4 research outputs found
A Tool for Computing and Estimating the Volume of the Solution Space of SMT(LA)
There are already quite a few tools for solving the Satisfiability Modulo
Theories (SMT) problems. In this paper, we present \texttt{VolCE}, a tool for
counting the solutions of SMT constraints, or in other words, for computing the
volume of the solution space. Its input is essentially a set of Boolean
combinations of linear constraints, where the numeric variables are either all
integers or all reals, and each variable is bounded. The tool extends SMT
solving with integer solution counting and volume computation/estimation for
convex polytopes. Effective heuristics are adopted, which enable the tool to
deal with high-dimensional problem instances efficiently and accurately
Fair Division Minimizing Inequality
Behavioural economists have shown that people are often averse to inequality
and will make choices to avoid unequal outcomes. In this paper, we consider how
to allocate indivisible goods fairly so as to minimize inequality. We consider
how this interacts with axiomatic properties such as envy-freeness, Pareto
efficiency and strategy-proofness. We also consider the computational
complexity of computing allocations minimizing inequality. Unfortunately, this
is computationally intractable in general so we consider several tractable
greedy online mechanisms that minimize inequality. Finally, we run experiments
to explore the performance of these methods
A Fast and Practical Method to Estimate Volumes of Convex Polytopes
The volume is an important attribute of a convex body. In general, it is
quite difficult to calculate the exact volume. But in many cases, it suffices
to have an approximate value. Volume estimation methods for convex bodies have
been extensively studied in theory, however, there is still a lack of practical
implementations of such methods. In this paper, we present an efficient method
which is based on the Multiphase Monte-Carlo algorithm to estimate volumes of
convex polytopes. It uses the coordinate directions hit-and-run method, and
employs a technique of reutilizing sample points. The experiments show that our
method can efficiently handle instances with dozens of dimensions with high
accuracy
Counting the Number of Solutions to Constraints
Compared with constraint satisfaction problems, counting problems have
received less attention. In this paper, we survey research works on the
problems of counting the number of solutions to constraints. The constraints
may take various forms, including, formulas in the propositional logic, linear
inequalities over the reals or integers, Boolean combination of linear
constraints. We describe some techniques and tools for solving the counting
problems, as well as some applications (e.g., applications to automated
reasoning, program analysis, formal verification and information security)