can get within a constant factor of optimal, or even within a factor of n^1-ε for any ε > 0. I think you are best off going with a heuristic like Chaitin’s algorithm, or finding a different way of thinking about this problem.

Sorry for the negative result, but hopefully this helps!