Parallel symbolic state-space exploration is difficult, but what is the alternative?

State-space exploration is an essential step in many modeling and analysis problems. Its goal is to find the states reachable from the initial state of a discrete-state model described. The state space can used to answer important questions, e.g., "Is there a dead state?" and "Can N become negative?", or as a starting point for sophisticated investigations expressed in temporal logic. Unfortunately, the state space is often so large that ordinary explicit data structures and sequential algorithms cannot cope, prompting the exploration of (1) parallel approaches using multiple processors, from simple workstation networks to shared-memory supercomputers, to satisfy large memory and runtime requirements and (2) symbolic approaches using decision diagrams to encode the large structured sets and relations manipulated during state-space generation. Both approaches have merits and limitations. Parallel explicit state-space generation is challenging, but almost linear speedup can be achieved; however, the analysis is ultimately limited by the memory and processors available. Symbolic methods are a heuristic that can efficiently encode many, but not all, functions over a structured and exponentially large domain; here the pitfalls are subtler: their performance varies widely depending on the class of decision diagram chosen, the state variable order, and obscure algorithmic parameters. As symbolic approaches are often much more efficient than explicit ones for many practical models, we argue for the need to parallelize symbolic state-space generation algorithms, so that we can realize the advantage of both approaches. This is a challenging endeavor, as the most efficient symbolic algorithm, Saturation, is inherently sequential. We conclude by discussing challenges, efforts, and promising directions toward this goal

Lex-Partitioning: A New Option for BDD Search

For the exploration of large state spaces, symbolic search using binary decision diagrams (BDDs) can save huge amounts of memory and computation time. State sets are represented and modified by accessing and manipulating their characteristic functions. BDD partitioning is used to compute the image as the disjunction of smaller subimages. In this paper, we propose a novel BDD partitioning option. The partitioning is lexicographical in the binary representation of the states contained in the set that is represented by a BDD and uniform with respect to the number of states represented. The motivation of controlling the state set sizes in the partitioning is to eventually bridge the gap between explicit and symbolic search. Let n be the size of the binary state vector. We propose an O(n) ranking and unranking scheme that supports negated edges and operates on top of precomputed satcount values. For the uniform split of a BDD, we then use unranking to provide paths along which we partition the BDDs. In a shared BDD representation the efforts are O(n). The algorithms are fully integrated in the CUDD library and evaluated in strongly solving general game playing benchmarks.Comment: In Proceedings GRAPHITE 2012, arXiv:1210.611

Improved Optimal and Approximate Power Graph Compression for Clearer Visualisation of Dense Graphs

Drawings of highly connected (dense) graphs can be very difficult to read. Power Graph Analysis offers an alternate way to draw a graph in which sets of nodes with common neighbours are shown grouped into modules. An edge connected to the module then implies a connection to each member of the module. Thus, the entire graph may be represented with much less clutter and without loss of detail. A recent experimental study has shown that such lossless compression of dense graphs makes it easier to follow paths. However, computing optimal power graphs is difficult. In this paper, we show that computing the optimal power-graph with only one module is NP-hard and therefore likely NP-hard in the general case. We give an ILP model for power graph computation and discuss why ILP and CP techniques are poorly suited to the problem. Instead, we are able to find optimal solutions much more quickly using a custom search method. We also show how to restrict this type of search to allow only limited back-tracking to provide a heuristic that has better speed and better results than previously known heuristics.Comment: Extended technical report accompanying the PacificVis 2013 paper of the same nam

An extensive English language bibliography on graph theory and its applications, supplement 1

Graph theory and its applications - bibliography, supplement

Optimizing the Efficiency of the United States Organ Allocation System through Region Reorganization

Allocating organs for transplantation has been controversial in the United States for decades. Two main allocation approaches developed in the past are (1) to allocate organs to patients with higher priority at the same locale; (2) to allocate organs to patients with the greatest medical need regardless of their locations. To balance these two allocation preferences, the U.S. organ transplantation and allocation network has lately implemented a three-tier hierarchical allocation system, dividing the U.S. into 11 regions, composed of 59 Organ Procurement Organizations (OPOs). At present, an procured organ is offered first at the local level, and then regionally and nationally. The purpose of allocating organs at the regional level is to increase the likelihood that a donor-recipient match exists, compared to the former allocation approach, and to increase the quality of the match, compared to the latter approach. However, the question of which regional configuration is the most efficient remains unanswered. This dissertation develops several integer programming models to find the most efficient set of regions. Unlike previous efforts, our model addresses efficient region design for the entire hierarchical system given the existing allocation policy. To measure allocation efficiency, we use the intra-regional transplant cardinality. Two estimates are developed in this dissertation. One is a population-based estimate; the other is an estimate based on the situation where there is only one waiting list nationwide. The latter estimate is a refinement of the former one in that it captures the effect of national-level allocation and heterogeneity of clinical and demographic characteristics among donors and patients. To model national-level allocation, we apply a modeling technique similar to spill-and-recapture in the airline fleet assignment problem. A clinically based simulation model is used in this dissertation to estimate several necessary parameters in the analytic model and to verify the optimal regional configuration obtained from the analytic model. The resulting optimal region design problem is a large-scale set-partitioning problem in whichthere are too many columns to handle explicitly. Given this challenge, we adapt branch and price in this dissertation. We develop a mixed-integer programming pricing problem that is both theoretically and practically hard to solve. To alleviate this existing computational difficulty, we apply geographic decomposition to solve many smaller-scale pricing problems based on pre-specified subsets of OPOs instead of a big pricing problem. When solving each smaller-scale pricing problem, we also generate multiple ``promising' regions that are not necessarily optimal to the pricing problem. In addition, we attempt to develop more efficient solutions for the pricing problem by studying alternative formulations and developing strong valid inequalities. The computational studies in this dissertation use clinical data and show that (1) regional reorganization is beneficial; (2) our branch-and-price application is effective in solving the optimal region design problem

OBOE: Collaborative Filtering for AutoML Model Selection

Algorithm selection and hyperparameter tuning remain two of the most challenging tasks in machine learning. Automated machine learning (AutoML) seeks to automate these tasks to enable widespread use of machine learning by non-experts. This paper introduces OBOE, a collaborative filtering method for time-constrained model selection and hyperparameter tuning. OBOE forms a matrix of the cross-validated errors of a large number of supervised learning models (algorithms together with hyperparameters) on a large number of datasets, and fits a low rank model to learn the low-dimensional feature vectors for the models and datasets that best predict the cross-validated errors. To find promising models for a new dataset, OBOE runs a set of fast but informative algorithms on the new dataset and uses their cross-validated errors to infer the feature vector for the new dataset. OBOE can find good models under constraints on the number of models fit or the total time budget. To this end, this paper develops a new heuristic for active learning in time-constrained matrix completion based on optimal experiment design. Our experiments demonstrate that OBOE delivers state-of-the-art performance faster than competing approaches on a test bed of supervised learning problems. Moreover, the success of the bilinear model used by OBOE suggests that AutoML may be simpler than was previously understood