  In this paper, we resolve the computational complexity of a number of
outstanding open problems with practical applications. Here is the list of
problems we show to be PPAD-complete, along with the domains of practical
significance: Fractional Stable Paths Problem (FSPP) [21] - Internet routing;
Core of Balanced Games [41] - Economics and Game theory; Scarf's Lemma [41] -
Combinatorics; Hypergraph Matching [1]- Social Choice and Preference Systems;
Fractional Bounded Budget Connection Games (FBBC) [30] - Social networks; and
Strong Fractional Kernel [2]- Graph Theory. In fact, we show that no fully
polynomial-time approximation schemes exist (unless PPAD is in FP).
  This paper is entirely a series of reductions that build in nontrivial ways
on the framework established in previous work. In the course of deriving these
reductions, we created two new concepts - preference games and personalized
equilibria. The entire set of new reductions can be presented as a lattice with
the above problems sandwiched between preference games (at the "easy" end) and
personalized equilibria (at the "hard" end). Our completeness results extend to
natural approximate versions of most of these problems. On a technical note, we
wish to highlight our novel "continuous-to-discrete" reduction from exact
personalized equilibria to approximate personalized equilibria using a linear
program augmented with an exponential number of "min" constraints of a specific
form. In addition to enhancing our repertoire of PPAD-complete problems, we
expect the concepts and techniques in this paper to find future use in
algorithmic game theory.
