Retrieve all product from one category (tree model)

Question

I'm using MySQL, have categories in a tree model. One product belongs to only one (leaf) category.

What I want to accomplish is, given one category (any, could or not be a leaf or a root), return all products in a GROUP BY sentence.

I have managed to make the query, but the category table is about 100.000+ rows so it is very slow to execute.

SELECT
        Publication._id, GROUP_CONCAT(title) as title,
        GROUP_CONCAT(sold_quantity order by Scrap.stamp) as sold,
        GROUP_CONCAT(Scrap.stamp ORDER BY Scrap.stamp) as dates,
        GROUP_CONCAT(price ORDER BY Scrap.stamp) as prices,
        Category.path_from_root
    FROM
        Publication, Scrap, Category
    WHERE
       (path_from_root LIKE '%" + cat + "%' OR 
        path_from_root LIKE '%" + cat + "%') AND
        Category._id = Publication.category_id AND
        Scrap.id = Publication.scrap
    GROUP BY(Publication._id)
    ORDER BY Publication._id
    LIMIT 1,10

If I remove the category table, the query is super fast. I have the ability to change the database structure. Category structure:

CREATE TABLE `Category` (
 `id` int(11) NOT NULL AUTO_INCREMENT,
 `_id` varchar(300) NOT NULL,
 `name` varchar(300) DEFAULT NULL,
 `status` tinyint(4) NOT NULL DEFAULT '1',
 `stamp` timestamp NOT NULL DEFAULT CURRENT_TIMESTAMP ON UPDATE CURRENT_TIMESTAMP,
 `total_items` int(11) NOT NULL DEFAULT '0',
 `active` tinyint(4) NOT NULL DEFAULT '1',
 `parse` tinyint(4) NOT NULL DEFAULT '0',
 `path_from_root` varchar(500) DEFAULT NULL,
 PRIMARY KEY (`id`)
)