ODIn/paper-BagRelationalPDBsAreHard
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Added some more to Intro rewrite.

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Aaron Huber 2020-11-23 19:47:16 -05:00
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@ -76,7 +76,7 @@ Note that such a query in set semantics is indeed \#-P hard, since it is a query
However, in the bag setting, $\expct\pbox{\poly(\prob_1,\ldots, \prob_\numvar)}$ is indeed linear in the size of the output polynomial as the number of operations in the computation is \textit{exactly} the number of output polynomial operations.
Now, consider query $\poly^2() := \left(\rel(A), E(A, B), R(B)\right) \times \left(\rel(A), E(A, B), R(B)\right)$. Abusing notation again, the output polynomial will be $\left(ab + bc + ca\right) \cdot \left(ab + bc + ca\right)$. Now, assume the restriction of all variables $X \in \vct{X}$ set to $\prob$. Here, again, in the setting of bag semantics, we have a query that is linear in the size of the expanded output polynomial, however it is not readily obvious that we achieve linearity for the factorized version of the polynomial as well. But if we think of this query in a graph theoretic setting, one can see that we end up with
Now, consider query $\poly^2() := \left(\rel(A), E(A, B), R(B)\right) \times \left(\rel(A), E(A, B), R(B)\right)$. Abusing notation again, the output polynomial will be $\left(ab + bc + ca\right) \cdot \left(ab + bc + ca\right)$. Now, assume the following restrictions. First, all variables $X \in \vct{X}$ are set to $\prob$. Second, all exponents $e > 1$ in the expanded polynomial are set to $1$. Call this modified polynomial $\rpoly(\prob,\ldots, \prob)$. We show that $\expct\pbox{\poly(\prob,\ldots, \prob)} = $\rpoly(\prob,\ldots, \prob)$. Here, again, in the setting of bag semantics, we have a query that is linear in the size of the expanded output polynomial, however it is not readily obvious that we achieve linearity for the factorized version of the polynomial as well. But if we think of this query in a graph theoretic setting, one can see that we end up with
\[\sum\limits_{(i, j) \in E}X_iX_j + \sum\limits_{\substack{(i, j), (i \ell) \in E,\\ i \neq \ell}}X_iX_jX_\ell + \sum\limits_{\substack{(i, j), (k, \ell) \in E,\\ i\neq j\neq k \neq \ell}}X_iX_jX_kX_\ell.\]