Marzullo's algorithm based on https://gist.github.com/3684493

marzullo.php

<?php
/**
 * Marzullo's algorithm
 *
 * @see http://en.wikipedia.org/wiki/Marzullo%27s_algorithm
 * @see https://gist.github.com/3684493
 */
function marzullo_algorithm($ranges)
{
    $table = array();
    foreach ($ranges as $range) {
        $table[] = array($range[0], -1);
        $table[] = array($range[1], +1);
    }
    usort($table, function($lhs, $rhs) {
        $result = intval($lhs[0]) - intval($rhs[0]);
        if ($result === 0) {
            $result = intval($rhs[1]) - intval($lhs[1]); // to exclude 'pathological overlaps'
        }
        return $result;
    });
    
    $best = 0;
    $count = 0;
    $len = count($table) - 1;
    for ($i = 0; $i < $len; $i++) {
        $count = $count - $table[$i][1];
        if ($best < $count) {
            $best = $count;
            $beststart = $table[$i][0];
            $bestend = $table[$i + 1][0];
        }
    }
    return array($best, $beststart, $bestend);
}

function test($tests) {
    foreach ($tests as $test) {
        $result = marzullo_algorithm($test['input']);
        if ($result == $test['expected']) {
            echo "Test OK\n";
        } else {
            echo 'Test failed: input=' . print_r($test['input']) . ', expected=' . print_r($test['expected']) . ', actual=' . print_r($result) . "\n";
        }
    }
}

$test_data = [
    ['input' => [[8,12],[11,13],[10,12]], 'expected' => [3,11,12]],
    ['input' => [[8,12],[11,13],[14,15]], 'expected' => [2,11,12]],
    ['input' => [[8,9],[8,12],[10,12]], 'expected' => [2,8,9]],
    ['input' => [[8,12],[9,10],[11,13],[10,12]], 'expected' => [3,11,12]],
    ['input' => [[8,12],[9,10],[11,13],[14,15]], 'expected' => [2,9,10]],
    ['input' => [[11,15],[8,15],[9,11],[10,14],[11,14],[9,10],[9,13],[12,15],[8,11],[14,15]], 'expected' => [6,12,13]],
];
test($test_data);