This work is motivated by a common urban renewal process called Reconstruct and Divide. It involves the demolition of old buildings and the construction of new ones. Original homeowners are compensated with upgraded apartments, while surplus units are sold for profit, so theoretically it is a win-win project for all parties involved. However, many Reconstruct and Divide projects are withheld or delayed due to disagreements over the assignment of new apartments, claiming they are not fair. The goal of this research is to develop algorithms for envy-free assignment of the new apartments, possibly using monetary payments to reduce envy. In contrast to previous works on envy-free assignment, in our setting the envy depends also on the value of the old apartments, as people with more valuable old apartments expect to get more valuable new apartments. This presents two challenges. First, in some cases, no assignment and payment-vector satisfy the common fairness notions of envy-freeness and proportionality. Hence, we focus on minimizing the envy and the disproportionality (the distance between an agent's value and their proportional share). We present a strongly polynomial-time algorithm that, for a given assignment, finds a payment vector that minimizes the maximum pairwise-envy. We also present a strongly polynomial-time algorithm that computes an assignment and payment-vector that together minimize the maximum disproportionality. Second, directly asking the agents for their subjective valuations for their old apartments is infeasible, as it is a dominant strategy for them to report very high values for their old apartments. We introduce a novel method to elicit agents' valuations indirectly. Using this method, we identify conditions under which our Minimum Disproportionality algorithm is risk-averse truthful.

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