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Let $G = (X\cup Y, E)$ be a bipartite graph. Suppose $X$ contains people, $Y$ contains seats in a theatre, and each edge $(x,y)$ has a weight representing how much person $x$ is willing to pay for seat $y$. The goal is to find a maximum-weight matching (to maximize the total profit).

But now there are social distancing constraints: if some seat $y_k$ is matched, then seats $y_{k+1}$ and $y_{k-1}$ must remain unmatched. Is there a polynomial-time algorithm for finding a maximum-weight matching?

WHAT I TRIED

  1. In the special case in which $|Y|=2|X|-1$ and we also require the matching to also have maximum cardinality, we must match only the odd-indexed vertices of $Y$, so the problem reduces to simple maximum-weight matching. But I am interested in maximum weight matching, and in the general case in which $|Y|$ may be arbitrary.

  2. Suppose all weights are 0 or 1. Without social distancing, the graph can be converted into a flow-network by adding a source node before $X$, adding a sink node after $Y$, and making the capacity of all edges equal to 1. Then, an integral flow corresponds to a matching. But I do not see how to convert the social distancing constraints to flow constraints.

  3. A common technique for finding a maximum-cardinality mathcing is to iteratively apply augmenting paths. But here, an augmenting path might break the distance constraint.

  4. Another common technique is to solve a linear program with a variable $z_{x,y}$ for each pair $x\in X, y\in Y$. Without distance constraints, the problem is to maximize $\sum_{x\in X, y\in Y} w_{x,y} z_{x,y}$ subject to the following constraints:

  • $\sum_{y \in Y} z_{x,y} \leq 1$ for all $x \in X$
  • $\sum_{x \in X} z_{x,y} \leq 1$ for all $y \in Y$
  • $0 \leq z_{x,y}\leq 1$ for all $x \in X$ and $y \in Y $

The constraint matrix is known to be totally-unimodular, so an integer solution exists. But With gaps, we have to replace the second set of constraints with:

  • $\sum_{x \in X} [z_{x,y} + z_{x,y+1}] \leq 1$ for all $y \in Y$

and I do not know whether the resulting matrix is totally-unimodular.

Is anything else known about this problem?

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