\(
\def\<#1>{\left<#1\right>}
\let\geq\geqslant
\let\leq\leqslant
% an undirected version of \rightarrow:
\newcommand{\mathdash}{\relbar\mkern-9mu\relbar}
\def\deg#1{\mathrm{deg}(#1)}
\newcommand{\dg}[1]{d_{#1}}
\newcommand{\Norm}{\mathrm{N}}
\newcommand{\const}[1]{c_{#1}}
\newcommand{\cconst}[1]{\alpha_{#1}}
\newcommand{\Exp}[1]{E_{#1}}
\newcommand*{\ppr}{\mathbin{\ensuremath{\otimes}}}
\newcommand*{\su}{\mathbin{\ensuremath{\oplus}}}
\newcommand{\nulis}{\vmathbb{0}} %{\mathbf{0}}
\newcommand{\vienas}{\vmathbb{1}}
\newcommand{\Up}[1]{#1^{\uparrow}} %{#1^{\vartriangle}}
\newcommand{\Down}[1]{#1^{\downarrow}} %{#1^{\triangledown}}
\newcommand{\lant}[1]{#1_{\mathrm{la}}} % lower antichain
\newcommand{\uant}[1]{#1_{\mathrm{ua}}} % upper antichain
\newcommand{\skal}[1]{\langle #1\rangle}
\newcommand{\NN}{\mathbb{N}} % natural numbers
\newcommand{\RR}{\mathbb{R}}
\newcommand{\minTrop}{\mathbb{T}_{\mbox{\rm\footnotesize min}}}
\newcommand{\maxTrop}{\mathbb{T}_{\mbox{\rm\footnotesize max}}}
\newcommand{\FF}{\mathbb{F}}
\newcommand{\pRR}{\mathbb{R}_{\mbox{\tiny $+$}}}
\newcommand{\QQ}{\mathbb{Q}}
\newcommand{\ZZ}{\mathbb{Z}}
\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}}
\newcommand{\A}{{\mathcal A}}
\newcommand{\B}{{\mathcal B}}
\newcommand{\C}{{\mathcal C}}
\newcommand{\R}{{\mathcal R}}
\newcommand{\MPS}[1]{f_{#1}} % matrix multiplication
\newcommand{\ddeg}[2]{\#_{#2}(#1)}
\newcommand{\length}[1]{|#1|}
\DeclareMathOperator{\support}{sup}
\newcommand{\supp}[1]{\support(#1)}
\DeclareMathOperator{\Support}{sup}
\newcommand{\spp}{\Support}
\newcommand{\Supp}[1]{\mathrm{Sup}(#1)} %{\mathcal{S}_{#1}}
\newcommand{\lenv}[1]{\lfloor #1\rfloor}
\newcommand{\henv}[1]{\lceil#1\rceil}
\newcommand{\homm}[2]{{#1}^{\langle #2\rangle}}
\let\daug\odot
\let\suma\oplus
\newcommand{\compl}[1]{Y_{#1}}
\newcommand{\pr}[1]{X_{#1}}
\newcommand{\xcompl}[1]{Y'_{#1}}
\newcommand{\xpr}[1]{X'_{#1}}
\newcommand{\cont}[1]{A_{#1}} % content
\def\fontas#1{\mathsf{#1}} %{\mathrm{#1}} %{\mathtt{#1}} %
\newcommand{\arithm}[1]{\fontas{Arith}(#1)}
\newcommand{\Bool}[1]{\fontas{Bool}(#1)}
\newcommand{\linBool}[1]{\fontas{Bool}_{\mathrm{lin}}(#1)}
\newcommand{\rBool}[2]{\fontas{Bool}_{#2}(#1)}
\newcommand{\BBool}[2]{\fontas{Bool}_{#2}(#1)}
\newcommand{\MMin}[1]{\fontas{Min}(#1)}
\newcommand{\MMax}[1]{\fontas{Max}(#1)}
\newcommand{\negMin}[1]{\fontas{Min}^{-}(#1)}
\newcommand{\negMax}[1]{\fontas{Max}^{-}(#1)}
\newcommand{\Min}[2]{\fontas{Min}_{#2}(#1)}
\newcommand{\Max}[2]{\fontas{Max}_{#2}(#1)}
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\newcommand{\Un}[1]{\fontas{L}(#1)}
\newcommand{\kUn}[2]{\fontas{L}_{#2}(#1)}
\newcommand{\Nor}{\mu} % norm without argument
\newcommand{\nor}[1]{\Nor(#1)}
\newcommand{\bool}[1]{\hat{#1}} % Boolean version of f
\newcommand{\bphi}{\phi} % boolean circuit
\newcommand{\xf}{\boldsymbol{\mathcal{F}}}
\newcommand{\euler}{\mathrm{e}}
\newcommand{\ee}{f} % other element
\newcommand{\exchange}[3]{{#1}-{#2}+{#3}}
\newcommand{\dist}[2]{{#2}[#1]}
\newcommand{\Dist}[1]{\mathrm{dist}(#1)}
\newcommand{\mdist}[2]{\dist{#1}{#2}} % min-max dist.
\newcommand{\matching}{\mathcal{M}}
\renewcommand{\E}{A}
\newcommand{\F}{\mathcal{F}}
\newcommand{\set}{W}
\newcommand{\Deg}[1]{\mathrm{deg}(#1)}
\newcommand{\mtree}{MST}
\newcommand{\stree}{{\cal T}}
\newcommand{\dstree}{\vec{\cal T}}
\newcommand{\Rich}{U_0}
\newcommand{\Prob}[1]{\ensuremath{\mathrm{Pr}\left\{{#1}\right\}}}
\newcommand{\xI}{\boldsymbol{I}}
\newcommand{\plus}{\mbox{\tiny $+$}}
\newcommand{\sgn}[1]{\left[#1\right]}
\newcommand{\ccompl}[1]{{#1}^*}
\newcommand{\contr}[1]{[#1]}
\newcommand{\harm}[2]{{#1}\,\#\,{#2}} %{{#1}\,\oplus\,{#2}}
\newcommand{\hharm}{\#} %{\oplus}
\newcommand{\rec}[1]{1/#1}
\newcommand{\rrec}[1]{{#1}^{-1}}
\DeclareRobustCommand{\bigO}{%
\text{\usefont{OMS}{cmsy}{m}{n}O}}
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\newcommand{\thr}[2]{\mathrm{Th}^{#1}_{#2}}
\newcommand{\rectbound}{h}
\newcommand{\pol}[3]{\sum_{#1\in #2}{#3}_{#1}\prod_{i=1}^n x_i^{#1_i}}
\newcommand{\tpol}[2]{\min_{#1\in #2}\left\{\skal{#1,x}+\const{#1}\right\}}
\newcommand{\comp}{\circ} % composition
\newcommand{\0}{\vec{0}}
\newcommand{\drops}[1]{\tau(#1)}
\newcommand{\HY}[2]{F^{#2}_{#1}}
\newcommand{\hy}[1]{f_{#1}}
\newcommand{\hh}{h}
\newcommand{\hymin}[1]{f_{#1}^{\mathrm{min}}}
\newcommand{\hymax}[1]{f_{#1}^{\mathrm{max}}}
\newcommand{\ebound}[2]{\partial_{#2}(#1)}
\newcommand{\Lpure}{L_{\mathrm{pure}}}
\newcommand{\Vpure}{V_{\mathrm{pure}}}
\newcommand{\Lred}{L_1} %L_{\mathrm{red}}}
\newcommand{\Lblue}{L_0} %{L_{\mathrm{blue}}}
\newcommand{\epr}[1]{z_{#1}}
\newcommand{\wCut}[1]{w(#1)}
\newcommand{\cut}[2]{w_{#2}(#1)}
\newcommand{\Length}[1]{l(#1)}
\newcommand{\Sup}[1]{\mathrm{Sup}(#1)}
\newcommand{\ddist}[1]{d_{#1}}
\newcommand{\sym}[2]{S_{#1,#2}}
\newcommand{\minsum}[2]{\mathrm{MinS}^{#1}_{#2}}
\newcommand{\maxsum}[2]{\mathrm{MaxS}^{#1}_{#2}} % top k-of-n function
\newcommand{\cirsel}[2]{\Phi^{#1}_{#2}} % its circuit
\newcommand{\sel}[2]{\sym{#1}{#2}} % symmetric pol.
\newcommand{\cf}[1]{{#1}^{o}}
\newcommand{\Item}[1]{\item[\mbox{\rm (#1)}]} % item in roman
\newcommand{\bbar}[1]{\underline{#1}}
\newcommand{\Narrow}[1]{\mathrm{Narrow}(#1)}
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\newcommand{\eepsil}{\varepsilon}
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\newcommand{\mon}[1]{\mathrm{mon}(#1)}
\newcommand{\mmon}{\alpha}
\newcommand{\gmon}{\alpha}
\newcommand{\hmon}{\beta}
\newcommand{\nnor}[1]{\|#1\|}
\newcommand{\inorm}[1]{\left\|#1\right\|_{\mbox{\tiny $\infty$}}}
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\newcommand{\coset}[1]{\textup{co-}{#1}}
\newcommand{\spol}[1]{\mathrm{ST}_{#1}}
\newcommand{\cayley}[1]{\mathrm{C}_{#1}}
\newcommand{\SQUARE}[1]{\mathrm{SQ}_{#1}}
\newcommand{\STCONN}[1]{\mathrm{STCON}_{#1}}
\newcommand{\STPATH}[1]{\mathrm{PATH}_{#1}}
\newcommand{\SSSP}[1]{\mathrm{SSSP}(#1)}
\newcommand{\APSP}[1]{\mathrm{APSP}(#1)}
\newcommand{\MP}[1]{\mathrm{MP}_{#1}}
\newcommand{\CONN}[1]{\mathrm{CONN}_{#1}}
\newcommand{\PERM}[1]{\mathrm{PER}_{#1}}
\newcommand{\mst}[2]{\tau_{#1}(#2)}
\newcommand{\MST}[1]{\mathrm{MST}_{#1}}
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\newcommand{\node}[1]{v_{#1}}
\newcommand{\BF}[2]{W_{#2}[#1]} % Bellman-Ford
\newcommand{\FW}[3]{W_{#1}[#2,#3]} % Floyd-Washall
\newcommand{\HK}[1]{W[#1]} % Held-Karp
\newcommand{\WW}[1]{W[#1]}
\newcommand{\pWW}[1]{W^{+}[#1]}
\newcommand{\nWW}[1]{W^-[#1]}
\newcommand{\knap}[2]{W_{#2}[#1]}
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\newcommand{\econt}[1]{C_{#1}}
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\newcommand{\rUn}[1]{\fontas{L}_{r}(#1)}
\newcommand{\copath}{\mathrm{co}\text{-}\mathrm{Path}_n}
\newcommand{\Path}{\mathrm{Path}_n}
\)
Can reciprocal inputs speed up (max,+) circuits?
By Theorem 6.11 in the book, (tropical) reciprocal inputs $-x_1,\ldots,-x_n$ (in addition to input variables $x_1,\ldots,x_n$) cannot substantially decrease
the size of tropical $(\min,+)$ circuits: the gap $(\min,+)/(\min,+,-x_i)$ is never larger than quadratic.
As mentioned in the book (remark 6.20),
the same holds also for $(\max,+)$ circuits as long as the computed $(\max,+)$ polynomial $f(x)=\max_{a\in A}\skal{a,x}+\const{a}$ is homogeneous, that is, if $a_1+\cdots+a_n=m$ holds for some $m\in \NN$ and all $a\in A$; the proof is now given in this comment.
However, the case of non-homogeneous $(\max,+)$ polynomials remains unclear.
Problem 1: Can the gap $(\max,+)/(\max,+,-x_i)$ be super-polynomial
for non-homogeneous $(\max,+)$ polynomials?
As a possible candidate for a $(\max,+)$ polynomial showing that this gap can be large
one could consider the
heaviest co-path
polynomial:
$\copath(x)= \max_{P}\ \sum_{e\not\in P} x_e$,
where the maximum is over all simple paths $P$ in $K_n$ from the
vertex $s=1$ to the vertex $t=n$; as before, we view paths as sets of
their edges. That is, $\copath(x)$ is the maximum weight of a subgraph of $K_n$ obtained by removing all edges of some $s$-$t$ path from $K_n$.
Note that this polynomial is not homogeneous: the degrees
of its monomials vary between $\binom{n}{2}-n+1$ and $\binom{n}{2}-1$.
The dual of a
$(\min,+)$ polynomial $f(x)=\min_{a\in A}\ \skal{a,x}$ with
$A\subseteq\{0,1\}^n$ is the $(\max,+)$ polynomial
$\ccompl{f}(x):=\max_{a\in A}\ \skal{\vec{1}-a,x}$. That is, feasible solution of $\ccompl{f}$ are complements $\vec{1}-a$ of the feasible solutions $a\in A$ of $f$.
Note that the tropical $(\max,+)$ polynomial $\copath$ is the dual of the tropical $(\min,+)$ shortest $s$-$t$ path polynomial
$\Path(x)= \min_{P}\ \sum_{e\in P} x_e$, which can be computed by a $(\min,+)$ circuit of size $O(n^3)$ resulting from the Bellman-Ford-Moore dynamic programming algorithm for the shortest $s$-$t$ path problem
(Example 1.7 in the book). So,
by Theorem B(1), the polynomial $\copath$ can be computed by a $(\max,+,-x_i)$ circuit
using $O(n^3)$ gates.
Problem 2:
Prove or disprove that the $(\max,+)$ polynomial $\copath$ requires
$(\max,+)$ circuits of super-polynomial size.
Footnotes:
(1)
Theorem B: Let $A\subseteq \{0,1\}^n$ be an antichain. If a $(\min,+)$
polynomial $f(x)=\min_{a\in A}\ \skal{a,x}$ can be computed by a
$(\min,+)$ circuit of size $s$, then its dual
$\ccompl{f}(x)=\max_{a\in A}\ \skal{\vec{1}-a,x}$ can be computed by
a $(\max,+,-x_i)$ circuit of size $n+s$. If $f$ is homogeneous, then $\ccompl{f}(x)$ can be computed by a $(\max,+)$ circuit of size $O(ns^2+n^3)$.
See this comment for the proof.
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