2 research outputs found
Minimizing Symmetric Set Functions Faster
We describe a combinatorial algorithm which, given a monotone and consistent
symmetric set function d on a finite set V in the sense of Rizzi, constructs a
non trivial set S minimizing d(S,V-S). This includes the possibility for the
minimization of symmetric submodular functions. The presented algorithm
requires at most as much time as the one described by Rizzi, but depending on
the function d, it may allow several improvements.Comment: 9 page
Combinatorial Optimization Algorithms via Polymorphisms
An elegant characterization of the complexity of constraint satisfaction
problems has emerged in the form of the the algebraic dichotomy conjecture of
[BKJ00]. Roughly speaking, the characterization asserts that a CSP {\Lambda} is
tractable if and only if there exist certain non-trivial operations known as
polymorphisms to combine solutions to {\Lambda} to create new ones. In an
entirely separate line of work, the unique games conjecture yields a
characterization of approximability of Max-CSPs. Surprisingly, this
characterization for Max-CSPs can also be reformulated in the language of
polymorphisms.
In this work, we study whether existence of non-trivial polymorphisms implies
tractability beyond the realm of constraint satisfaction problems, namely in
the value-oracle model. Specifically, given a function f in the value-oracle
model along with an appropriate operation that never increases the value of f ,
we design algorithms to minimize f . In particular, we design a randomized
algorithm to minimize a function f that admits a fractional polymorphism which
is measure preserving and has a transitive symmetry.
We also reinterpret known results on MaxCSPs and thereby reformulate the
unique games conjecture as a characterization of approximability of max-CSPs in
terms of their approximate polymorphisms