\(
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% an undirected version of \rightarrow:
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\newcommand{\Up}[1]{#1^{\uparrow}} %{#1^{\vartriangle}}
\newcommand{\Down}[1]{#1^{\downarrow}} %{#1^{\triangledown}}
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\newcommand{\uant}[1]{#1_{\mathrm{ua}}} % upper antichain
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\newcommand{\gf}[1]{GF(#1)}
\newcommand{\conv}[1]{\mathrm{Conv}(#1)}
\newcommand{\vvec}[2]{\vec{#1}_{#2}}
\newcommand{\f}{{\mathcal F}}
\newcommand{\h}{{\mathcal H}}
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\DeclareMathOperator{\Support}{sup}
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\let\daug\odot
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\)
What input weights do we used to prove lower bounds?
All upper bounds of the size of $(\min,+)$ and $(\max,+)$
circuits hold when the circuits are required to solve (or to approximate)
a given optimization problem on all nonnegative input weightings $x\in\RR_+^n$.
That is, the obtained circuits are correct on all nonnegative real input weightings.
When proving lower bounds, we also assumed that circuits are correct on all nonnegative real input weightings.
But a reader might wonder whether we really need all these input weightings $x\in\RR_+^n$
in the proofs of the lower bounds?
The answer is: no! Namely, all lower bounds proved in the book also hold when the circuits
are only required to be $D$-correct, that is, must be correct on all input weightings
$x\in D^n$ from much smaller domains $D\subset\RR_+$; on other input weightings, the circuits
can output arbitrary values.
Namely, it is implicit in the book that already the domain
$D=\NN$ suffices for this purpose in the case of $(\min,+)$ circuits, and that even the Boolean domain $D=\{0,1\}$ suffices
in the case of $(\max,+)$ circuits.
The goal of this comment is to make this aspect a bit more explicit.
Namely, observe that the domain $D$ (from which input weights can come) was
only important in two results established in Chapter 1:
- Structure: What domain $D$ was actually used to prove
the structural properties of sets $B\subseteq\NN^n$ of feasible solutions produced by tropical circuits (Lemmas 1.23 and 1.24).
- Eliminate constant inputs: What domain $D$ was
actually used to prove that, when dealing with tropical circuits solving
or approximating $0/1$ optimization problems, we can safely restrict
ourselves to constant-free circuits (Lemma 1.29).
Since all our subsequent proofs of lower bounds only rely only on these two facts,
all they hold when the circuits are required to bo correct on the input weightings form the following domains $D$.
(max,+) circuits: $D=\{0,1\}$
In the proof of the structural properties of sets $B\subseteq\NN^n$ of feasible solutions produced by $(\max,+)$ circuits (Lemma 1.24)
only $0$-$1$ input weightings were used. For the elimination of constant inputs from
$(\max,+)$ circuits (Lemma 1.29) only the input all-$0$ weighting $x=\vec{0}$ was sufficient.
All lower bounds for $(\max,+)$ circuits proved in the book hold when circuits are only required to be correct on $0$-$1$ input weightings $x\in\{0,1\}^n$.
(min,+) circuits: $D=\{0,1,2,\ldots\}$
In the proof of the structural properties of sets $B\subseteq\NN^n$ of feasible solutions
produced by $(\min,+)$ circuits (Lemma 1.23)
only input weightings $x\in\NN^n$ were used. For the elimination of constant inputs from $(\min,+)$ circuits (Lemma 1.29), it is enough to use ceilings
(take $\lambda:=2\lceil c/d\rceil$ in the proof of Lemma 1.29, instead of just $\lambda:=2c/d$).
All lower bounds for $(\min,+)$ circuits proved in the book hold when circuits are only required to be correct on all nonnegative integer input weightings $x\in\NN^n$.
Remark 6.13(1)) could be relevant in this context.
Footnotes:
(1)
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