Conjunctive query evaluation — the fundamental operation in database theory — has complexity controlled by a parameter called submodular width. The existing optimal algorithm (PANDA) achieves the submodular-width bound by working in dual space: it manipulates entropic certificates and polymatroids, then reconstructs the query answer from the dual representation. The approach is correct and optimal but opaque — the dual-space manipulations obscure what's happening to the actual data.
Jaguar works in primal space. Instead of constructing dual certificates, it directly computes a sequence of guided joins — each join chosen to minimize the intermediate result size while maintaining the submodular-width bound. The algorithm maintains a polymatroid that tracks the available join orderings and selects the next join adaptively, based on the data it's seen so far.
The runtime matches PANDA: O(N^(subw(Q)+ε)) for any Boolean conjunctive query Q. But the algorithm is significantly simpler because it operates on the objects you care about (relations, tuples, joins) rather than on their duals (entropic vectors, polymatroids). The simplicity isn't just aesthetic — it makes the algorithm easier to implement, easier to extend to degree constraints, and easier to combine with other database optimizations.
The structural point: the same complexity bound can be achieved from opposite sides of a duality. The dual approach (PANDA) proves the bound by showing that any query can be decomposed into a bounded number of entropic pieces. The primal approach (Jaguar) proves the bound by showing that adaptive joins never exceed the submodular-width cost. The duality is mathematical; the implementations differ in what they touch — data or certificates — and the data-touching version is the one practitioners can use.