The infinitive (or perhaps it should be analyzed as the subjunctive?) gives a sense of completion, while the gerund gives more of a sense of process. So "There is a new approach to solving homelessness" indicates that the approach will help work on the problem of homelessness, while "There is a new approach to solve homelessness" implies that it will ...
With a syntactically close example
(2a) He was great at buying pictures that brightened up the rooms
it is clear that not all such sentences sound natural (or even acceptable) when using the reduced form
(2b) ?/*He was great at buying pictures brightening up the rooms.
(3a) She had the knack of buying cardigans which matched her skirts