I have two tables:
CREATE TABLE `articles` (
  `id` int(11) NOT NULL AUTO_INCREMENT,
  `title` varchar(1000) DEFAULT NULL,
  `last_updated` datetime DEFAULT NULL         
        
If you have lots of categories, this query cannot be made efficient. No single index can cover two tables at once in MySQL.
You have to do denormalization: add last_updated, has_comments and deleted into article_categories:
CREATE TABLE `article_categories` (
  `article_id` int(11) NOT NULL DEFAULT '0',
  `category_id` int(11) NOT NULL DEFAULT '0',
  `last_updated` timestamp NOT NULL,
  `has_comments` boolean NOT NULL,
  `deleted` boolean NOT NULL,
  PRIMARY KEY (`article_id`,`category_id`),
  KEY `category_id` (`category_id`),
  KEY `ix_articlecategories_category_comments_deleted_updated` (category_id, has_comments, deleted, last_updated)
) ENGINE=InnoDB DEFAULT CHARSET=utf8
and run this query:
SELECT  *
FROM    (
        SELECT  article_id
        FROM    article_categories
        WHERE   (category_id, has_comments, deleted) = (78, 1, 0)
        ORDER BY
                last_updated DESC
        LIMIT   100, 20
        ) q
JOIN    articles a
ON      a.id = q.article_id
Of course you should update article_categories as well whenever you update relevant columns in article. This can be done in a trigger.
Note that the column has_comments is boolean: this will allow using an equality predicate to make a single range scan over the index.
Also note that the LIMIT goes into the subquery. This makes MySQL use late row lookups which it does not use by default. See this article in my blog about why do they increase performance:
If you were on SQL Server, you could make an indexable view over your query, which essentially would make a denormalized indexed copy of article_categories with the additional fields, automatically mainained by the server.
Unfortunately, MySQL does not support this and you will have to create such a table manually and write additional code to keep it in sync with the base tables.