The minimum spanning tree problem with conflict constraints and its variations

AbstractWe consider the minimum spanning tree problem with conflict constraints (MSTC). The problem is known to be strongly NP-hard and computing even a feasible solution is NP-hard. When the underlying graph is a cactus, we show that the feasibility problem is polynomially bounded whereas the optimization version is still NP-hard. When the conflict graph is a collection of disjoint cliques, (equivalently, when the conflict relation is transitive) we observe that MSTC can be solved in polynomial time. We also identify other special cases of MSTC that can be solved in polynomial time. Exploiting these polynomially solvable special cases we derive strong lower bounds. Also, various heuristic algorithms and feasibility tests are discussed along with preliminary experimental results. As a byproduct of this investigation, we show that if an ϵ-optimal solution to the maximum clique problem can be obtained in polynomial time, then a (3ϵ−1)-optimal solution to the maximum edge clique partitioning (Max-ECP) problem can be obtained in polynomial time. As a consequence, we have a polynomial time approximation algorithm for the Max-ECP with performance ratio O(n(loglogn)2log3n), improving the best previously known bound of O(n)

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

Principles of Dataset Versioning: Exploring the Recreation/Storage Tradeoff

The relative ease of collaborative data science and analysis has led to a proliferation of many thousands or millions of $versions$ of the same datasets in many scientific and commercial domains, acquired or constructed at various stages of data analysis across many users, and often over long periods of time. Managing, storing, and recreating these dataset versions is a non-trivial task. The fundamental challenge here is the $storage-recreation\;trade-off$: the more storage we use, the faster it is to recreate or retrieve versions, while the less storage we use, the slower it is to recreate or retrieve versions. Despite the fundamental nature of this problem, there has been a surprisingly little amount of work on it. In this paper, we study this trade-off in a principled manner: we formulate six problems under various settings, trading off these quantities in various ways, demonstrate that most of the problems are intractable, and propose a suite of inexpensive heuristics drawing from techniques in delay-constrained scheduling, and spanning tree literature, to solve these problems. We have built a prototype version management system, that aims to serve as a foundation to our DATAHUB system for facilitating collaborative data science. We demonstrate, via extensive experiments, that our proposed heuristics provide efficient solutions in practical dataset versioning scenarios