Algorithm 1: Greedy Cooperative Sensor Node Pre-Selection Heuristic Based on G-GDOP

Input: C, $G_C=(C,{\mathcal{E}}_C)$, K, $B_{\mathrm{link}}$, $B_{\mathrm{slot}}$, $B_{\mathrm{res}}$, $E_{\min }$, $\left\{E_i\right\}$, $\left\{s_e\right\}$, $\left\{c_i^{\mathrm{node}}\right\}$, $\left\{c_e^{\mathrm{link}}\right\}$, $\widehat{\mathbf{x}}$, $\left\{{\widehat{w}}_e\right\}$, ${\tau }_J$, and ${ϵ}_{\mathrm{tie}}$; use $+\infty$ to disable an upper budget.     Output: $status$, $S_{\mathrm{out}}$, $\mathcal{E}\left(S_{\mathrm{out}}\right)$, M, $T_{\mathrm{slot}}$, and $C_{\mathrm{res}}$.

1: Compute $Q_o$, $d_g$, $d_o$, and the projected weighted link-information vectors ${\left\{a_e\right\}}_{e\in {\mathcal{E}}_C}$.   2:  if preprocessing fails then return SELECTION_NUMERICAL_FAILURE.   3:  $S\leftarrow ⌀$; $\mathcal{E}\leftarrow ⌀$; $M\leftarrow 0$; $T_{\mathrm{slot}}\leftarrow 0$; $C_{\mathrm{res}}\leftarrow 0$; $J_o\leftarrow 0$.   4:  function ${\mathrm{S}}_{\mathrm{TATS}}(i,S)$.   5:    $\Delta {\mathcal{E}}_i\leftarrow \mathcal{E}(S\cup \left\{i\right\})\setminus \mathcal{E}\left(S\right)$.   6:    $\Delta M_i\leftarrow |\Delta {\mathcal{E}}_i|$; $\Delta T_i\leftarrow {\sum }_{e\in \Delta {\mathcal{E}}_i}{s}_e$.   7:    $\Delta C_i\leftarrow c_i^{\mathrm{node}}+{\sum }_{e\in \Delta {\mathcal{E}}_i}{c}_e^{\mathrm{link}}$.   8:    $\Delta J_{o,i}\leftarrow {\sum }_{e\in \Delta {\mathcal{E}}_i}{a}_e{a}_e^{\mathsf{T}}$.   9:    return $(\Delta {\mathcal{E}}_i,\Delta M_i,\Delta T_i,\Delta C_i,\Delta J_{o,i})$. 10:  end function. 11:  function ${\mathrm{F}}_{\mathrm{EASIBLE}}(i,\Delta M_i,\Delta T_i,\Delta C_i)$. 12:    return $(E_i\ge E_{\min })\wedge \left(\right|S|+1\le K)\wedge (M+\Delta M_i\le B_{\mathrm{link}})\wedge (T_{\mathrm{slot}}+\Delta T_i\le B_{\mathrm{slot}})\wedge (C_{\mathrm{res}}+\Delta C_i\le B_{\mathrm{res}})$. 13:  end function.        Stage I: observable-seed construction 14:  while ${\mathrm{rank}}_{{\tau }_J}\left(J_o\right)<d_o$ do. 15:    if $\left|S\right|\ge K$ then return UNOBSERVABLE. 16:    $C_{\mathrm{rem}}\leftarrow C\setminus S$. 17:    if $C_{\mathrm{rem}}=⌀$ then return UNOBSERVABLE. 18:    ${\mathcal{F}}_{\mathrm{seed}}\leftarrow ⌀$; $i^{\star }\leftarrow ⌀$; ${\kappa }^{\star }\leftarrow ⌀$; $r_{\mathrm{cur}}\leftarrow {\mathrm{rank}}_{{\tau }_J}\left(J_o\right)$. 19:    for each $i\in C_{\mathrm{rem}}$ in increasing node-ID order do. 20:         $(\Delta {\mathcal{E}}_i,\Delta M_i,\Delta T_i,\Delta C_i,\Delta J_{o,i})\leftarrow {\mathrm{S}}_{\mathrm{TATS}}(i,S)$. 21:         if $\neg {\mathrm{F}}_{\mathrm{EASIBLE}}(i,\Delta M_i,\Delta T_i,\Delta C_i)$ then continue. 22:         ${\mathcal{F}}_{\mathrm{seed}}\leftarrow {\mathcal{F}}_{\mathrm{seed}}\cup \left\{i\right\}$; $J_o^{+i}\leftarrow J_o+\Delta J_{o,i}$. 23:         $r_i\leftarrow {\mathrm{rank}}_{{\tau }_J}\left(J_o^{+i}\right)$; compute the retained ${\lambda }_{\min }^{+i}$ of $J_o^{+i}$. 24:         ${\kappa }_i\leftarrow (r_i-r_{\mathrm{cur}},{\lambda }_{\min }^{+i},-\Delta T_i,-\Delta C_i,-i)$. 25:         if $i^{\star }=⌀$ or ${\kappa }_i{\succ }_{\mathrm{lex}}{\kappa }^{\star }$ then $i^{\star }\leftarrow i$ and ${\kappa }^{\star }\leftarrow {\kappa }_i$. 26:    end for. 27:    if ${\mathcal{F}}_{\mathrm{seed}}=⌀$ then return RESOURCE_INFEASIBLE. 28:    $(\Delta {\mathcal{E}}_{i^{\star }},\Delta M_{i^{\star }},\Delta T_{i^{\star }},\Delta C_{i^{\star }},\Delta J_{o,i^{\star }})\leftarrow {\mathrm{S}}_{\mathrm{TATS}}(i^{\star },S)$; $S\leftarrow S\cup \left\{i^{\star }\right\}$; $\mathcal{E}\leftarrow \mathcal{E}\cup \Delta {\mathcal{E}}_{i^{\star }}$. 29:    $M\leftarrow M+\Delta M_{i^{\star }}$; $T_{\mathrm{slot}}\leftarrow T_{\mathrm{slot}}+\Delta T_{i^{\star }}$; $C_{\mathrm{res}}\leftarrow C_{\mathrm{res}}+\Delta C_{i^{\star }}$; $J_o\leftarrow J_o+\Delta J_{o,i^{\star }}$. 30:  end while. 31:  $S_{\mathrm{seed}}\leftarrow S$; $k_{\mathrm{seed}}\leftarrow \left|S_{\mathrm{seed}}\right|$. 32:  if $J_o^{-1}$ cannot be evaluated then return SELECTION_NUMERICAL_FAILURE. 33:  $P\leftarrow J_o^{-1}$.        Stage II: post-seed A-optimal refinement 34:  while $\left|S\right|<K$ do. 35:    $i^{\star }\leftarrow ⌀$; $g^{\star }\leftarrow -\infty$; $\Delta T^{\star }\leftarrow +\infty$; $\Delta C^{\star }\leftarrow +\infty$; $P^{\star }\leftarrow P$. 36:    for each $i\in C\setminus S$ in increasing node-ID order do. 37:         $(\Delta {\mathcal{E}}_i,\Delta M_i,\Delta T_i,\Delta C_i,\Delta J_{o,i})\leftarrow {\mathrm{S}}_{\mathrm{TATS}}(i,S)$. 38:         if $\neg {\mathrm{F}}_{\mathrm{EASIBLE}}(i,\Delta M_i,\Delta T_i,\Delta C_i)$ then continue. 39:         if $\Delta {\mathcal{E}}_i=⌀$ then $P^{\left(i\right)}\leftarrow P$; else compute $P^{\left(i\right)}$ using Equation (37). 40:         $g_i\leftarrow \mathrm{tr}\left(P\right)-\mathrm{tr}\left(P^{\left(i\right)}\right)$. 41:         if $i^{\star }=⌀$ or $g_i>g^{\star }+{ϵ}_{\mathrm{tie}}$ then store $(i,g_i,\Delta T_i,\Delta C_i,P^{\left(i\right)})$ as the current candidate. 42:         else if $|g_i-g^{\star }|\le {ϵ}_{\mathrm{tie}}$ and $\Delta T_i<\Delta T^{\star }-{ϵ}_{\mathrm{tie}}$ then store $(i,g_i,\Delta T_i,\Delta C_i,P^{\left(i\right)})$. 43:         else if $|g_i-g^{\star }|\le {ϵ}_{\mathrm{tie}}$, $|\Delta T_i-\Delta T^{\star }|\le {ϵ}_{\mathrm{tie}}$, and $\Delta C_i<\Delta C^{\star }-{ϵ}_{\mathrm{tie}}$ then store $(i,g_i,\Delta T_i,\Delta C_i,P^{\left(i\right)})$. 44:         else if $|g_i-g^{\star }|\le {ϵ}_{\mathrm{tie}}$, $|\Delta T_i-\Delta T^{\star }|\le {ϵ}_{\mathrm{tie}}$, $|\Delta C_i-\Delta C^{\star }|\le {ϵ}_{\mathrm{tie}}$, and $i<i^{\star }$ then store $(i,g_i,\Delta T_i,\Delta C_i,P^{\left(i\right)})$. 45:    end for. 46:    if $i^{\star }=⌀$ then break. 47:    $(\Delta {\mathcal{E}}_{i^{\star }},\Delta M_{i^{\star }},\Delta T_{i^{\star }},\Delta C_{i^{\star }},\Delta J_{o,i^{\star }})\leftarrow {\mathrm{S}}_{\mathrm{TATS}}(i^{\star },S)$. 48:    $S\leftarrow S\cup \left\{i^{\star }\right\}$; $\mathcal{E}\leftarrow \mathcal{E}\cup \Delta {\mathcal{E}}_{i^{\star }}$. 49:    $M\leftarrow M+\Delta M_{i^{\star }}$; $T_{\mathrm{slot}}\leftarrow T_{\mathrm{slot}}+\Delta T_{i^{\star }}$; $C_{\mathrm{res}}\leftarrow C_{\mathrm{res}}+\Delta C_{i^{\star }}$. 50:    $J_o\leftarrow J_o+\Delta J_{o,i^{\star }}$; $P\leftarrow P^{\star }$. 51:  end while. 52:  if exact-K selection is required and $\left|S\right|<K$ then return RESOURCE_INFEASIBLE. 53:  if ${\mathrm{rank}}_{{\tau }_J}\left(J_o\right)<d_o$ then return UNOBSERVABLE. 54:  Evaluate the final A-optimal metric and resource statistics. 55:  if the final metric evaluation fails then return SELECTION_NUMERICAL_FAILURE. 56:  $S_{\mathrm{out}}\leftarrow S$. 57:  return $\mathtt{SUCCESS},S_{\mathrm{out}},\mathcal{E}\left(S_{\mathrm{out}}\right),M,T_{\mathrm{slot}},C_{\mathrm{res}}$.