# Preventing oversell, allocation of limited resources with overlapping properties

I am trying to solve problem of preventing oversell of limited resources.

Consider resources (people) who are described by set of properties where each property belongs to different category (example properties from four categories: male, age 25-30, 2 children, interested in games).

Buyers want to allocate access to resources. Buyers can specify subset of categories and one property from each category (example: allocate 1000 males, age 25-30 or allocate 100 females, age 25-30, interested in music).

In my real life example I have 6m+ possible set of properties (profiles) where for each set of properties I know how many profiles exists.

My initial approach was to build a graph like one below:

and then traverse using edge weights, for instance validating if demand for 100 females, age2 can be satisfied:

1. check if size(female, age2) < 100
2. for each parent:
1. check if size(parent) < 100 and go to 2.
3. for each child:
1. check if size(child) < 100 * weight(edge(node, child)) go to 1.

(above algorithm is simplified as does not prevent visiting same node multiple times)

It all works fine when graph is small, however when number of nodes and edges (dependencies) between nodes (profile universe groups) grows it does not scale very well.

Consider example:

• large graph, 6m nodes, 20m+ edges
• buyer wants to allocate 1000 males (and there are only males and females in gender category)

algorithm would start with top-level 'male' node which probably has 10m+ outgoing edges and 10m+ checks would be required (and probably each of those 10m outgoing edges has incoming edges which need to be checked as well).

I was trying to find different approach but failed. I was trying to google out existing solutions but seems like I am unable to even name problem properly. Any reference to what is this problem similar to would be good for me as a starting point.

Two more graphs to present exponential growth of the graph: 3 categories

4 categories

Update

Regarding size, assuming 8 categories of properties where each category has: 2, 6, 6, 6, 6, 8, 1140, 150 values respectively then estimated number of profiles: 2*6^4*8*1140*150 ~= 3.5 * 10^9. Number of nodes in graph: at least 7 * 10^9, number of edges in graph: at least 140 * 10^9.

Update #2

Formula for number of nodes is:

$\sum_{i<n}\prod_{k<i \atop j_1, j_2, ..., j_k < n} s_{j_{1}} ... s_{j_{n}}$

where $n$ is number of categories and $s_x$ is size of category $x$.

So in my example there would be 11'169'108'657 nodes.

Update #3

As per @Raphael advice - I have reduced number of nodes and now formula is:

$\sum_{i<n-M}\prod_{k<i \atop j_1, j_2, ..., j_k < n} s_{j_{1}} ... s_{j_{n}}$

where $M<n$ and assumed that distribution of resources across smallest slices of universe is equal. At the same time removed lot of edges from graph.

Example of sub-graph size reduction:

• Could you give a little more detail on what the problem you're trying to solve is, maybe in more mathematical/graph-theoretic terms? I suspect you will be a lot more likely to receive an answer on this particular site if you can separate the problem from the domain-specific knowledge as much as possible. Is there a particular quantity you're trying to maximize, or a particular set of constraints you need to satisfy? – jmite May 27 '13 at 14:19
• @jmite thanks for the comment, I guess my problem is that graph-based solution is 'too perfect' and when graph is large it is not applicable in real life. I was trying to define my problem and maybe someone will have some tips on how to solve it using different approach (graphs are not required). – Maciej Łopaciński May 27 '13 at 15:07
• When the graph is large, there could still be approximate solutions which run very fast. 11 billion is not intractably hard if you have a linear algorithm. – jmite May 27 '13 at 15:40
• why do you create a node for each possible profile, and not just for the profiles present in your data? – Sasho Nikolov May 27 '13 at 16:55
• I don't understand your algorithm. What is the function of the weights? – Raphael May 28 '13 at 6:58

So the data structure holding pointers for every possible combination of classifiers is huge. Sure, but why build it at all? Don't overengineer this!

Just store the profiles in a database and do one (linear time) filtering sweep for each query, i.e. select/count on demand. For a few millions of records, that should require no further preprocessing.

If the number of requests is large and/or you need really small response times, you can think about caching, or creating equivalence classes along some popular classifiers, or along classifiers with few large classes. Then, the linear sweep has to be done only on small lists.

For example, you can divide your database along gender and age (assuming these are included in most customer queries)

$\qquad \{m, w, o\} \times \{0..5, 10..15, \dots, 95..100\}$

where the values obviously depend on your data. Then, each query will require only few of these small lists, and can even parallelise if you store the individual chunks separately.

• Thanks @Raphael. Decreasing number of classifiers is one of the options (but business driven so need to stay open here). Regarding select/count on demand - will think about it. I did not mention that it all happens in time and where multiple allocation demands overlaps, number of resources changes over time and distribution of resources changes as well. Still you might be right re over-engineering this ;) – Maciej Łopaciński May 28 '13 at 11:39
• simple select/count on demand works in case of simple validation (is there enough males age2?). However when it comes to allocation (allocate 100 males for buyer 1) then system need to preserve "universe" distribution - so allocating 100 males means 30 of them must be age1, 40 must be age2 and remaining 30 must be age3. The goal is to prevent buyers from allocating uncommon group in unfair way. Example: assume there is uncommon group (which is 1% of males) - when buyer wants to allocate 1000 males then he must get exactly 1000*1%=10 males from this uncommon group. – Maciej Łopaciński May 28 '13 at 14:05
• @MaciejŁopaciński Sampling uniformly from the whole set of feasible candidates solves that problem for you, at least for large sets and samples (I seem to remember that sets sizes in the low hundreds and samples sizes in the dozens gets you close to fair, statistically, but better check that.) That is, you may get a non-representative set, but probability is small. Note furthermore that there may not be a set of feasible candidates that is of sufficient size and representative w.r.t all other classifiers. – Raphael May 28 '13 at 17:31
• Not sure if I get it right - but still it led me to some optimizations: (1) I reduced number of edges by at least 10 times and top level nodes like "male" instead of having 700'000 edges now has 260; (2) I have removed most detailed "slices" so instead of having node "gender/age/children" I left "gender/age" and "age/children" and assumed equal distribution across "gender/age/children" (in case of eight categories I assumed it for two) and I think this is what you meant by sampling. – Maciej Łopaciński May 29 '13 at 11:56