SAT Modulo Monotonic Theories

We define the concept of a monotonic theory and show how to build efficient SMT (SAT Modulo Theory) solvers, including effective theory propagation and clause learning, for such theories. We present examples showing that monotonic theories arise from many common problems, e.g., graph properties such as reachability, shortest paths, connected components, minimum spanning tree, and max-flow/min-cut, and then demonstrate our framework by building SMT solvers for each of these theories. We apply these solvers to procedural content generation problems, demonstrating major speed-ups over state-of-the-art approaches based on SAT or Answer Set Programming, and easily solving several instances that were previously impractical to solve

Stability of shortest paths in complex networks with random edge weights

We study shortest paths and spanning trees of complex networks with random edge weights. Edges which do not belong to the spanning tree are inactive in a transport process within the network. The introduction of quenched disorder modifies the spanning tree such that some edges are activated and the network diameter is increased. With analytic random-walk mappings and numerical analysis, we find that the spanning tree is unstable to the introduction of disorder and displays a phase-transition-like behavior at zero disorder strength $\epsilon=0$. In the infinite network-size limit ($N\to \infty$), we obtain a continuous transition with the density of activated edges $\Phi$ growing like $\Phi \sim \epsilon^1$ and with the diameter-expansion coefficient $\Upsilon$ growing like $\Upsilon\sim \epsilon^2$ in the regular network, and first-order transitions with discontinuous jumps in $\Phi$ and $\Upsilon$ at $\epsilon=0$ for the small-world (SW) network and the Barab\'asi-Albert scale-free (SF) network. The asymptotic scaling behavior sets in when $N\gg N_c$, where the crossover size scales as $N_c\sim \epsilon^{-2}$ for the regular network, $N_c \sim \exp[\alpha \epsilon^{-2}]$ for the SW network, and $N_c \sim \exp[\alpha |\ln \epsilon| \epsilon^{-2}]$ for the SF network. In a transient regime with $N\ll N_c$, there is an infinite-order transition with $\Phi\sim \Upsilon \sim \exp[-\alpha / (\epsilon^2 \ln N)]$ for the SW network and $\sim \exp[ -\alpha / (\epsilon^2 \ln N/\ln\ln N)]$ for the SF network. It shows that the transport pattern is practically most stable in the SF network.Comment: 9 pages, 7 figur

Setting Parameters by Example

We introduce a class of "inverse parametric optimization" problems, in which one is given both a parametric optimization problem and a desired optimal solution; the task is to determine parameter values that lead to the given solution. We describe algorithms for solving such problems for minimum spanning trees, shortest paths, and other "optimal subgraph" problems, and discuss applications in multicast routing, vehicle path planning, resource allocation, and board game programming.Comment: 13 pages, 3 figures. To be presented at 40th IEEE Symp. Foundations of Computer Science (FOCS '99