How can both of them be paying more than half the household costs?  It's possible to have two households under one roof, but not in situations like this.  

Bob...I've never encountered this situation, and my initial reaction was no.  With that said, I did research and found this article.  That's why I'm seeking input.  

https://turbotax.intuit.com/tax-tips/family/guide-to-filing-taxes-as-head-of-household/L4Nx6DYu9

Can two people file Head of Household on their return?

Two people cannot file as Head of Household on the same return. If they are married, then they typically have to either file as Married Filing Jointly on the same return or as Married Filing Separately on separate returns.

Two people can both claim Head of Household filing status while living in the same home. However, both need to meet the criteria necessary to be eligible for Head of Household status:

• You both are unmarried.
• You both are able to claim your own qualifying dependent.
• That dependent resides at the same residence for more than half the year. (In the case of a parent, they can live elsewhere. However, you still have to provide them with at least half of their support.)
• They both have to pay more than half of the cost of keeping up the portion of the home for themselves and their dependent(s).

I think they are referring to cases where there are more than one family in a home.  In your case you only have one family and therefore you can only have one head.

Nice try, but you didn't answer my question.  How do both of them pay more than half the household costs?

OK, I'll give you a hint.  Mary is a teen-ager who like her mother is a vegan.  All that designer food is expensive.  Dad refuses to pay for it.  So Mary and Mom do their own shopping and Mom pays for all of their groceries.  Meanwhile, Dad buys all the food for son John.  Lots of meat and dairy products.  They have separate refrigerators.  

And those separate refrigerators have locks on them (like they do at Fire Stations, for the different shifts..;-)

 think they are referring to cases where there are more than one family in a home

This!

@abctax55 But they do that at fire stations only in case there is an audit, because meals at the firehouse are deductible, remember.  

@GodFather Godfather, quick finder and the tax book have excellent flow charts, etc on this, take a look and you will see that Bob and Iron Man are correct.

@BobKamman 

Not going there...

And nope, they do it so the other shifts don't eat all their goodies 😉

Living in one home doesn't disqualify it, but they would need to have "separate households".

https://www.irs.gov/pub/irs-wd/1998-041.pdf

Shared children is the sticking point here. If they were unmarried and lived together with 2 kids where each kid was from a previous relationship - then you could make a case for HOH but could have a hard time justifying it if it was ever flagged. But since from your info the kids are from their relationship together they won't qualify.