minicomplexity
relational zero-match
definition

Given two binary relations on $[h]$, check that it is possible to start from $0$ and return to it after an even number of alternate applications of the relations.

details

For every $h\geq1$, relational zero-match$h$ (or just rzmatch$h$) is defined over the alphabet

$\mathbb{P}([h]{\times}[h])\times\{\mathtt{l},\mathtt{r}\}$
of all pairs consisting of a binary relation on $[h]=\{0,1,\dots,h{-}1\}$ and of a side tag. Every instance of rzmatch$h$ is of the form
$(A,\mathtt{l})(B,\mathtt{r})\,$,
for some $A,B\subseteq[h]{\times}[h]$.

Each symbol of an instance is interpreted as an $h$-tall $2$-column directed graph, in which the direction of the arrows is selected by the tag ($\mathtt{l}$: left-to-right; $\mathtt{r}$: right-to-left) and the end-points of the arrows are selected by the relation. E.g., if $h=5$ and

\begin{aligned} A&=\{(0,2),(1,1),(1,3),(2,3),(2,4),(4,4)\} \\ B&=\{(0,1),(1,0),(1,3),(1,4),(2,2),(4,1)\} \,, \end{aligned}
then the symbols $(A,\mathtt{l})$ and $(B,\mathtt{r})$ above represent respectively the graphs: two input symbols when h=5 The full instance is interpreted as the graph that we get by overlaying the two graphs of the individual symbols. E.g., for the symbols above, the interpretation is: a positive instance when h=5 If this graph contains a cycle through the top vertex of the left column, we say it is a zero-match.

An instance is positive if its graph is a zero-match. E.g., the above instance of rzmatch$5$ is positive, because of the bold cycle. In contrast, an instance whose symbols represent the next two graphs on the left a negative instance when h=5 is negative, because none of the cycles contained in their overlay on the right pass through the top left vertex.

notes

Introduced by Berman Lingas 1977, as a problem in 2N\1D which witnesses the asymptotic value of the trade-off in the conversion from 2NFAs to 1DFAs. (Actually defined over a $4$-symbol alphabet.)

The name “relational zero-match” was suggested by Kapoutsis Pighizzini 2011.

See relational match for the variant where we ask for any cycle, and relational path for a variant where we ask for a path. See functional zero-match for the restriction to functions.

1N co-1N f1D/uny
1N co-1N f1D/uny 1D re-1D re-1N co-1D rc-1D rc-1N 1D/uny 21D/uny e1D/uny r1D/uny 1N/uny 21N/uny e1N/uny r1N/uny f1N/uny RD/uny 2RD/uny eRD/uny rRD/uny fRD/uny RN/uny 2RN/uny eRN/uny rRN/uny fRN/uny SD/uny 2SD/uny eSD/uny rSD/uny fSD/uny SN/uny 2SN/uny eSN/uny rSN/uny fSN/uny 2D/uny 22D/uny e2D/uny r2D/uny f2D/uny 2N/uny 22N/uny e2N/uny r2N/uny f2N/uny re-1D/uny re-21D/uny re-e1D/uny re-r1D/uny re-f1D/uny re-1N/uny re-21N/uny re-e1N/uny re-r1N/uny re-f1N/uny re-RD/uny re-2RD/uny re-eRD/uny re-rRD/uny re-fRD/uny re-RN/uny re-2RN/uny re-eRN/uny re-rRN/uny re-fRN/uny re-SD/uny re-2SD/uny re-eSD/uny re-rSD/uny re-fSD/uny re-SN/uny re-2SN/uny re-eSN/uny re-rSN/uny re-fSN/uny re-2D/uny re-22D/uny re-e2D/uny re-r2D/uny re-f2D/uny re-2N/uny re-22N/uny re-e2N/uny re-r2N/uny re-f2N/uny co-1D/uny co-21D/uny co-e1D/uny co-r1D/uny co-f1D/uny co-1N/uny co-21N/uny co-e1N/uny co-r1N/uny co-f1N/uny co-RD/uny co-2RD/uny co-eRD/uny co-rRD/uny co-fRD/uny co-RN/uny co-2RN/uny co-eRN/uny co-rRN/uny co-fRN/uny co-SD/uny co-2SD/uny co-eSD/uny co-rSD/uny co-fSD/uny co-SN/uny co-2SN/uny co-eSN/uny co-rSN/uny co-fSN/uny co-2D/uny co-22D/uny co-e2D/uny co-r2D/uny co-f2D/uny co-2N/uny co-22N/uny co-e2N/uny co-r2N/uny co-f2N/uny rc-1D/uny rc-21D/uny rc-e1D/uny rc-r1D/uny rc-f1D/uny rc-1N/uny rc-21N/uny rc-e1N/uny rc-r1N/uny rc-f1N/uny rc-RD/uny rc-2RD/uny rc-eRD/uny rc-rRD/uny rc-fRD/uny rc-RN/uny rc-2RN/uny rc-eRN/uny rc-rRN/uny rc-fRN/uny rc-SD/uny rc-2SD/uny rc-eSD/uny rc-rSD/uny rc-fSD/uny rc-SN/uny rc-2SN/uny rc-eSN/uny rc-rSN/uny rc-fSN/uny rc-2D/uny rc-22D/uny rc-e2D/uny rc-r2D/uny rc-f2D/uny rc-2N/uny rc-22N/uny rc-e2N/uny rc-r2N/uny rc-f2N/uny
RN re-RN co-RN rc-RN
RN re-RN co-RN rc-RN 21D e1D r1D f1D 21N e1N r1N f1N 2RD eRD rRD fRD 2RN eRN rRN fRN 2SD eSD rSD fSD SN 2SN eSN rSN fSN 22D e2D r2D f2D 2N 22N e2N r2N f2N re-21D re-e1D re-r1D re-f1D re-21N re-e1N re-r1N re-f1N re-2RD re-eRD re-rRD re-fRD re-2RN re-eRN re-rRN re-fRN re-2SD re-eSD re-rSD re-fSD re-SN re-2SN re-eSN re-rSN re-fSN re-22D re-e2D re-r2D re-f2D re-2N re-22N re-e2N re-r2N re-f2N co-21D co-e1D co-r1D co-f1D co-21N co-e1N co-r1N co-f1N co-2RD co-eRD co-rRD co-fRD co-2RN co-eRN co-rRN co-fRN co-2SD co-eSD co-rSD co-fSD co-SN co-2SN co-eSN co-rSN co-fSN co-22D co-e2D co-r2D co-f2D co-2N co-22N co-e2N co-r2N co-f2N rc-21D rc-e1D rc-r1D rc-f1D rc-21N rc-e1N rc-r1N rc-f1N rc-2RD rc-eRD rc-rRD rc-fRD rc-2RN rc-eRN rc-rRN rc-fRN rc-2SD rc-eSD rc-rSD rc-fSD rc-SN rc-2SN rc-eSN rc-rSN rc-fSN rc-22D rc-e2D rc-r2D rc-f2D rc-2N rc-22N rc-e2N rc-r2N rc-f2N
complete for:
– no record –
also hard for:
– no record –
cowl:$\leq^\text{lac}_\text{1D}$ cowl$^r$:$\leq^\text{lac}_\text{1D}$ disj$^c$:$\leq^\text{lac}_\text{1D}$ fcomp:$\leq^\text{lac}_\text{1D}$ fcomp$^r$:$\leq^\text{lac}_\text{RD}$ fcomp$^c$:$\leq^\text{lac}_\text{1D}$ fcomp$^{rc}$:$\leq^\text{lac}_\text{RD}$ fmatch:$\leq^\text{t}_\text{h}$ fmatch:$\leq^\text{lac}_\text{1D}$ fmatch$^r$:$\leq^\text{t}_\text{h}$ fmatch$^r$:$\leq^\text{lac}_\text{1D}$ fpath:$\leq^\text{lac}_\text{RD}$ fpath:$\leq^\text{lac}_\text{1N}$ fpath$^r$:$\leq^\text{lac}_\text{RD}$ fpath$^r$:$\leq^\text{lac}_\text{1N}$ fzmatch:$\leq_\text{id}$ fzmatch$^r$:$\leq^\text{t}_\text{h}$ fzmatch$^r$:$\leq^\text{lac}_\text{1D}$ incl$^c$:$\leq^\text{lac}_\text{1D}$ incl$^{rc}$:$\leq^\text{lac}_\text{1D}$ mem:$\leq^\text{lac}_\text{1D}$ mem$^r$:$\leq^\text{lac}_\text{1D}$ mem$^c$:$\leq^\text{lac}_\text{1D}$ mem$^{rc}$:$\leq^\text{lac}_\text{1D}$ proj:$\leq^\text{lac}_\text{RD}$ proj$^r$:$\leq^\text{lac}_\text{1D}$ proj$^c$:$\leq^\text{lac}_\text{RD}$ proj$^{rc}$:$\leq^\text{lac}_\text{1D}$
cowl:$\leq^\text{lac}_\text{1D}$ cowl$^r$:$\leq^\text{lac}_\text{1D}$ disj$^c$:$\leq^\text{lac}_\text{1D}$ fcomp:$\leq^\text{lac}_\text{1D}$ fcomp$^r$:$\leq^\text{lac}_\text{RD}$ fcomp$^c$:$\leq^\text{lac}_\text{1D}$ fcomp$^{rc}$:$\leq^\text{lac}_\text{RD}$ fmatch:$\leq^\text{t}_\text{h}$ fmatch:$\leq^\text{lac}_\text{1D}$ fmatch$^r$:$\leq^\text{t}_\text{h}$ fmatch$^r$:$\leq^\text{lac}_\text{1D}$ fpath:$\leq^\text{lac}_\text{RD}$ fpath:$\leq^\text{lac}_\text{1N}$ fpath$^r$:$\leq^\text{lac}_\text{RD}$ fpath$^r$:$\leq^\text{lac}_\text{1N}$ fzmatch:$\leq_\text{id}$ fzmatch$^r$:$\leq^\text{t}_\text{h}$ fzmatch$^r$:$\leq^\text{lac}_\text{1D}$ incl$^c$:$\leq^\text{lac}_\text{1D}$ incl$^{rc}$:$\leq^\text{lac}_\text{1D}$ mem:$\leq^\text{lac}_\text{1D}$ mem$^r$:$\leq^\text{lac}_\text{1D}$ mem$^c$:$\leq^\text{lac}_\text{1D}$ mem$^{rc}$:$\leq^\text{lac}_\text{1D}$ proj:$\leq^\text{lac}_\text{RD}$ proj$^r$:$\leq^\text{lac}_\text{1D}$ proj$^c$:$\leq^\text{lac}_\text{RD}$ proj$^{rc}$:$\leq^\text{lac}_\text{1D}$ cowl:$\leq_\text{1D}$ cowl:$\leq^\text{lac}_\text{RD}$ cowl:$\leq_\text{RD}$ cowl:$\leq^\text{lac}_\text{2D}$ cowl:$\leq_\text{2D}$ cowl:$\leq^\text{lac}_\text{1N}$ cowl:$\leq_\text{1N}$ cowl$^r$:$\leq_\text{1D}$ cowl$^r$:$\leq^\text{lac}_\text{RD}$ cowl$^r$:$\leq_\text{RD}$ cowl$^r$:$\leq^\text{lac}_\text{2D}$ cowl$^r$:$\leq_\text{2D}$ cowl$^r$:$\leq^\text{lac}_\text{1N}$ cowl$^r$:$\leq_\text{1N}$ disj$^c$:$\leq_\text{1D}$ disj$^c$:$\leq^\text{lac}_\text{RD}$ disj$^c$:$\leq_\text{RD}$ disj$^c$:$\leq^\text{lac}_\text{2D}$ disj$^c$:$\leq_\text{2D}$ disj$^c$:$\leq^\text{lac}_\text{1N}$ disj$^c$:$\leq_\text{1N}$ disj$^{rc}$:$\leq^\text{lac}_\text{1D}$ disj$^{rc}$:$\leq_\text{1D}$ disj$^{rc}$:$\leq^\text{lac}_\text{RD}$ disj$^{rc}$:$\leq_\text{RD}$ disj$^{rc}$:$\leq^\text{lac}_\text{2D}$ disj$^{rc}$:$\leq_\text{2D}$ disj$^{rc}$:$\leq^\text{lac}_\text{1N}$ disj$^{rc}$:$\leq_\text{1N}$ fcomp:$\leq_\text{1D}$ fcomp:$\leq^\text{lac}_\text{RD}$ fcomp:$\leq_\text{RD}$ fcomp:$\leq^\text{lac}_\text{2D}$ fcomp:$\leq_\text{2D}$ fcomp:$\leq^\text{lac}_\text{1N}$ fcomp:$\leq_\text{1N}$ fcomp$^r$:$\leq_\text{RD}$ fcomp$^r$:$\leq^\text{lac}_\text{2D}$ fcomp$^r$:$\leq_\text{2D}$ fcomp$^c$:$\leq_\text{1D}$ fcomp$^c$:$\leq^\text{lac}_\text{RD}$ fcomp$^c$:$\leq_\text{RD}$ fcomp$^c$:$\leq^\text{lac}_\text{2D}$ fcomp$^c$:$\leq_\text{2D}$ fcomp$^c$:$\leq^\text{lac}_\text{1N}$ fcomp$^c$:$\leq_\text{1N}$ fcomp$^{rc}$:$\leq_\text{RD}$ fcomp$^{rc}$:$\leq^\text{lac}_\text{2D}$ fcomp$^{rc}$:$\leq_\text{2D}$ fmatch:$\leq_\text{h}$ fmatch:$\leq_\text{1D}$ fmatch:$\leq^\text{lac}_\text{RD}$ fmatch:$\leq_\text{RD}$ fmatch:$\leq^\text{lac}_\text{2D}$ fmatch:$\leq_\text{2D}$ fmatch:$\leq^\text{lac}_\text{1N}$ fmatch:$\leq_\text{1N}$ fmatch$^r$:$\leq_\text{h}$ fmatch$^r$:$\leq_\text{1D}$ fmatch$^r$:$\leq^\text{lac}_\text{RD}$ fmatch$^r$:$\leq_\text{RD}$ fmatch$^r$:$\leq^\text{lac}_\text{2D}$ fmatch$^r$:$\leq_\text{2D}$ fmatch$^r$:$\leq^\text{lac}_\text{1N}$ fmatch$^r$:$\leq_\text{1N}$ fpath:$\leq_\text{RD}$ fpath:$\leq^\text{lac}_\text{2D}$ fpath:$\leq_\text{2D}$ fpath:$\leq_\text{1N}$ fpath$^r$:$\leq_\text{RD}$ fpath$^r$:$\leq^\text{lac}_\text{2D}$ fpath$^r$:$\leq_\text{2D}$ fpath$^r$:$\leq_\text{1N}$ fzmatch:$\leq^\text{t}_\text{h}$ fzmatch:$\leq_\text{h}$ fzmatch:$\leq^\text{lac}_\text{1D}$ fzmatch:$\leq_\text{1D}$ fzmatch:$\leq^\text{lac}_\text{RD}$ fzmatch:$\leq_\text{RD}$ fzmatch:$\leq^\text{lac}_\text{2D}$ fzmatch:$\leq_\text{2D}$ fzmatch:$\leq^\text{lac}_\text{1N}$ fzmatch:$\leq_\text{1N}$ fzmatch$^r$:$\leq_\text{h}$ fzmatch$^r$:$\leq_\text{1D}$ fzmatch$^r$:$\leq^\text{lac}_\text{RD}$ fzmatch$^r$:$\leq_\text{RD}$ fzmatch$^r$:$\leq^\text{lac}_\text{2D}$ fzmatch$^r$:$\leq_\text{2D}$ fzmatch$^r$:$\leq^\text{lac}_\text{1N}$ fzmatch$^r$:$\leq_\text{1N}$ incl$^c$:$\leq_\text{1D}$ incl$^c$:$\leq^\text{lac}_\text{RD}$ incl$^c$:$\leq_\text{RD}$ incl$^c$:$\leq^\text{lac}_\text{2D}$ incl$^c$:$\leq_\text{2D}$ incl$^c$:$\leq^\text{lac}_\text{1N}$ incl$^c$:$\leq_\text{1N}$ incl$^{rc}$:$\leq_\text{1D}$ incl$^{rc}$:$\leq^\text{lac}_\text{RD}$ incl$^{rc}$:$\leq_\text{RD}$ incl$^{rc}$:$\leq^\text{lac}_\text{2D}$ incl$^{rc}$:$\leq_\text{2D}$ incl$^{rc}$:$\leq^\text{lac}_\text{1N}$ incl$^{rc}$:$\leq_\text{1N}$ mem:$\leq_\text{1D}$ mem:$\leq^\text{lac}_\text{RD}$ mem:$\leq_\text{RD}$ mem:$\leq^\text{lac}_\text{2D}$ mem:$\leq_\text{2D}$ mem:$\leq^\text{lac}_\text{1N}$ mem:$\leq_\text{1N}$ mem$^r$:$\leq_\text{1D}$ mem$^r$:$\leq^\text{lac}_\text{RD}$ mem$^r$:$\leq_\text{RD}$ mem$^r$:$\leq^\text{lac}_\text{2D}$ mem$^r$:$\leq_\text{2D}$ mem$^r$:$\leq^\text{lac}_\text{1N}$ mem$^r$:$\leq_\text{1N}$ mem$^c$:$\leq_\text{1D}$ mem$^c$:$\leq^\text{lac}_\text{RD}$ mem$^c$:$\leq_\text{RD}$ mem$^c$:$\leq^\text{lac}_\text{2D}$ mem$^c$:$\leq_\text{2D}$ mem$^c$:$\leq^\text{lac}_\text{1N}$ mem$^c$:$\leq_\text{1N}$ mem$^{rc}$:$\leq_\text{1D}$ mem$^{rc}$:$\leq^\text{lac}_\text{RD}$ mem$^{rc}$:$\leq_\text{RD}$ mem$^{rc}$:$\leq^\text{lac}_\text{2D}$ mem$^{rc}$:$\leq_\text{2D}$ mem$^{rc}$:$\leq^\text{lac}_\text{1N}$ mem$^{rc}$:$\leq_\text{1N}$ proj:$\leq_\text{RD}$ proj:$\leq^\text{lac}_\text{2D}$ proj:$\leq_\text{2D}$ proj$^r$:$\leq_\text{1D}$ proj$^r$:$\leq^\text{lac}_\text{RD}$ proj$^r$:$\leq_\text{RD}$ proj$^r$:$\leq^\text{lac}_\text{2D}$ proj$^r$:$\leq_\text{2D}$ proj$^r$:$\leq^\text{lac}_\text{1N}$ proj$^r$:$\leq_\text{1N}$ proj$^c$:$\leq_\text{RD}$ proj$^c$:$\leq^\text{lac}_\text{2D}$ proj$^c$:$\leq_\text{2D}$ proj$^{rc}$:$\leq_\text{1D}$ proj$^{rc}$:$\leq^\text{lac}_\text{RD}$ proj$^{rc}$:$\leq_\text{RD}$ proj$^{rc}$:$\leq^\text{lac}_\text{2D}$ proj$^{rc}$:$\leq_\text{2D}$ proj$^{rc}$:$\leq^\text{lac}_\text{1N}$ proj$^{rc}$:$\leq_\text{1N}$
ctwl:$\leq^\text{lac}_\text{1D}$ ctwl$^r$:$\leq^\text{lac}_\text{1D}$ rmatch:$\leq^\text{t}_\text{h}$ rmatch:$\leq^\text{lac}_\text{1D}$ rmatch$^r$:$\leq^\text{t}_\text{h}$ rmatch$^r$:$\leq^\text{lac}_\text{1D}$ rpath:$\leq^\text{lac}_\text{RD}$ rpath:$\leq^\text{lac}_\text{1N}$ rpath$^r$:$\leq^\text{lac}_\text{RD}$ rpath$^r$:$\leq^\text{lac}_\text{1N}$ rzmatch$^r$:$\leq^\text{t}_\text{h}$ rzmatch$^r$:$\leq^\text{lac}_\text{1D}$
ctwl:$\leq^\text{lac}_\text{1D}$ ctwl$^r$:$\leq^\text{lac}_\text{1D}$ rmatch:$\leq^\text{t}_\text{h}$ rmatch:$\leq^\text{lac}_\text{1D}$ rmatch$^r$:$\leq^\text{t}_\text{h}$ rmatch$^r$:$\leq^\text{lac}_\text{1D}$ rpath:$\leq^\text{lac}_\text{RD}$ rpath:$\leq^\text{lac}_\text{1N}$ rpath$^r$:$\leq^\text{lac}_\text{RD}$ rpath$^r$:$\leq^\text{lac}_\text{1N}$ rzmatch$^r$:$\leq^\text{t}_\text{h}$ rzmatch$^r$:$\leq^\text{lac}_\text{1D}$ ctwl:$\leq_\text{1D}$ ctwl:$\leq^\text{lac}_\text{RD}$ ctwl:$\leq_\text{RD}$ ctwl:$\leq^\text{lac}_\text{2D}$ ctwl:$\leq_\text{2D}$ ctwl:$\leq^\text{lac}_\text{1N}$ ctwl:$\leq_\text{1N}$ ctwl$^r$:$\leq_\text{1D}$ ctwl$^r$:$\leq^\text{lac}_\text{RD}$ ctwl$^r$:$\leq_\text{RD}$ ctwl$^r$:$\leq^\text{lac}_\text{2D}$ ctwl$^r$:$\leq_\text{2D}$ ctwl$^r$:$\leq^\text{lac}_\text{1N}$ ctwl$^r$:$\leq_\text{1N}$ rmatch:$\leq_\text{h}$ rmatch:$\leq_\text{1D}$ rmatch:$\leq^\text{lac}_\text{RD}$ rmatch:$\leq_\text{RD}$ rmatch:$\leq^\text{lac}_\text{2D}$ rmatch:$\leq_\text{2D}$ rmatch:$\leq^\text{lac}_\text{1N}$ rmatch:$\leq_\text{1N}$ rmatch$^r$:$\leq_\text{h}$ rmatch$^r$:$\leq_\text{1D}$ rmatch$^r$:$\leq^\text{lac}_\text{RD}$ rmatch$^r$:$\leq_\text{RD}$ rmatch$^r$:$\leq^\text{lac}_\text{2D}$ rmatch$^r$:$\leq_\text{2D}$ rmatch$^r$:$\leq^\text{lac}_\text{1N}$ rmatch$^r$:$\leq_\text{1N}$ rpath:$\leq_\text{RD}$ rpath:$\leq^\text{lac}_\text{2D}$ rpath:$\leq_\text{2D}$ rpath:$\leq_\text{1N}$ rpath$^r$:$\leq_\text{RD}$ rpath$^r$:$\leq^\text{lac}_\text{2D}$ rpath$^r$:$\leq_\text{2D}$ rpath$^r$:$\leq_\text{1N}$ rzmatch$^r$:$\leq_\text{h}$ rzmatch$^r$:$\leq_\text{1D}$ rzmatch$^r$:$\leq^\text{lac}_\text{RD}$ rzmatch$^r$:$\leq_\text{RD}$ rzmatch$^r$:$\leq^\text{lac}_\text{2D}$ rzmatch$^r$:$\leq_\text{2D}$ rzmatch$^r$:$\leq^\text{lac}_\text{1N}$ rzmatch$^r$:$\leq_\text{1N}$
ctwl:$\leq^\text{t}_\text{h}$ ctwl$^r$:$\leq^\text{t}_\text{h}$ rpath:$\leq^\text{t}_\text{h}$ rpath:$\leq^\text{lac}_\text{1D}$ rpath$^r$:$\leq^\text{t}_\text{h}$ rpath$^r$:$\leq^\text{lac}_\text{1D}$ twl:$\leq^\text{lac}_\text{1D}$ twl$^r$:$\leq^\text{lac}_\text{1D}$ twl$^c$:$\leq^\text{t}_\text{h}$ twl$^c$:$\leq^\text{lac}_\text{1D}$ twl$^{rc}$:$\leq^\text{t}_\text{h}$ twl$^{rc}$:$\leq^\text{lac}_\text{1D}$
ctwl:$\leq^\text{t}_\text{h}$ ctwl$^r$:$\leq^\text{t}_\text{h}$ rpath:$\leq^\text{t}_\text{h}$ rpath:$\leq^\text{lac}_\text{1D}$ rpath$^r$:$\leq^\text{t}_\text{h}$ rpath$^r$:$\leq^\text{lac}_\text{1D}$ twl:$\leq^\text{lac}_\text{1D}$ twl$^r$:$\leq^\text{lac}_\text{1D}$ twl$^c$:$\leq^\text{t}_\text{h}$ twl$^c$:$\leq^\text{lac}_\text{1D}$ twl$^{rc}$:$\leq^\text{t}_\text{h}$ twl$^{rc}$:$\leq^\text{lac}_\text{1D}$ ctwl:$\leq_\text{h}$ ctwl$^r$:$\leq_\text{h}$ rpath:$\leq_\text{h}$ rpath:$\leq_\text{1D}$ rpath$^r$:$\leq_\text{h}$ rpath$^r$:$\leq_\text{1D}$ twl:$\leq^\text{t}_\text{h}$ twl:$\leq_\text{h}$ twl:$\leq_\text{1D}$ twl:$\leq^\text{lac}_\text{RD}$ twl:$\leq_\text{RD}$ twl:$\leq^\text{lac}_\text{2D}$ twl:$\leq_\text{2D}$ twl:$\leq^\text{lac}_\text{1N}$ twl:$\leq_\text{1N}$ twl$^r$:$\leq^\text{t}_\text{h}$ twl$^r$:$\leq_\text{h}$ twl$^r$:$\leq_\text{1D}$ twl$^r$:$\leq^\text{lac}_\text{RD}$ twl$^r$:$\leq_\text{RD}$ twl$^r$:$\leq^\text{lac}_\text{2D}$ twl$^r$:$\leq_\text{2D}$ twl$^r$:$\leq^\text{lac}_\text{1N}$ twl$^r$:$\leq_\text{1N}$ twl$^c$:$\leq_\text{h}$ twl$^c$:$\leq_\text{1D}$ twl$^c$:$\leq^\text{lac}_\text{RD}$ twl$^c$:$\leq_\text{RD}$ twl$^c$:$\leq^\text{lac}_\text{2D}$ twl$^c$:$\leq_\text{2D}$ twl$^c$:$\leq^\text{lac}_\text{1N}$ twl$^c$:$\leq_\text{1N}$ twl$^{rc}$:$\leq_\text{h}$ twl$^{rc}$:$\leq_\text{1D}$ twl$^{rc}$:$\leq^\text{lac}_\text{RD}$ twl$^{rc}$:$\leq_\text{RD}$ twl$^{rc}$:$\leq^\text{lac}_\text{2D}$ twl$^{rc}$:$\leq_\text{2D}$ twl$^{rc}$:$\leq^\text{lac}_\text{1N}$ twl$^{rc}$:$\leq_\text{1N}$