In this paper, we study the structure of set-multilinear arithmetic circuits and set-multilinear branching programs with the aim of
showing lower bound results. We define some natural restrictions of these models for which we are able to show lower bound results. Some
of our results extend existing lower bounds, while others are new and raise open questions. Specifically, our main results are the
following:

(1) We observe that set-multilinear arithmetic circuits can be transformed into shallow set-multilinear circuits efficiently, following \cite{VSBR83,RY08}. Hence, polynomial size set-multilinear circuits have quasi-polynomial size set-multilinear branching programs.

(2) We show that \emph{k-narrow} set-multilinear ABPs computing the Permanent polynomial
$\mathrm{PER}_n$ (or determinant $\mathrm{DET}_n$) require $2^{\Omega(k)}$
size. As a consequence, we show that sum of $r$ read-once oblivious ABPs computing $\mathrm{PER}_n$ requires size $2^{\Omega(\frac{n}{r})}$.

(3) We also show that set-multilinear branching programs are exponentially more powerful than \emph{interval} multilinear circuits (where the index sets for each gate is restricted to be an
interval w.r.t.\ some ordering), assuming the sum-of-squares conjecture. This further underlines the power of set-multilinear
branching programs.

(4) Finally, we show exponential lower bounds for set-multilinear circuits with restrictions on the number of parse trees of monomials
and prove exponential lower bounds results.

Some of proofs are rewritten and parameters are improved.

In this paper, we study the structure of set-multilinear arithmetic
circuits and set-multilinear branching programs with the aim of
showing lower bound results. We define some natural restrictions of
these models for which we are able to show lower bound results. Some
of our results extend existing lower bounds, while others are new and
raise open questions. More specifically, our main results are the
following:

(1) We observe that set-multilinear arithmetic circuits can be
transformed into shallow set-multilinear circuits efficiently,
similar to depth reduction results of [VSBR83,RY08] for more
general commutative circuits. As a consequence, we note that
polynomial size set-multilinear circuits have quasi-polynomial size
set-multilinear branching programs. We show that \emph{narrow}
set-multilinear ABPs (with a restricted number of set types)
computing the Permanent polynomial $\mathrm{PER}_n$ require
$2^{n^{\Omega(1)}}$ size. A similar result for general
set-multilinear ABPs appears difficult as it would imply that the
Permanent requires superpolynomial size set-multilinear circuits. It
would also imply that the noncommutative Permanent requires
superpolynomial size noncommutative arithmetic circuits.

(2) Indeed, we also show that set-multilinear branching programs are
exponentially more powerful than \emph{interval} multilinear
circuits (where the index sets for each gate is restricted to be an
interval w.r.t.\ some ordering), assuming the sum-of-squares
conjecture. This further underlines the power of set-multilinear
branching programs.

(3) Finally, we consider set-multilinear circuits with restrictions
on the number of proof trees of monomials computed by it, and prove
exponential lower bounds results. This raises some new lower bound
questions.