https://stat.duke.edu/~sayan/npcomplete.pdf

The computational complexity of probabilistic inference using Bayesian belief networks (research note)

R J Baxter 1982 London: Academic xii + 486 pp price £43.60 Over the past few years there has been a growing belief that all the twodimensional lattice statistical models will eventually be solved and that it will be Professor Baxter who solves them. Baxter has inherited the mantle of Onsager who started the process by solving exactly the two-dimensional Ising model in 1944.

Exactly Solved Models in Statistical Mechanics

Resource-constrained project scheduling: Notation, classification, models, and methods

A k-tree is a graph that can be reduced to the k-complete graph by a sequence of removals of a degree k vertex with completely connected neighbors. We address the problem of determining whether a graph is a partial graph of a k-tree. This problem is motivated by the existence of polynomial time algorithms for many combinatorial problems on graphs when the graph is constrained to be a partial k-tree for fixed k. These algorithms have practical applications in areas such as reliability, concurrent broadcasting and evaluation of queries in a relational database system. We determine the complexity status of two problems related to finding the smallest number k such that a given graph is a partial k-tree. First, the corresponding decision problem is NP-complete. Second, for a fixed (predetermined) value of k, we present an algorithm with polynomially bounded (but exponential in k) worst case time complexity. Previously, this problem had only been solved for $k = 1,2,3$.

Complexity of finding embeddings in a k -tree

We describe a system that supports arbitrarily complex SQL queries on probabilistic databases. The query semantics is based on a probabilistic model and the results are ranked, much like in Information Retrieval. Our main focus is efficient query evaluation, a problem that has not received attention in the past. We describe an optimization algorithm that can compute efficiently most queries. We show, however, that the data complexity of some queries is #P-complete, which implies that these queries do not admit any efficient evaluation methods. For these queries we describe both an approximation algorithm and a Monte-Carlo simulation algorithm.

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Efficient query evaluation on probabilistic databases

Several enumeration and reliability problems are shown to be # P-complete, and hence, at least as hard as NP-complete problems. Included are important problems in network reliability analysis, namely, computing the probability that a graph is connected and counting the number of minimum cardinality $(s,t)$-cuts or directed network cuts. Also shown to be # P-complete are counting vertex covers in a bipartite graph, counting antichains in a partial order, and approximating the probability that a graph is connected and the probability that a pair of vertices is connected.

The Complexity of Counting Cuts and of Computing the Probability that a Graph is Connected

We introduce the property of k-decomposability for simplicial complexes which, if satisfied by the dual simplicial complex of a convex polyhedron, implies that the diameter of the polyhedron is bounded by a polynomial of degree k + 1 in the number of facets. These properties form a hierarchy, each one implying the next. The strongest, vertex decomposability, implies the Hirsch conjecture; the weakest is equivalent to shellability. We show that several cases in which the Hirsch conjecture has been verified can be handled by these methods, which also give the shellability of a number of simplicial complexes of combinatorial interest. We conclude with a strengthened form of the property of shellability which would imply the Hirsch conjecture for polytopes.

Decompositions of Simplicial Complexes Related to Diameters of Convex Polyhedra

Comparing and computing distances between phylogenetic trees are important biological problems, especially for models where edge lengths play an important role. The geodesic distance measure between two phylogenetic trees with edge lengths is the length of the shortest path between them in the continuous tree space introduced by Billera, Holmes, and Vogtmann. This tree space provides a powerful tool for studying and comparing phylogenetic trees, both in exhibiting a natural distance measure and in providing a euclidean-like structure for solving optimization problems on trees. An important open problem is to find a polynomial time algorithm for finding geodesics in tree space. This paper gives such an algorithm, which starts with a simple initial path and moves through a series of successively shorter paths until the geodesic is attained.

/pdf/a-fast-algorithm-for-computing-geodesic-distances-in-tree-26k2ny5jjb.pdf

A Fast Algorithm for Computing Geodesic Distances in Tree Space

Comparing and computing distances between phylogenetic trees are important biological problems, especially for models where edge lengths play an important role. The geodesic distance measure between two phylogenetic trees with edge lengths is the length of the shortest path between them in the continuous tree space introduced by Billera, Holmes, and Vogtmann. This tree space provides a powerful tool for studying and comparing phylogenetic trees, both in exhibiting a natural distance measure and in providing a Euclidean-like structure for solving optimization problems on trees. An important open problem is to find a polynomial time algorithm for finding geodesics in tree space. This paper gives such an algorithm, which starts with a simple initial path and moves through a series of successively shorter paths until the geodesic is attained.

In this article, computational procedures are presented for generating bounds on measures of network reliability. The two measures considered, reachability and connectedness, are the probability that there is an operating path from a node to all other nodes in a directed (respectively undirected) stochastic network. Our bounds, which are given in terms of polynomials in p, the common arc failure probability, are based on recent bounding results developed by the authors for the class of shellable independence systems. Two pairs of bounds are given: weaker bounds whose computation time is bounded by a polynomial in the size of the network and tighter bounds whose computation time is bounded by a polynomial in the size of the network and the number of minimum-cardinality network cuts. Computational results are also given which evaluate the quality of the bounds. The generation of the bounds involves several interesting path and cut counting problems.

J. Scott Provan

Papers

The Complexity of Counting Cuts and of Computing the Probability that a Graph is Connected

Decompositions of Simplicial Complexes Related to Diameters of Convex Polyhedra

A Fast Algorithm for Computing Geodesic Distances in Tree Space

A Fast Algorithm for Computing Geodesic Distances in Tree Space

Calculating bounds on reachability and connectedness in stochastic networks