Combinatorial explosion

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In mathematics, a combinatorial explosion is the rapid growth of the complexity of a problem due to how the combinatorics of the problem is affected by the input, constraints, and bounds of the problem. Combinatorial explosion is sometimes used to justify the intractability of certain problems.^[1][2] Examples of such problems include certain mathematical functions, the analysis of some puzzles and games, and some pathological examples which can be modelled as the Ackermann function.

Examples

Latin squares

A Latin square of order n is an n × n array with entries from a set of n elements with the property that each element of the set occurs exactly once in each row and each column of the array. An example of a Latin square of order three is given by,

1	2	3
2	3	1
3	1	2

A common example of a Latin square would be a completed Sudoku puzzle.^[3] A Latin square is a combinatorial object (as opposed to an algebraic object) since only the arrangement of entries matters and not what the entries actually are. The number of Latin squares as a function of the order (independent of the set from which the entries are drawn) (sequence A002860 in the OEIS) provides an example of combinatorial explosion as illustrated by the following table.

Sudoku

A combinatorial explosion can also occur in some puzzles played on a grid, such as Sudoku.^[2] A Sudoku is a type of Latin square with the additional property that each element occurs exactly once in sub-sections of size √n × √n (called boxes). Combinatorial explosion occurs as n increases, creating limits to the properties of Sudokus that can be constructed, analyzed, and solved, as illustrated in the following table.

Games

One example in a game where combinatorial complexity leads to a solvability limit is in solving chess (a game with 64 squares and 32 pieces). Chess is not a solved game. In 2005 all chess game endings with six pieces or fewer were solved, showing the result of each position if played perfectly. It took ten more years to complete the tablebase with one more chess piece added, thus completing a 7-piece tablebase. Adding one more piece to a chess ending (thus making an 8-piece tablebase) is considered intractable due to the added combinatorial complexity.^[6][7]

Furthermore, the prospect of solving larger chess-like games becomes more difficult as the board-size is increased, such as in large chess variants, and infinite chess.^[8]

Computing

Combinatorial explosion can occur in computing environments in a way analogous to communications and

An example is a taxonomy where different vegetables inherit from their ancestor species. Attempting to compare the tastiness of each vegetable with the others becomes intractable since the hierarchy only contains information about genetics and makes no mention of tastiness. However, instead of having to write comparisons for carrot/carrot, carrot/potato, carrot/sprout, potato/potato, potato/sprout, sprout/sprout, they can all multiply inherit from a separate class of tasty whilst keeping their current ancestor-based hierarchy, then all of the above can be implemented with only a tasty/tasty comparison.

Arithmetic

Suppose we take the factorial of n:

${\displaystyle n!=n\cdot (n-1)\cdot \ldots \cdot 2\cdot 1}$

Then 1! = 1, 2! = 2, 3! = 6, and 4! = 24. However, we quickly get to extremely large numbers, even for relatively small n. For example, 100! ≈ 9.33262154×10¹⁵⁷, a number so large that it cannot be displayed on most calculators, and vastly larger than the estimated number of fundamental particles in the observable universe.^[9]

Communication

In administration and computing, a combinatorial explosion is the rapidly accelerating increase in communication lines as organizations are added in a process. (This growth is often casually described as "exponential" but is actually polynomial.)

If two organizations need to communicate about a particular topic, it may be easiest to communicate directly in an ad hoc manner—only one channel of communication is required. However, if a third organization is added, three separate channels are required. Adding a fourth organization requires six channels; five, ten; six, fifteen; etc.

In general, it will take ${\displaystyle l={\frac {n(n-1)}{2}}={n \choose 2}}$ communication lines for n organizations, which is just the number of 2-combinations of n elements (see also Binomial coefficient).^[10]

The alternative approach is to realize when this communication will not be a one-off requirement, and produce a generic or intermediate way of passing information. The drawback is that this requires more work for the first pair, since each must convert its internal approach to the common one, rather than the superficially easier approach of just understanding the other.

See also

References