santa.js

function santa(data) {
    // get the names from the first index of the array
    var names = data.map(a => a.shift()),
    result = {};
    // build the data structure as an object of gifter => giftees
    names.map((name, i) => result[name] = data[i].map((v, j) => v ? names[j] : false).filter(v => v));

    // iterate around each name
    while (names.length) {
        // make sure we start with the name that has the fewest choices
        var name;
        names = names.sort((a, b) => result[a].length - result[b].length);
        name = names.shift();

        // if we don't have any names left, it must be impossible
        if (result[name].length == 0) {
            return false
        }
        else {
            // pick a name at random from the list
            var selectedName = result[name][Math.floor(result[name].length * Math.random())];
            // and set it as the object
            result[name] = selectedName;

            // if it's not just a string now...
            if (typeof result[selectedName] == 'object') {
                // filter the current name from the list
                result[selectedName] = result[selectedName].filter(value => value != name);
            }

            // filter all the remaining arrays to ensure we can't pick the same one again
            names.map(currentName => result[currentName] = result[currentName].filter(value => value != selectedName));
        }
    }

    return result;
}

/* usage:
santa([
    ["Alice", 0, 0, 1, 1, 1, 1, 1],
    ["Bob",   0, 0, 0, 0, 0, 1, 0],
    ["Carla", 1, 1, 0, 0, 0, 1, 1],
    ["Dan",   1, 1, 0, 0, 0, 1, 1],
    ["Erin",  1, 1, 0, 0, 0, 1, 1],
    ["Frank", 1, 1, 1, 1, 1, 0, 1],
    ["Gary",  1, 1, 1, 1, 1, 1, 0]
]);
// might return something like:
[
    ["Alice", "Dan"],
    ["Bob", "Erin"],
    ["Carla", "Bob"],
    ["Dan", "Alice"],
    ["Erin", "Frank"],
    ["Frank", "Carla"]
]

santa([
    ["Alice", 0, 1, 1, 1, 0, 1],
    ["Bob",   0, 0, 1, 1, 0, 1],
    ["Carla", 0, 1, 0, 1, 0, 1],
    ["Dan",   0, 1, 1, 0, 0, 0],
    ["Erin",  0, 0, 1, 0, 0, 1],
    ["Frank", 1, 1, 1, 1, 1, 0]
]);
// would always return
false

santa([
    ["Alice", 0, 1, 1],
    ["Bob",   1, 0, 1],
    ["Carla", 1, 1, 0]
]);
// might return something like:
[
    ["Alice", "Bob"],
    ["Bob", "Carla"],
    ["Carla", "Alice"]
]
*/