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## Alternative criterion for approximate maximum-weight perfect matching algorithms

Is there any literature on approximate maximum-weight perfect matchings where the approximation criterion is not the factor between the approximate and exact weight sum achieved by each solution, but instead the cardinality of the intersection of the edge sets in the approximate and an optimal solution? The relevant class of graphs $$G(V_1\cup V_2,E)$$ is bipartite and assumed to have perfect matchings, i.e., $$|V_1| = |V_2| = N$$.

To be clear:

An $$(1-\epsilon)$$ approximation algorithm obtains a solution, i.e., a perfect matching $$E’\subset E$$, for which, for some $$\epsilon > 0$$, the sum of edge weights is equal or greater than $$(1-\epsilon)$$MWM, where MWM is the maximal sum of edge weights among all perfect matchings. Assume that $$\mathcal{E}$$ is the set of optimal solutions. I seek an algorithm which produces a perfect matching $$E”$$ satisfying $$\max_{E’\in\mathcal{E}}|E”\cap E’|\geq(1-\epsilon)N$$. I am willing to restrict the problem to the class of graphs for which the optimal solution is unique, and therefore the complicating term “$$\max_{E’\in\mathcal{E}}$$” can be removed from the problem definition.

Randomized algorithms, in the sense that output satisfies the required inequality with probability $$1 – \delta$$, for some $$\delta > 0$$, are also welcome.

I am familiar with Duan and Pettie’s work, and most references therein.

## How one should know about perfect font size?

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Let $$X$$ be a quasi-compact and quasi-separated scheme, and $$U\subseteq X$$ be a quasi-compact open subscheme. Then we can consider $$Rj_*\mathcal{O}_U$$ the (derived) pushforward of the structure sheaf of $$U$$. This is a coconnective sheaf of rings (in fact it is locally of Tor amplitude in $$(-∞,0]$$, since locally it can be represented as a finite limit of sheaves of the form $$\mathcal{O}_X[f^{-1}]$$).

Q: Can $$Rj_*\mathcal{O}_U$$ be represented as a filtered colimit of uniformly bounded below perfect complexes?

Note: I am using the homological grading convention, so, e.g., for me every connective sheaf is bounded below.

The result is true if $$X$$ has an ample family of line bundles, because then I could just use the colimit of Koszul complexes, but I’d like it to be true in more generality.