20 research outputs found
Decomposition of Trees and Paths via Correlation
We study the problem of decomposing (clustering) a tree with respect to costs
attributed to pairs of nodes, so as to minimize the sum of costs for those
pairs of nodes that are in the same component (cluster). For the general case
and for the special case of the tree being a star, we show that the problem is
NP-hard. For the special case of the tree being a path, this problem is known
to be polynomial time solvable. We characterize several classes of facets of
the combinatorial polytope associated with a formulation of this clustering
problem in terms of lifted multicuts. In particular, our results yield a
complete totally dual integral (TDI) description of the lifted multicut
polytope for paths, which establishes a connection to the combinatorial
properties of alternative formulations such as set partitioning.Comment: v2 is a complete revisio
Quantum annealing for systems of polynomial equations
Numerous scientific and engineering applications require numerically solving
systems of equations. Classically solving a general set of polynomial equations
requires iterative solvers, while linear equations may be solved either by
direct matrix inversion or iteratively with judicious preconditioning. However,
the convergence of iterative algorithms is highly variable and depends, in
part, on the condition number. We present a direct method for solving general
systems of polynomial equations based on quantum annealing, and we validate
this method using a system of second-order polynomial equations solved on a
commercially available quantum annealer. We then demonstrate applications for
linear regression, and discuss in more detail the scaling behavior for general
systems of linear equations with respect to problem size, condition number, and
search precision. Finally, we define an iterative annealing process and
demonstrate its efficacy in solving a linear system to a tolerance of
.Comment: 11 pages, 4 figures. Added example for a system of quadratic
equations. Supporting code is available at
https://github.com/cchang5/quantum_poly_solver . This is a post-peer-review,
pre-copyedit version of an article published in Scientific Reports. The final
authenticated version is available online at:
https://www.nature.com/articles/s41598-019-46729-
On Symmetric Pseudo-Boolean Functions: Factorization, Kernels and Applications
A symmetric pseudo-Boolean function is a map from Boolean tuples to real
numbers which is invariant under input variable interchange. We prove that any
such function can be equivalently expressed as a power series or factorized.
The kernel of a pseudo-Boolean function is the set of all inputs that cause the
function to vanish identically. Any -variable symmetric pseudo-Boolean
function has a kernel corresponding to at least one
-affine hyperplane, each hyperplane is given by a constraint for constant. We use these results to
analyze symmetric pseudo-Boolean functions appearing in the literature of spin
glass energy functions (Ising models), quantum information and tensor networks.Comment: 10 page
Quadratization of Symmetric Pseudo-Boolean Functions
A pseudo-Boolean function is a real-valued function
of binary variables; that is, a mapping from
to . For a pseudo-Boolean function on
, we say that is a quadratization of if is a
quadratic polynomial depending on and on auxiliary binary variables
such that for
all . By means of quadratizations, minimization of is
reduced to minimization (over its extended set of variables) of the quadratic
function . This is of some practical interest because minimization of
quadratic functions has been thoroughly studied for the last few decades, and
much progress has been made in solving such problems exactly or heuristically.
A related paper \cite{ABCG} initiated a systematic study of the minimum number
of auxiliary -variables required in a quadratization of an arbitrary
function (a natural question, since the complexity of minimizing the
quadratic function depends, among other factors, on the number of
binary variables). In this paper, we determine more precisely the number of
auxiliary variables required by quadratizations of symmetric pseudo-Boolean
functions , those functions whose value depends only on the Hamming
weight of the input (the number of variables equal to ).Comment: 17 page