Alkyd resin paints usually contain solvents such as petroleum spirit without expecting them to shorten the time of drying. They just help in enhancing drying tolerance under adverse conditions (such as low temperature, high humidity). The main purpose for using these solvents is to get easier & wider application properties and to have a higher level of performance on difficult substrates (such as heavily stained or powdery surfaces).
To fasten drying these paints, paint drying catalysts ( or driers) are added. These driers are metal soaps or coordination compounds which shorten the total drying time to far less than the undesirable time (over 24 hours in most applications).