I've gotten this to work, API definition can be found here https://us1.api.mailchimp.com/schema/3.0/Segments/Merge/InterestSegment.json
Interests have to be grouped under interest categories (called 'Groups' in some parts of the UI).
Here is the JSON for the segment_opts member of the recipients array:
"segment_opts": {
"match": "any",
"conditions": [{
"condition_type": "Interests",
"field": "interests-31f7aec0ec",
"op": "interestcontains",
"value": ["a9014571b8", "5e824ac953"]
}]
}
Here is the PHP array version with comments. The 'match' member refers to the the rules in the array of 'conditions'. The segment can match any, all, or none of the conditions. This example has only one condition, but others can be added as additional arrays in the 'conditions' array:
$segment_opts = array(
'match' => 'any', // or 'all' or 'none'
'conditions' => array (
array(
'condition_type' => 'Interests', // note capital I
'field' => 'interests-31f7aec0ec', // ID of interest category
// This ID is tricky: it is
// the string "interests-" +
// the ID of interest category
// that you get from MailChimp
// API (31f7aec0ec)
'op' => 'interestcontains', // or interestcontainsall, interestcontainsnone
'value' => array (
'a9014571b8', // ID of interest in that category
'5e824ac953' // ID of another interest in that category
)
)
)
);