Although in many cases they get bad results, the specific case on which I am focusing is the situation where the public school is about average, but is placed in a high-cost urban area (e.g. New York City, San Francisco, Washington DC) and so the per-pupil costs escalate to $20K and beyond.
Mass home school: At those prices, once you have 3 kids, it no longer makes sense to work for a middle class income. The school district should be paying the parents directly and taking a cut for standardized testing and infrequent (once a year) checkups. High school may still be indicated for some specialized classes (e.g. for chemistry labs) and certain parents (e.g. ESL) may not be capable, so it is not applicable for every family. However to save facility costs, you can have on and off days for students, especially in rural areas where transportation times are long.
Charter bidding: specify a percentage to perform on standardized tests and a student selection procedure (e.g. random lottery, opt-in, special education, ESL) and have each charter school take a block of children on a flat rate. Should work well if there are multiple established charters or private schools in the area; will work less well in rural areas.
Shuttling outside the urban center: most of the difference of school costs are driven by salaries, which in turn are driven by housing/child care. Instead of trying to school the children in the high cost urban area, have them boarding school or shuttle to suburban area where the labor pool can live further out. Basically meeting in the middle where the facility cost may still be somewhat expensive, but the labor rates should be lower.
Sell the school buildings and go all-tutor: instead of trying to hold onto expensive facilities and working bus schedules, push most or all of the children into private homes of ~10 or less students. In some rural areas or in the suburbs where there may be a number of small underutilized spaces, it may be correct. For some suburbs or urban pockets where the facilities are becoming underutilized, it may allow the school district to sell the school parcel outright instead of uneconomic renovate/remodel, bringing down overhead costs.
Boarding school during the week (or working parent lodging in the city during the week): when all else fails, do you want your kids to learn at an affordable price, or do you want them to fail and grow up to be gangsters? Strongly recommended in failing urban schools in bad neighborhoods.