Paper solves NP-hard projection for three-view cardinality constraints.
problem NP-hard projection onto cardinality constraint sets.
method Formulates projection as linear programming with TVCS.
result Linear programming solution matches original projection.
Efficiently solves high-dimensional regression with overlapping groups using greedy hard-thresholding.
problem High-dimensional regression problems with overlapping groups of relevant features.
method Greedy hard-thresholding combined with submodular optimization to avoid NP-hard projections.
result Strong theoretical guarantees even with poorly conditioned data and overlapping features.
New method efficiently solves multi-matching problems with geometric consistency.
problem Robustly matching multiple objects in challenging real-world settings.
method Higher-order Projected Power Iteration method that incorporates geometric consistency.
result Guaranteed cycle-consistent multi-matchings with theoretical convergence.
Dual IHT algorithm solves NP-hard non-convex sparse minimization problems.
problem Non-convex sparse minimization with ℓ2-regularized loss function. method Developed a dual IHT algorithm for maximizing the non-smooth dual objective.
result Sparse recovery performance is invariant to RIP, superior to primal IHT algorithms.
Moving between 3-manifold triangulations is NP-hard
problem Moving between two triangulations of a 3-manifold
method Showing that the number of bistellar moves and sparse degree-two edge collapses is NP-hard
result First NP-hardness result concerning moves between two triangulations of a 3-manifold
Link crossing number problem is NP-hard.
problem Determining the crossing number of a link.
method NP-hardness proof.
result Link crossing number problem is NP-hard.
Proving NP-hardness of shellability and decomposability for simplicial complexes.
problem Determining shellability and decomposability of simplicial complexes.
method Reduction to prove NP-hardness for shellability and decomposability problems.
result Proving NP-hardness and NP-completeness for shellability and decomposability problems in simplicial complexes.
Knot theory problems are proven NP-hard or NP-complete.
problem Problems in knot theory, like diagram transformations and unlinking, are computationally hard.
method Proved NP-hard or NP-complete using Reidemeister moves and sublink analysis.
result Certain knot theory problems are computationally intractable.
Trading system uses NP-hard optimization to select stocks for high Sharpe ratio trading.
problem Finding profitable, uncorrelated stocks for high Sharpe ratio trading.
method NP-hard combinatorial optimization using Ising machine and simulated bifurcation algorithm.
result Trading strategy with FPGA-based system achieves 164 μs response latency.
Detecting convexity in polynomials over boxes is NP-hard even for degree 3.
problem Detecting convexity in polynomials over compact regions, especially boxes.
method Proof by reduction to the NP-hard problem of testing global convexity of polynomials of degree four.
result The problem of testing convexity over a box is strongly NP-hard even for polynomials of degree 3.
Paper tackles graph structure learning via spectral constraints.
problem Learning graphs with specific structures from data.
method Convert structural constraints to Laplacian eigenvalue constraints, integrate with Gaussian graphical modeling.
result Unified framework for learning various graph structures, convergent and scalable.
Proving NP-hardness of embedding 3-manifolds with boundary tori into R^3.
problem Deciding embeddability of 3-manifolds with boundary tori into R^3.
method Reduction from satisfiability problem, using techniques from low-dimensional topology, particularly Dehn fillings on link complements.
result Proving NP-hardness of the embeddability problem for 3-manifolds with boundary tori.
Proving NP-hardness of unknotting and related link problems.
problem Determining if a knot can be untangled with a limited number of moves.
method Proving NP-hardness through reductions to known hard problems.
result Several link problems are proven to be NP-hard.
Hard problem to determine if 3D shape can be split into genus-g parts.
problem Deciding if a 3-manifold has a Heegaard splitting of genus g or less.
method Reduction from NP-complete CNF-SAT problem to Heegaard Genus ≤ g problem.
result Computing Heegaard genus is NP-hard.
We describe ways to define and calculate L1-norm signal subspaces which are less sensitive to outlying data than L2-calculated subspaces. We focus on the computation of the L1 maximum-projection principal component of a data matrix containing N signal samples of dimension D and conclude that the general proble…
Computing PL geometric category in 2D is NP-hard.
problem Determining the PL geometric category of 2D polyhedra.
method Reduction from shellability of 2-complexes, which is known to be NP-hard.
result It is NP-hard to decide whether the PL geometric category of a 2D polyhedron is at most 2.
Optimal CL requires perfect memory and is NP-hard.
problem Designing CL algorithms that perform reliably and avoid catastrophic forgetting.
method Theoretical approach to derive computational properties of optimal CL algorithms.
result Optimal CL algorithms generally solve an NP-hard problem and require perfect memory.
Hardness result for approximating manifold radius.
problem Approximating the radius of triangulated manifolds.
method Proving NP-hardness for almost-polynomial approximation.
result It is NP-hard to approximate the hyperspherical radius up to an almost-polynomial factor.
Deep learning synthesizes diverse solutions for NP-hard problems.
problem Finding optimal solutions for NP-hard problems.
method Graph convolutional network trained to estimate solution likelihood; guided tree search.
result Substantial performance improvement over recent deep learning work.
Researchers prove NP-hardness of learning parameter-bounded Bayes nets.
problem Learning parameter-bounded Bayes nets is computationally hard.
method Proved NP-hardness of learning parameter-bounded Bayes nets and a promise search variant.
result Proved NP-hardness of a promise search variant of LEARN.
This work tackles fast and accurate low-rank factorization of compressed data.
problem Accurately and efficiently computing low-rank matrix or tensor factorizations from compressed data.
method Factorization in the compressed domain followed by reconstruction of original factors.
result Provable recovery of original factors under certain conditions.
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.
problem Solving NP-hard reasoning problems from natural inputs.
method Scalable neural architecture and loss function for discrete Graphical Models.
result Empirically shows efficient learning of NP-hard problems.
NP-hard problem found for non-orientable surfaces in 3D manifolds.
problem Finding non-orientable surfaces in triangulated 3-manifolds.
method Proved NP-hardness and provided an explicit algorithm for odd Euler genus.
result Proved NP-hardness and provided an explicit algorithm for odd Euler genus.
The paper proves shellability is hard for d-balls when d is at least 3.
problem Shellability for d-balls is NP-hard when d ≥ 3.
method NP-hardness proof for triangulated d-balls and d-manifolds/d-pseudomanifolds with boundary.
result Shellability is NP-hard for triangulated d-balls when d ≥ 3.
Transformers struggle to learn Markovian dynamics, showing NP-hard optimization challenges.
problem Understanding transformers' limitations in learning Markovian dynamical functions.
method Investigated through a structured ICL setup, analyzing loss landscapes and parameter optimization.
result Recovering optimal transformer parameters for Markovian functions is NP-hard.
Two DRL policies collaborate to solve NP-hard routing problems.
problem Solving complex routing problems like TSP without expert knowledge.
method Learning Collaborative Policies (LCP) using seeder and reviser policies.
result Improves solution quality over single-policy DRL on various NP-hard routing problems.
New method improves solving NP-hard problems on graphs.
problem Solving combinatorial optimization problems on graphs.
method A novel reinforcement learning strategy based on AlphaGo Zero for graph embeddings.
result Our method generalizes better to various graphs than S2V-DQN.
Bailouts in financial networks are hard to optimize due to NP-hardness.
problem Optimizing bailouts in a network of insolvent banks.
method Modeling bailouts as an optimization problem, proving NP-hardness and inapproximability.
result Banks can strategically alter debt contracts to increase their market value in the event of a bailout.
New algorithms for ranking data under two distance measures.
problem Rank aggregation under Kendall τ and Spearman footrule distances.
method Constant-approximation algorithms for NP-hard problems.
result Illustrative applications on the Mallows model and genomic data.
Study shows optimal RL with transition look-ahead is NP-hard for ℓ≥2.
problem Optimal reinforcement learning with transition look-ahead is computationally hard.
method Proved NP-hardness for ℓ≥2 using linear programming. result There is a precise boundary between tractable and intractable cases for RL with look-ahead.
Paper tackles NP-hard multi-agent planning with reinforcement learning.
problem Solving NP-hard multi-agent, multi-task planning problems with time-dependent rewards.
method Developed a reinforcement learning framework using mean-field inference and auction-based selection.
result Achieved near-optimality and transferability in solving MRRC and IPMS problems.
Paper proves MDS NP-hard and provides a PTAS.
problem Theoretical limitations of MDS objective function.
method Proves NP-hardness and provides a PTAS approximation algorithm.
result Minimizing Kamada-Kawai objective is NP-hard.
Equity-Transformer solves NP-hard min-max routing problems efficiently.
problem Min-max routing problems with multiple agents and large-scale applications.
method Sequential planning approach with Transformer and equitable workload distribution inductive biases.
result Significant runtime and cost reductions in min-max mTSP and min-max mPDP tasks.
Solves NP-hard MAP inference by avoiding fractional vertices.
problem Finding the most likely configuration in graphical models.
method Proves polynomial-time MAP inference under a condition on fractional vertices.
result Can provably perform MAP inference in polynomial time when a bounded number of fractional vertices are present.
Boolean logic used for neural network training and inference, with convergence analysis.
problem Discrete optimization in neural networks with Boolean logic.
method Boolean logic backpropagation with convergence analysis.
result First convergence analysis for Boolean logic in neural networks.
New method uses reinforcement learning to improve Simulated Annealing.
problem Optimization problems with unknown cost functions.
method Replaces Metropolis engine with Macau Algorithm.
result Effective heuristic for unknown cost functions.
This research uses reinforcement learning to find optimal emission offsets in greenhouse gas markets.
problem Finding optimal emission offsets in greenhouse gas markets to control excess emissions.
method Utilized reinforcement learning, specifically Nash-DQN, to estimate market Nash equilibria.
result Emitting firms can achieve significant financial savings by abiding by the Nash equilibria found in the market.
Efficiently solves tensor nonnegative rank-one approximation problem.
problem Nonnegative tensor nonnegative rank-one approximation.
method Closed-form KL principal component and iterative EM algorithm.
result The KL principal component minimizes generalized Kullback-Leibler divergence efficiently.
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.
problem Training two-layer neural networks is computationally hard in fixed dimensions.
method Parameterized complexity analysis considering dimension and number of neurons.
result Training two-layer neural networks is NP-hard for two dimensions.
Paper develops zeroth and first order stochastic Frank-Wolfe algorithms for constrained optimization.
problem Optimization problems with difficult-to-project deterministic constraints and efficient projection constraints.
method Stochastic Frank-Wolfe algorithms with momentum and trimmed variants.
result Guaranteed fast convergence rates comparable to unconstrained problems.
Deep neural networks approximate solutions to NP-hard problems.
problem Approximating solutions to NP-hard combinatorial optimization problems.
method Homotopic recurrent neural networks combined with reinforcement learning.
result Homotopic RNNs improve the quality of solutions compared to vanilla RNNs.
Paper solves NP-hard haplotyping problem using matrix completion.
problem Reconstructing inherited genetic variations from DNA sequencing data.
method Binary matrix factorization and alternating minimization.
result The proposed technique achieves lower haplotype reconstruction error.
New proof shows a link problem is hard without complex links.
problem Deciding if a link contains a trivial sublink is hard.
method Reduces from Independent Set Problem, avoiding Brunnian links.
result The Trivial Sublink Problem is NP-hard due to mod 2 linking.
This article contains detailed proofs and additional examples related to the UAI-2013 submission `Learning Sparse Causal Models is not NP-hard'. It describes the FCI+ algorithm: a method for sound and complete causal model discovery in the presence of latent confounders and/or selection bias, that has worst case polyno…
New MIP formulations for neural network Lipschitz constant estimation.
problem Ensuring robustness of neural networks by calculating their Lipschitz constant.
method Reformulating the neural network Lipschitz estimation problem as a Quadratically Constrained MIP (MIQCQP) problem.
result Solutions of the MIQCQP formulations provide bounds on the Lipschitz constant, with conditions for exactness.
EPMF factorizes matrices by adjusting their entries to match a specified power.
problem Factorizing matrices with adjusted entries to match a specified power.
method Analyzes the computational complexity of exact and approximate EPMF problems.
result Exact EPMF is strongly NP-hard, but can be solved in polynomial time when rank is fixed.