Fig. 5
On top, we have a signed instance \((\pi ,\breve{\pi },\breve{\iota })\) of the \({\textsc {SbIRT}}\) problem, with \(\pi =({-2}~{+1}~{-3})\), \(\breve{\pi }=(5,3,7,2)\), and \(\breve{\iota }=(4,4,8,1)\), which is mapped by the \({\mathcal {F}}\) function into an unsigned instance \((\pi ^{\prime },\breve{\pi }^{\prime },\breve{\iota }^{\prime })\) (at the bottom), such that \(\pi ^{\prime }=(4~3~1~2~6~5)\), \(\breve{\pi }^{\prime }=(5,0,3,0,7,0,2)\), and \(\breve{\iota }^{\prime }=(4,0,4,0,8,0,1)\). The \({\mathcal {G}}\) function maps a solution \(S^{\prime }=\big (\tau ^{(1,3,5)}_{(4,3,0)},\rho ^{(3,4)}_{(1,3)},\rho ^{(5,6)}_{(7,1)}\big )\) for the instance \((\pi ^{\prime },\breve{\pi }^{\prime },\breve{\iota }^{\prime })\) into a valid solution \(S=\big (\tau ^{(1,2,3)}_{(4,3,0)},\rho ^{(2,2)}_{(1,3)},\rho ^{(3,3)}_{(7,1)}\big )\), with same size, for the instance \((\pi ,\breve{\pi },\breve{\iota })\)