Algebraic formulation for packing problem
It is possible to prove the fixed-parameter tractability of various graph-theoretical problems (e.g., finding a path of length k or finding k disjoint triangles) using algebraic techniques by reducing the problem to questions about certain polynomials. See for example the following very readable paper by Ryan Williams:

http://arxiv.org/abs/0807.3026v3

and some other related papers:

http://www.cs.cmu.edu/~jkoutis/papers/MultilinearDetection.pdf

http://doi.ieeecomputersociety.org/10.1109/FOCS.2010.24

http://arxiv.org/abs/0912.2371

About your meta question: as the formulation of graph-theoretical problems is usually "combinatorial," it is not surprising to me that most of the natural approaches are "combinatorial" rather than "algebraic." The "algebraic" approaches work only if the problem has some unexpected connection with algebra, which is probably quite rare for natural "combinatorial" problems. In the above-mentioned papers, such a connection is discovered, allowing us to solve the problem using algebraic techniques. Interestingly, for these problems the algebraic algorithms are faster (have better bounds) than the combinatorial ones.