P versus NP

A global semicircle law and the exact extreme eigenvalues are proved for biased orientations of random graphs, revealing multiple large eigenvalues not present in the unbiased case.

arxiv:2611.10862 · math.CO · 2026-11

A charged version of the blackfold approximation yields new families of rotating charged black holes and string-dipole rings in five or more dimensions, matching every exact solution available for comparison.

arxiv:1012.4517 · hep-th · 2010-12

The paper gives a complete complexity classification for contingency-based causality in Datalog with negation under the well-founded semantics, covering all six transitions among true, false, and undefined.

arxiv:2611.11756 · cs.DB · 2026-11

Binary direct tokenization is M^{1-ε}-inapproximable but admits an O(M/log M) approximation; fixed-alphabet compression savings have an EPTAS; unary merge models are APX-hard or strongly NP-complete.

arxiv:2611.12348 · cs.DS · 2026-11

The problem of optimally increasing capacities to minimize steady-state congestion is weakly NP-hard on series-parallel networks and strongly NP-hard on general networks; the paper gives exact and approximate algorithms for series-parallel networks.

arxiv:2611.12486 · cs.DS · 2026-11

Type-logical grammars with product, associativity, extraction, and infixation generate at least the quasi-linear languages (NLIN) and have NP-complete fixed recognition.

arxiv:2611.12618 · cs.CC · 2026-11

Monotone Circuit Satisfiability and Hamiltonian Cycle are shown complete for mNP under new positive reductions, and posP ⊊ mNP is shown not to relativize while a weak diagonalization proof exists.

arxiv:2611.09622 · cs.CC · 2026-11

Matching prescribed PageRank scores via B edge additions is NP-complete and constant-factor inapproximable; three heuristics are proposed and evaluated.

arxiv:2611.06782 · cs.SI · 2026-11

A conditional oracle separation shows that NP∩coNP contains problems hard for quantum computers, assuming UP is not in BQP, and a black-box impossibility result limits how such hardness can be derived from standard cryptographic primitives.

arxiv:2611.07413 · quant-ph · 2026-11

SoS refutation certificates for semirandom k-XOR are converted, via a learned decomposition and a sample-splitting voting trick, into an n^{O(ell)}-time algorithm recovering planted solutions to semirandom Boolean CSPs at the (n/ell)^{k/2}ell constraint threshold.

arxiv:2611.08114 · cs.DS · 2026-11

Cyclic integer systems with two nonzeros per row and column, coefficients of absolute value 2 or 3, and right-hand sides in {-1,0,1,2} are strongly NP-complete.

arxiv:2611.08304 · math.OC · 2026-11

Four primitives (everlasting-secure 2QKE, standard 2QKE, EV-StatePuzzs, and one-time secure quantum signatures) are existentially equivalent.

arxiv:2611.05985 · quant-ph · 2026-11

Improved NP-hardness bounds for Max-3-Cut and Max-k-Cut for small k via a negation trick and a general soundness criterion for gadget reductions from Håstad's equations.

arxiv:2611.01402 · cs.CC · 2026-11

Relative to constructed oracles, proofs of quantumness can coexist with P = NP, and quantum-verifier oblivious state preparation can coexist with BQP = QCMA, blocking relativizing cryptographic constructions.

arxiv:2611.05303 · quant-ph · 2026-11

A single score coordinate of a strongly log-concave Gaussian mixture built from a Boolean formula is polynomial-time approximable if and only if P = NP.

arxiv:2611.02346 · cs.CC · 2026-11

For Weil-Petersson random surfaces of genus g with n = o(√g) cusps, the expected number of eigenvalues below 1/4 is O((n+1)^2/g), so the spectral gap converges to 1/4.

arxiv:2611.01477 · math.DG · 2026-11

Unweighted partial interval multicover is NP-complete for unit intervals.

arxiv:2610.41182 · cs.DM · 2026-10

Computing treedepth is NP-hard for planar graphs of maximum degree 3, under polynomial-time Turing reductions, unless P=NP.

arxiv:2610.42033 · cs.DM · 2026-10

Label-tree-based correlation clustering is hard to approximate within a logarithmic factor, yet a greedy recursive splitter with carefully maintained counters solves it quickly in practice.

arxiv:2610.38667 · cs.LG · 2026-10

Training the parameters of variational quantum circuits is NP-hard even for systems with logarithmically many qubits or free fermions, so the hardness is intrinsic to the classical optimization rather than inherited from the ground-state problem.

arxiv:2101.07267 · quant-ph · 2021-01

The epimorphism problem from finitely presented groups onto a fixed finite non-nilpotent dihedral group is NP-complete.

arxiv:2610.39487 · math.GR · 2026-10

In Erdős-Rényi random graphs, every online algorithm and every low-degree polynomial algorithm provably stops at half of the largest possible induced matching, and greedy achieves that half.

arxiv:2610.40369 · cs.DS · 2026-10

Two-photon decay of excited hydrogen, fine structure, and spin-flip energy exchange reduce the Wouthuysen-Field coupling strength by up to a factor of two, and fitting formulae are supplied for 21 cm reionization codes.

arxiv:astro-ph/0507102 · astro-ph · 2005-07

Efficient classical sampling from the output of the p=1 QAOA would collapse the Polynomial Hierarchy, making QAOA a candidate for quantum supremacy.

arxiv:1602.07674 · quant-ph · 2016-02

In general-sum games, mixing does not remove the computational hardness of strategizing against a known no-regret learner; in zero-sum games, pure play already recovers the best known surplus guarantees.

arxiv:2610.36261 · cs.GT · 2026-10

Analytical scaling laws for primordial black hole clustering show that, for a large dark matter fraction, late-universe binary mergers could reach the LIGO/Virgo detection band, though the result hinges on an uncertain halo core radius.

arxiv:2009.04731 · astro-ph.CO · 2020-09

Wrapping GPT-3.5 in a backtracking tree search with a rule-based Sudoku checker raises solve rates on 3x3 to 5x5 puzzles from 0.4-0.5 to 0.8-1.0, at up to 100 LLM calls per puzzle.

arxiv:2305.08291 · cs.AI · 2023-05

For identical parallel machine scheduling with assignment costs c_ij = a_i w_j + b_j, we give (1+ε,1)-approximation algorithms for three problems, including a PTAS for minimizing makespan under a cost budget.

arxiv:2610.33666 · cs.DS · 2026-10

For quadratic minimization on a simplex, the rule of repeatedly removing the most negative coordinate is correct for up to 4 variables but fails for 5, and determining its global correctness is coNP-complete.

arxiv:2610.33908 · math.OC · 2026-10

For any constant α ≥ 1, we construct an α-navigable graph in O~(n^2 log Δ) time that is simultaneously a constant-factor (α, 2α+O(1))-bicriteria approximation and an O(log n) (α, α+O(1))-bicriteria approximation, with a matching hardness result.

arxiv:2610.34302 · cs.DS · 2026-10

S6-coloring of cubic graphs is NP-complete, and H-coloring for every simple connected cubic H can be solved in O*(sqrt(8)^n) time.

arxiv:2610.35794 · cs.CC · 2026-10

For spherical-linear Bayesian optimization on a cube, Thompson sampling can be globally optimized with Dinkelbach iteration, and Surface Thompson Sampling matches Gaussian-process Bayesian optimization in accuracy while cutting acquisition walltime.

arxiv:2610.32628 · stat.ML · 2026-10

For any finite graph family H with min chromatic number > s, approximating ex(G, K_s, H) within additive error n^{s-delta} is NP-hard.

arxiv:2610.30654 · math.CO · 2026-10

Single-edge carbon-aware preemptive scheduling with arbitrary precedence is polynomial-time exact, cloud outsourcing has an FPTAS via a weighted-tardiness reduction, and two non-migrating edge servers are inapproximable.

arxiv:2610.29945 · cs.DS · 2026-10

The paper proves that Sustainable Envy-Free Multi-Period Allocation is NP-hard in nearly every parameter setting, with the only tractable cases occurring for few agents and periods jointly or for unary-encoded weights under very limited conditions.

arxiv:2610.26032 · cs.CC · 2026-10

Liveness of ring Petri nets is coNP-complete, and the halting problem for chip-firing on simple digraphs is NP-complete.

arxiv:2610.23523 · cs.CC · 2026-10

For any fixed number of polygonal robots, a single framework computes (1+ε)-optimal collision-free plans for total distance, maximum distance, and makespan in polynomial time.

arxiv:2610.20244 · cs.CG · 2026-10

The authors provide the first formal proof that the Gromov-Wasserstein distance is NP-hard and establish inapproximability for maximum, local, and global graph distortion distances.

arxiv:2610.16926 · cs.DM · 2026-10

Capacitated stack and queue layouts with colored edges are NP-complete for two fixed pages, and their parameterized complexity with respect to pages, colors, and conflict-graph structure is fully characterized.

arxiv:2610.18071 · cs.DS · 2026-10

Inclusion-wise extremal knapsack problems are inapproximable even with unit costs or profits, but regain approximation schemes under weight or value restrictions.

arxiv:2610.18017 · cs.DS · 2026-10

For fixed dimension or doubling metrics, a bicriteria approximation scheme for minimum well-separated pair decomposition exists; in general metrics, relaxing separation by a factor below 2 remains NP-hard.

arxiv:2610.13650 · cs.CG · 2026-10

PID on split graphs is O(n^4)-time solvable, NP-complete on graphs one vertex away from split, and classified across pendant-star extensions.

arxiv:2610.09968 · cs.DM · 2026-10

Static output feedback stabilization with rational data is ∃R-complete, and it is solvable in polynomial time whenever the effective gain dimension of the plant is fixed.

arxiv:2610.00390 · math.OC · 2026-09

Strong odd coloring is NP-complete on perfect elimination bipartite graphs for k≥3, inapproximable within factor O(n^(1/2-ε)), but admits a linear-time optimal algorithm on block graphs and an FPT algorithm by treewidth.

arxiv:2610.01441 · cs.DM · 2026-10

The decline of the .400 hitter is best explained by a modern trade-off between strikeout-avoidance talent and finishing talent, whose correlation moved from near zero to about -0.53, not by a general narrowing of hitter talent.

arxiv:2609.38693 · stat.ME · 2026-09

Fixed-interval scheduling with a per-machine idle break of prescribed length x is exactly solvable in polynomial time only for x≤1, NP-complete for every fixed x≥2, and yet admits a tight polynomial OPT+1 machine approximation.

arxiv:2609.34254 · cs.DS · 2026-09

Multi-agent optimal contracts are APX-hard for gross-substitutes (OXS) rewards, yet admit an EPTAS for weighted matroid rank functions and an FPTAS for partition matroids.

arxiv:2609.34665 · cs.GT · 2026-09

Strong and Semi-Strong Justified Representation committee selection is 1-1/e inapproximable in general but tractable on interval-structured inputs, and a new Distinctive Representation axiom is NP-hard yet has an efficient local variant.

arxiv:2609.29332 · cs.GT · 2026-09

The Iwata-Ravi 3/2 guarantee for unrestricted max-min T-join is tight under UGC, and exact-cardinality remote matching admits an explicit piecewise 7/2-and-up approximation profile depending on p = k/n.

arxiv:2609.26671 · cs.DS · 2026-09

Output feedback stabilization is strongly NP-hard, even for systems with a single output and polynomially bounded matrix entries.

arxiv:2609.20636 · math.OC · 2026-09

The paper gives the first total-function separation between BPP and PNP in communication complexity, using a cheat-sheet construction combined with fully linear PCPs.

arxiv:2609.20763 · cs.CC · 2026-09

Exact-cardinality, equally weighted one-factor portfolio selection is NP-complete when asset-specific variances vary, and not in nonuniform AC^0 even with identity noise and one negative loading.

arxiv:2609.17626 · econ.EM · 2026-09

Selecting optimal trigger words to minimize prefix-free graph size is NP-hard for any fixed trigger length and for alphabets as small as three symbols.

arxiv:2609.17353 · cs.CC · 2026-09

The non-resumable two-machine flow shop with a fixed unavailability interval on the second machine is pseudopolynomially solvable exactly but has no FPTAS unless P=NP.

arxiv:2609.14033 · cs.DM · 2026-09

Weak partition connectivity in hedgegraphs is NP-complete to decide, with a PTAS for full-support split systems and no FPTAS unless P=NP.

arxiv:2609.14932 · cs.DS · 2026-09

The paper proves impossibility of reconfiguration paths for envy-freeness up to k goods (EFk) for any fixed k, provides positive results for recursively balanced allocations, and establishes NP-hardness of deciding existence of EF1 reconfiguration paths.

arxiv:2609.15358 · cs.GT · 2026-09

The paper establishes DP-hardness for additive MMS existence, Δ2P-completeness for 2-additive valuations, and co-NP-hardness for three agents, all with polynomial-time reductions.

arxiv:2609.15841 · cs.GT · 2026-09

For every fixed integer p≥2, decision version of maximizing a linear function over integer points of an ℓ_p ball is NP-complete; exact algorithms and CVP connections are given for the Euclidean case.

arxiv:2609.15114 · math.OC · 2026-09

The paper shows that, under axioms of shared history and historical indistinguishability, embedded observers cannot soundly prove P equals NP, conditional on the existence of an observer-compatible evolutionary world.

arxiv:2609.10864 · math.LO · 2026-09

Assuming P != NP, the paper shows that finitely axiomatizable elementary modal logics have polynomial-time validity problems, and uses this to build infinitely many non-elementary K4-stable logics.

arxiv:2609.10872 · math.LO · 2026-09

Max Independent Set remains NP-hard in graphs excluding the planar 5x5 grid as an induced minor, via a reduction from Max Cut.

arxiv:2609.11285 · cs.CC · 2026-09

Under flatness and non-degeneracy assumptions, a minimizer can be expanded as a constant plus linear term plus a broken polynomial plus a small remainder that converges weakly to a function solving a specific PDE and is C^3 in half-balls.

arxiv:2609.11324 · math.AP · 2026-09

Exact planning with multi-step transition lookahead is NP-hard for every fixed discount factor, yet a randomized polynomial-time approximation scheme achieves near-optimal planning and learning.

arxiv:2609.11807 · stat.ML · 2026-09

Under PPAD and ETH conjectures, the paper claims optimal N^((log N/ε²)^(1-o(1))) lower bounds for ε-Nash equilibrium and free-game value estimation, but the Nash half has a load-bearing parameter miscalibration as written.

arxiv:2609.07136 · cs.GT · 2026-09

MaxMMP is APX-hard, and even with two colors it cannot be approximated arbitrarily well, but the paper gives polynomial-time approximation algorithms with ratios of order 1/c.

arxiv:2609.07193 · cs.DS · 2026-09

The Project Scheduling Interdiction Problem with Delay Groups (PSIP-DG) is introduced, proven NP-hard, and solved with effective heuristics.

arxiv:2608.27386 · cs.DM · 2026-08

Membership and uniqueness questions for optimal PAV committees are Δ₂^P-complete, and counting optimal committees is #·OptP-complete under metric reductions.

arxiv:2609.08884 · cs.GT · 2026-09

For every finite set of Boolean functions, the paper completely characterizes the complexity of fitting, empirical risk minimization, and PAC learning for formulas using those functions as connectives.

arxiv:2609.08961 · cs.LO · 2026-09

Ulam median and Ulam center are NP-hard to approximate within a factor of 51/50 - epsilon, and admit no additive approximation scheme, even for four input permutations.

arxiv:2608.29180 · cs.CC · 2026-08

The Frobenius number decision problem is Sigma_2^P-complete.

arxiv:2608.30036 · cs.CC · 2026-08

Recognizing exactness of the Max-Cut SDP relaxation is NP-complete for connected simple unweighted graphs.

arxiv:2609.03508 · math.OC · 2026-09

It is NP-hard to approximate the Ulam median or Ulam center within a factor of 35/34 - epsilon with only four input rankings.

arxiv:2609.05874 · cs.CC · 2026-09

Fair Top-k Katz Centrality Design: given a directed graph, target group proportion, and allowable edge additions, find the minimum edges to achieve the target in the Katz top-k ranking; the paper proves strong inapproximability and gives a scalable algorithm (BLADE).

arxiv:2609.03899 · cs.SI · 2026-09

The paper's attempted proof that PA has no non-standard models and P≠NP relies on conflating the provable statement 'every number is zero or has a predecessor' with the false claim that every element of every model is a natural number.

arxiv:math/0603605 · math.GM · 2006-03

Under cardinality-based repairs, static consistent query answering is P^NP(log n)-complete, while incremental versions become fixed-parameter tractable in the length of the update sequence.

arxiv:cs/0604002 · cs.DB · 2006-04

A survey of recent non-rigidity results for Hadamard, Fourier, and circulant matrices, and of new explicit constructions of rigid matrices in E^NP and P^NP.

arxiv:2009.09460 · cs.CC · 2020-09

Max Min FVS has polynomial-time approximability exactly n^{2/3} and an essentially tight sub-exponential time-approximation trade-off under ETH.

arxiv:2009.09971 · cs.CC · 2020-09

For online hypergraph matching with waiting delays, the paper proves an optimal competitive ratio of 1/d for utility maximization (k≥3) and a tight 3/2 deterministic ratio for pair cost minimization.

arxiv:2009.12022 · cs.DS · 2020-09

Robust network design with dynamic routing has no constant-factor approximation unless P=NP, and under ETH no Ω(log n / log log n) approximation, making the known O(log n) bound nearly tight.

arxiv:2009.12291 · cs.CC · 2020-09

A group of young researchers from around the world present artworks inspired by their own research, along with brief explanations of the mathematics involved.

arxiv:2009.14084 · math.HO · 2020-09

For the admissible-level affine sl2 Kazhdan-Lusztig categories, the McRae-Yang tensor functor is shown to be naturally isomorphic to quantum Drinfeld-Sokolov reduction, proving the conjecture.

arxiv:2608.13869 · math.RT · 2026-08

For single-machine fair repetitive scheduling, the max total completion time problem is weakly NP-hard on three days, and the max total waiting time and tardiness problems are weakly NP-hard on two days.

arxiv:2608.19952 · cs.DS · 2026-08

In stochastic multi-robot graph monitoring, repeating the best homogeneous strategy achieves a 1-(1-1/r)^r approximation, and no polynomial-time algorithm can beat 1-1/e unless P=NP.

arxiv:2608.20618 · cs.DS · 2026-08

A systematic complexity classification shows static, two-stage, and K-adaptability robust optimization range from P to undecidable, with decision-dependent uncertainty and unboundedness causing the hardest cases.

arxiv:2608.21574 · math.OC · 2026-08

A new predicate characterization of the counting class TotP via prefix-extension predicates, plus closure properties and an equivalence between closure under FP+ composition and P = PP.

arxiv:2504.20262 · cs.CC · 2025-04

Minimizing deterministic co-Buchi and history-deterministic Buchi automata with transition-based acceptance is NP-complete, completing the known boundary of polynomial-time minimization for these automata.

arxiv:2504.20553 · cs.FL · 2025-04

A polynomial-time algorithm for unsplittable flow routing in Clos networks guarantees congestion at most 9/5 of optimal, with lower bounds of 3/2 offline and 2 online.

arxiv:2505.03908 · cs.NI · 2025-05

Safety analysis in NGAC is claimed to be coNP-complete, with an MIS-enumeration algorithm that is polynomial when the constraint graph has few maximal independent sets.

arxiv:2505.06406 · cs.CC · 2025-05

Disk graphs can be subcolored with O(log^3 n) colors, admitting an O(log^2 n)-approximation, and several subcoloring problems are NP-hard on unit disk and co-comparability graphs.

arxiv:2506.19452 · cs.DS · 2025-06

Anisotropic p-Laplace equations with semilinear terms admit no positive solutions in the stated subcritical range and no nontrivial stable sign-changing solutions under the stated growth conditions.

arxiv:2507.20182 · math.AP · 2025-07

For 2D demand bin packing, short-height and square variants are NP-hard to approximate within any factor below 2, and a 3-approximation is claimed for the general case.

arxiv:2508.13347 · cs.DS · 2025-08

Spectra and energies of NCCC-graphs are computed for groups with central quotients Z_p×Z_p and D_{2m}, covering dihedral, dicyclic, semidihedral, and related families.

arxiv:2508.19616 · math.CO · 2025-08

Candidate control under greedy participatory budgeting rules is W[1]-hard in the number of controlled projects, but fixed-parameter tractable when combining cost diversity with voter count or controlled project count, and inapproximable within m^{1-epsilon}.

arxiv:2607.25723 · cs.GT · 2026-07

For a new family of subrecursive classes H^m_{n,l} defined by bounded recursion along generalized inverses, the paper proves a complete inclusion criterion for n,n' >= 2, with strides ordered by reverse divisibility, and shows H^2_{1,l} is a proper subclass of FP.

arxiv:2608.04871 · cs.LO · 2026-08

Cluster Deletion is NP-hard to approximate within sqrt(2)-epsilon, and UGC-hard within 2-epsilon, matching the known 2-approximation and answering an open question about Cluster Editing versus Bad Triangle Transversal.

arxiv:2608.04883 · cs.DS · 2026-08

The paper claims NP-completeness of proper conflict-free k-coloring for perfect elimination bipartite graphs for every k≥3, gives an inapproximability bound, and provides linear-time algorithms for block, proper interval, chain, and pseudo-split graphs.

arxiv:2608.10874 · cs.DM · 2026-08

It is NP-hard to approximate the hyperspherical radius of triangulated surfaces and triangulated high-dimensional spheres to within any almost-polynomial factor.

arxiv:1908.02824 · math.DG · 2019-08

In the K-Γ-Γ' model for α-RuCl3, the chiral Kitaev spin liquid survives only in a small corner, while new nematic paramagnet phases dominate the intermediate magnetic-field regime.

arxiv:1908.07671 · cond-mat.str-el · 2019-08

Inserting a prescribed set of edges into a simple drawing is NP-complete, the maximization version is APX-hard, and one-edge insertion is polynomial when the endpoints form a dominating set.

arxiv:1908.08129 · cs.CG · 2019-08

Efficient algorithms exist for optimal private Bayesian persuasion of voters under k-voting and plurality rules, but optimal public persuasion is NP-hard to approximate by any factor.

arxiv:1908.10620 · cs.GT · 2019-08