# Estimate complexity of integer sequence enumeration

I would like to seek help on the complexity of the following problem. Given positive integers $m$, $n$, $D_1$ and $D_2$, find all sequences $a_1\lt a_2 \lt \dots \lt a_n$ are there such that:

1. each $a_i\in\{1, \dots, m\}$
2. $\sum_{i=1}^n a_i = D_1$
3. $\sum_{i=1}^n a_i^2 = D_2$.

I want to know the complexity with respect to $m$, $n$, $D_1$ and $D_2$.

For example, when $m=12$, $n=4$, $D_1=26$ and $D_2=214$ the solutions are

$$3,5, 6, 12\qquad 2, 5, 8, 11\qquad 1, 7, 8, 10$$

• This seems as modification of subset sum for 1..m range. – rus9384 Aug 30 '17 at 11:15
• @rus9384 The simultaneous restriction on the sum of squares makes this look like a completely different problem, to me. – David Richerby Aug 30 '17 at 12:25
• @DavidRicherby, yet it does not mean that subset sum is not reducible to this. – rus9384 Aug 30 '17 at 12:43
• @rus9384 Sure. Any problem might reduce to it. Saying that alone doesn't help at all: the nonlinear nature of the constraints here make the reduction look really difficult. – David Richerby Aug 30 '17 at 12:54

Here is a simple recursive solution that runs in exponential time.

Geometrically you want to enumerate the lattice points which lies inside the first quadrant of the $n$-hypercube of height $m$ and on the intersection of the $D_1$-radius $L_1$ sphere and the $D_2$-radius $L_2$ sphere (let call them respectively $S_1$ and $S_2$), up to reordering of the coefficients.

Denoting by $V_n$ the volume of the $n$ unit dimensional ball, we have :

• $m^n/n!$ points in the intersection of the hypercube and the quadrant (let's call it $D$) with the ordering ,
• $\alpha = O\left(m^n e^{\frac{-(D_1-nm/6)^2}{3nm}}\right)$ in $S_1\cap D$ (see theorem 1.3 of http://www.doiserbia.nb.rs/img/doi/1452-8630/2008/1452-86300802222R.pdf), considering the ordering condition.
• $\beta = \frac{1}{n!2^{n}}V_n D_2^{n/2} + O\left(D_2^{n/2-1+\epsilon}\right)$ in $S_2$ (Generalized Gauss circle problem+quadrant condition+order condition). I'm not considering intersection with the hypercube here, so the global complexity will be an overestimation.

Depending on your parameters choose the minimum of $\alpha$ or $\beta$. If its $\alpha$ perform a recursive enumeration of the integers partitions of length $n$ with the restriction on $m$ and use early abort to discard solutions which are getting out $S_2$. In the other case, enumerate the lattice point on $S_2$ ( with recursive orthogonal projections for instance) and check for the appartenance in $S_1$ for each returned vector.

Hence the global complexity will be dominated by $min(\alpha, \beta)$.

• I think $D_1$-radius $L_1$ should be a hyperplane instead of a sphere. Though I've no idea whether it is computable when $m$ & $n$ are large (~hundreds). Thanks – Steve Yau Sep 1 '17 at 1:44
• The intersection of the hyperplane with the hypercube is the same as the intersection of the $L_1$ sphere and the hypercube. I've doubts on the tractability in high dimension indeed... Enumeration of the ball up to dimension lower than a hundred is feasible nonetheless. – Sn0w Sep 1 '17 at 15:46