arXiv research

Boolean logic used for neural network training and inference, with convergence analysis.

New method controls gradient error for sparse MRFs.

Efficient algorithm for Bayesian networks reduces marginal probability distribution computation.

Moving between 3-manifold triangulations is NP-hard

This paper explores the possibility of near-optimally solving multi-agent, multi-task NP-hard planning problems with time-dependent rewards using a learning-based algorithm. In particular, we consider a class of robot/machine scheduling problems called the multi-robot reward collection problem (MRRC). Such MRRC problem…

Trading system uses NP-hard optimization to select stocks for high Sharpe ratio trading.

We prove that for every $d\geq 2$, deciding if a pure, $d$-dimensional, simplicial complex is shellable is NP-hard, hence NP-complete. This resolves a question raised, e.g., by Danaraj and Klee in 1978. Our reduction also yields that for every $d \ge 2$ and $k \ge 0$, deciding if a pure, $d$-dimensional, simplicial com…

Bayesian inference for topics in documents with many potential causes.

We show that determining the crossing number of a link is NP-hard. For some weaker notions of link equivalence, we also show NP-completeness.

Inference in general Markov random fields (MRFs) is NP-hard, though identifying the maximum a posteriori (MAP) configuration of pairwise MRFs with submodular cost functions is efficiently solvable using graph cuts. Marginal inference, however, even for this restricted class, is in #P. We prove new formulations of deriv…

Computing PL geometric category in 2D is NP-hard.

Optimal CL requires perfect memory and is NP-hard.

Given a graphical model, one essential problem is MAP inference, that is, finding the most likely configuration of states according to the model. Although this problem is NP-hard, large instances can be solved in practice. A major open question is to explain why this is true. We give a natural condition under which we …

Researchers prove NP-hardness of learning parameter-bounded Bayes nets.

We give a reduction from {\sc clique} to establish that sparse PCA is NP-hard. The reduction has a gap which we use to exclude an FPTAS for sparse PCA (unless P=NP). Under weaker complexity assumptions, we also exclude polynomial constant-factor approximation algorithms.

Paper introduces scalable neural architecture for solving NP-hard problems.

Generative models (GMs) such as Generative Adversary Network (GAN) and Variational Auto-Encoder (VAE) have thrived these years and achieved high quality results in generating new samples. Especially in Computer Vision, GMs have been used in image inpainting, denoising and completion, which can be treated as the inferen…

We show that {\sc Heegaard Genus $\leq g$}, the problem of deciding whether a triangulated 3-manifold admits a Heegaard splitting of genus less than or equal to $g$, is NP-hard. The result follows from a quadratic time reduction of the NP-complete problem {\sc CNF-SAT} to {\sc Heegaard Genus $\leq g$}.

It has recently been shown that the problem of testing global convexity of polynomials of degree four is {strongly} NP-hard, answering an open question of N.Z. Shor. This result is minimal in the degree of the polynomial when global convexity is of concern. In a number of applications however, one is interested in test…

The paper proves shellability is hard for d-balls when d is at least 3.

We present a new family of zero-field Ising models over $N$ binary variables/spins obtained by consecutive "gluing" of planar and $O(1)$-sized components and subsets of at most three vertices into a tree. The polynomial-time algorithm of the dynamic programming type for solving exact inference (computing partition func…

Boolean matrix factorization and Boolean matrix completion from noisy observations are desirable unsupervised data-analysis methods due to their interpretability, but hard to perform due to their NP-hardness. We treat these problems as maximum a posteriori inference problems in a graphical model and present a message p…

Transformers struggle to learn Markovian dynamics, showing NP-hard optimization challenges.

Optimal intervention in economic networks modeled as influence maximization, with hard computational problems.

We prove that certain problems naturally arising in knot theory are NP--hard or NP--complete. These are the problems of obtaining one diagram from another one of a link in a bounded number of Reidemeister moves, determining whether a link has an unlinking or splitting number $k$, finding a $k$-component unlink as a sub…

Two DRL policies collaborate to solve NP-hard routing problems.

We prove that the problem of deciding whether a 2- or 3-dimensional simplicial complex embeds into $\mathbb{R}^3$ is NP-hard. Our construction also shows that deciding whether a 3-manifold with boundary tori admits an $\mathbb{S}^{3}$ filling is NP-hard. The former stands in contrast with the lower dimensional cases wh…

Neural model with parameterized algorithms improves graph CO problem solving.

Bailouts in financial networks are hard to optimize due to NP-hardness.

Study shows optimal RL with transition look-ahead is NP-hard for $\ell \geq 2$.

In this paper, we provide new complexity results for algorithms that learn discrete-variable Bayesian networks from data. Our results apply whenever the learning algorithm uses a scoring criterion that favors the simplest model able to represent the generative distribution exactly. Our results therefore hold whenever t…

Sparse linear regression is hard to solve efficiently, even with k-sparse solutions.

Paper proves MDS NP-hard and provides a PTAS.

Transformers improve solving mixed-integer programs, especially CLSP.

Equity-Transformer solves NP-hard min-max routing problems efficiently.

Bayesian networks (BNs) are graphical models that are useful for representing high-dimensional probability distributions. There has been a great deal of interest in recent years in the NP-hard problem of learning the structure of a BN from observed data. Typically, one assigns a score to various structures and the sear…

Given data over the joint distribution of two random variables $X$ and $Y$, we consider the problem of inferring the most likely causal direction between $X$ and $Y$. In particular, we consider the general case where both $X$ and $Y$ may be univariate or multivariate, and of the same or mixed data types. We take an inf…

We present a method for the reconstruction of networks, based on the order of nodes visited by a stochastic branching process. Our algorithm reconstructs a network of minimal size that ensures consistency with the data. Crucially, we show that global consistency with the data can be achieved through purely local consid…

New method uses reinforcement learning to improve Simulated Annealing.

We prove that deciding if a diagram of the unknot can be untangled using at most $k$ Riedemeister moves (where $k$ is part of the input) is NP-hard. We also prove that several natural questions regarding links in the $3$-sphere are NP-hard, including detecting whether a link contains a trivial sublink with $n$ componen…

New method improves MAP inference for CGMs on path graphs, avoiding approximation and maintaining integrality.

We present a learning-based approach to computing solutions for certain NP-hard problems. Our approach combines deep learning techniques with useful algorithmic elements from classic heuristics. The central component is a graph convolutional network that is trained to estimate the likelihood, for each vertex in a graph…

It is well known that Sparse PCA (Sparse Principal Component Analysis) is NP-hard to solve exactly on worst-case instances. What is the complexity of solving Sparse PCA approximately? Our contributions include: 1) a simple and efficient algorithm that achieves an $n^{-1/3}$-approximation; 2) NP-hardness of approximatio…

Training neural networks is hard in fixed dimensions.

StrNN uses neural network structures to learn conditional independencies.

In this paper we address the problem of finding the most probable state of a discrete Markov random field (MRF), also known as the MRF energy minimization problem. The task is known to be NP-hard in general and its practical importance motivates numerous approximate algorithms. We propose a submodular relaxation approa…

New proof shows a link problem is hard without complex links.

LGS-Net improves NCO performance on combinatorial optimization tasks.