Inapproximability of Treewidth and Related Problems

Graphical models, such as Bayesian Networks and Markov networks play an important role in artificial intelligence and machine learning. Inference is a central problem to be solved on these networks. This, and other problems on these graph models are often known to be hard to solve in general, but tractable on graphs with bounded Treewidth. Therefore, finding or approximating the Treewidth of a graph is a fundamental problem related to inference in graphical models. In this paper, we study the approximability of a number of graph problems: Treewidth and Pathwidth of graphs, Minimum Fill-In, One-Shot Black (and Black-White) pebbling costs of directed acyclic graphs, and a variety of different graph layout problems such as Minimum Cut Linear Arrangement and Interval Graph Completion. We show that, assuming the recently introduced Small Set Expansion Conjecture, all of these problems are NP-hard to approximate to within any constant factor in polynomial time.

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Adding Edges for Maximizing Weighted Reachability

Algorithms ◽

10.3390/a13030068 ◽

2020 ◽

Vol 13 (3) ◽

pp. 68 ◽

Cited By ~ 1

Author(s):

Federico Corò ◽

Gianlorenzo D'Angelo ◽

Cristina M. Pinotti

Keyword(s):

Polynomial Time ◽

Greedy Algorithms ◽

Dynamic Programming Algorithm ◽

Directed Acyclic Graphs ◽

Constant Factor ◽

Programming Algorithm ◽

Single Source ◽

Graph Augmentation ◽

Set Size ◽

Acyclic Graphs

In this paper, we consider the problem of improving the reachability of a graph. We approach the problem from a graph augmentation perspective, in which a limited set size of edges is added to the graph to increase the overall number of reachable nodes. We call this new problem the Maximum Connectivity Improvement (MCI) problem. We first show that, for the purpose of solve solving MCI, we can focus on Directed Acyclic Graphs (DAG) only. We show that approximating the MCI problem on DAG to within any constant factor greater than 1 − 1 / e is NP -hard even if we restrict to graphs with a single source or a single sink, and the problem remains NP -complete if we further restrict to unitary weights. Finally, this paper presents a dynamic programming algorithm for the MCI problem on trees with a single source that produces optimal solutions in polynomial time. Then, we propose two polynomial-time greedy algorithms that guarantee ( 1 − 1 / e ) -approximation ratio on DAGs with a single source, a single sink or two sources.

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Consistency Guarantees for Greedy Permutation-Based Causal Inference Algorithms

Biometrika ◽

10.1093/biomet/asaa104 ◽

2021 ◽

Author(s):

L Solus ◽

Y Wang ◽

C Uhler

Keyword(s):

Graphical Models ◽

Directed Acyclic Graph ◽

Real Data ◽

Directed Acyclic Graphs ◽

Equivalence Classes ◽

Acyclic Graph ◽

Basic Task ◽

Acyclic Graphs ◽

Edge Graph ◽

Markov Equivalence

Abstract Directed acyclic graphical models are widely used to represent complex causal systems. Since the basic task of learning such a model from data is NP-hard, a standard approach is greedy search over the space of directed acyclic graphs or Markov equivalence classes of directed acyclic graphs. As the space of directed acyclic graphs on p nodes and the associated space of Markov equivalence classes are both much larger than the space of permutations, it is desirable to consider permutation-based greedy searches. Here, we provide the first consistency guarantees, both uniform and high-dimensional, of a greedy permutation-based search. This search corresponds to a simplex-like algorithm operating over the edge-graph of a subpolytope of the permutohedron, called a directed acyclic graph associahedron. Every vertex in this polytope is associated with a directed acyclic graph, and hence with a collection of permutations that are consistent with the directed acyclic graph ordering. A walk is performed on the edges of the polytope maximizing the sparsity of the associated directed acyclic graphs. We show via simulated and real data that this permutation search is competitive with current approaches.

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Fairness Criteria through the Lens of Directed Acyclic Graphs

The Oxford Handbook of Ethics of AI ◽

10.1093/oxfordhb/9780190067397.013.31 ◽

2020 ◽

pp. 491-520

Author(s):

Benjamin R. Baer ◽

Daniel E. Gilbert ◽

Martin T. Wells

Keyword(s):

Graphical Models ◽

Directed Acyclic Graphs ◽

Substantial Portion ◽

Intuitive Appeal ◽

Acyclic Graphs ◽

Causal Criteria ◽

Alternate Source ◽

Significant Attention

This chapter provides an alternate source of intuition about fairness criteria using probabilistic directed acyclic graphical models. A substantial portion of the literature on fairness in algorithms proposes, analyzes, and operationalizes simple formulaic criteria for assessing fairness. Two of these criteria—Equalized Odds and Calibration by Group—have gained significant attention not only for their simplicity and intuitive appeal but also for their incompatibility. Graphical models have been used to motivate and expose fairness criteria in other works, especially those which work with explicitly causal criteria for fairness. The chapter then argues that graphical models provide an invaluable source of intuition even in noncausal scenarios and reveal the weakness of Equalized Odds.

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CONDITIONAL INDEPENDENCE STRUCTURES AND GRAPHICAL MODELS

International Journal of Uncertainty Fuzziness and Knowledge-Based Systems ◽

10.1142/s0218488503002326 ◽

2003 ◽

Vol 11 (05) ◽

pp. 545-571 ◽

Cited By ~ 11

Author(s):

BARBARA VANTAGGI

Keyword(s):

Graphical Models ◽

Conditional Independence ◽

Random Variables ◽

Directed Acyclic Graphs ◽

Lower Probability ◽

Conditional Probabilities ◽

Stochastic Independence ◽

Representation Problem ◽

Acyclic Graphs

In this paper we study conditional independence structures arising from conditional probabilities and lower conditional probabilities. Such models are based on notions of stochastic independence apt to manage also those situations where zero evaluations on possible events are present: this is particularly crucial for lower probability. The "graphoid" properties of such models are investigated, and the representation problem of conditional independence structures is dealt with by generalizing the wellknown classic separation criteria for undirected and directed acyclic graphs. Our graphical models describe the independence statements and the possible logical dependencies among the random variables.

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Propagating Distributions Up Directed Acyclic Graphs

Neural Computation ◽

10.1162/089976699300016881 ◽

1999 ◽

Vol 11 (1) ◽

pp. 215-227 ◽

Cited By ~ 1

Author(s):

Eric B. Baum ◽

Warren D. Smith

Keyword(s):

Graphical Models ◽

Directed Acyclic Graph ◽

Search Tree ◽

Directed Acyclic Graphs ◽

Game Tree ◽

Worst Case ◽

Exponential Time ◽

Acyclic Graphs ◽

Game Trees ◽

Directed Growth

In a previous article, we considered game trees as graphical models. Adopting an evaluation function that returned a probability distribution over values likely to be taken at a given position, we described how to build a model of uncertainty and use it for utility-directed growth of the search tree and for deciding on a move after search was completed. In some games, such as chess and Othello, the same position can occur more than once, collapsing the game tree to a directed acyclic graph (DAG). This induces correlations among the distributions at sibling nodes. This article discusses some issues that arise in extending our algorithms to a DAG. We give a simply described algorithm for correctly propagating distributions up a game DAG, taking account of dependencies induced by the DAG structure. This algorithm is exponential time in the worst case. We prove that it is #P complete to propagate distributions up a game DAG correctly. We suggest how our exact propagation algorithm can yield a fast but inexact heuristic.

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