Incorporating distinguishability

So far we only considered the case of a single attribute, which is the fully indistinguishable case. What about multiple attributes? We consider two important cases of distinguishability based on whether attributes get mixed up or not. If attributes are not intertwined, i.e., each $a_i$ in (1) is a single attribute, it is straightforward to see that for $m$ attributes, all we need to do is to run the algorithm for a single attribute $m$ times, once for each attribute. Additional computation can then be performed to calculate more complicated combinations. For example, if we want the lower and upper bounds on the number of all targets for a shadow, then we can simply add up individual lower and upper bounds.

For the second case in which we have multiple attributes grouped together, above approach does not work. Using the example from Fig. 11, suppose that there are two teams, red and blue, and the initial conditions of shadows at $t = t_0$ are of the form $(red \textrm{ or } blue, l_i, u_i)$ . Suppose that we want to get the lower and upper bound of the number of targets in $s_{19}$ again. For lower bounds, four computations are needed: first we set red capacities to 0 and blue capacities to $l_i$ for all edges starting from $S$ . The capacities for each color for edges ending in $T$ are set as before. Running two max-flow computations, one for red and one for blue, gives us one possible lower bound $l_{r1}, l_{b1}$ . Switching red and blue and repeat above procedure gives us another lower bound $l_{r2}, l_{b2}$ . We should have $l_{r1} + l_{b1} = l_{r2} + l_{b2}$ . The lower bound on $s_{19}$ is then $l_{r1} + l_{b1}$ red or blue targets with between $l_{r1}$ and $l_{r_2}$ red targets. The upper bound can be obtained similarly.