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NP-complete decision problems for quadratic polynomials

Kenneth Manders, Leonard Adleman
1976 Proceedings of the eighth annual ACM symposium on Theory of computing - STOC '76  
In this a r t i c l e we show the NP-completeness of some simple number-theoretic problems. Natural simplifications of these problems invariably are known to be in P.  ...  Unless P=NP this is impossible, even for a class of quadratic diophantine equations in two unknowns for which a decision procedure in the original sense is in fact a v a i l a b l e . .  ...  be NP-complete, we hope that the NP-completeness of these problems w i l l play a role in showing the NP-completeness of further problems of a numerical nature, much as the s a t i s f i a b i l i t y  ... 
doi:10.1145/800113.803627 dblp:conf/stoc/MandersA76 fatcat:sim4pz2f6ndkvmrc6vx66uur7a

Computational Complexity of Quadratic Unconstrained Binary Optimization [article]

Hirotoshi Yasuoka
2022 arXiv   pre-print
We also study the computational complexity of the decision version of the QUBO problem with integer coefficients. We prove that this problem is DP-complete.  ...  In this paper, we study the computational complexity of the quadratic unconstrained binary optimization (QUBO) problem under the functional problem FP^NP categorization.  ...  Acknowledgements We want to thank Akira Miki and Chih-Hong Cheng for useful advice.  ... 
arXiv:2109.10048v2 fatcat:pmovoz3y3jft7j42glozpcvgzm

On the complexity of quadratic programming in real number models of computation

K. Meer
1994 Theoretical Computer Science  
In particular we show that this problem is not NP-complete in the Koiran setting. Applications to the (full) BSS-model are discussed.  ...  On the complexity of quadratic programming in real number models of computation, Theoretical Computer Science 133 (1994) 85-94.  ...  I wish to express my thanks for the great hospitality and the excellent working conditions. Furthermore I want to thank F. Cucker and H.Th. Jongen for helpful discussions.  ... 
doi:10.1016/0304-3975(94)00070-0 fatcat:wx2tcrz475hutithkoz6abv24y

On NP complete problems I [article]

Minoru Fujimoto, Kunihiko Uehara
2008 arXiv   pre-print
We study the quadratic residue problem known as an NP complete problem by way of the prime number and show that a nondeterministic polynomial process does not belong to the class P because of a random  ...  distribution of solutions for the quadratic residue problem.  ...  Because any other NP complete problem can be reduce to the quadratic residue problem in polynomial time, it is possible to prove that all NP complete problems can not be solved in a polynomial calculated  ... 
arXiv:0809.0962v1 fatcat:ggl7byztyrgptfgaak3s4z2zey

Good neighbors are hard to find: computational complexity of network formation

Richard Baron, Jacques Durieu, Hans Haller, Rahul Savani, Philippe Solal
2008 Review of Economic Design  
We find that deciding if a player has a strategy that guarantees him a certain payoff against a given strategy profile of the other players is an NP-complete problem.  ...  Deciding if there exists a strategy profile that guarantees a certain aggregate payoff is also NP-complete.  ...  A decision problem is NP-complete if it is both in NP and NP-hard. Main results.  ... 
doi:10.1007/s10058-008-0043-x fatcat:7y5oisbivve2bn4rkn6cmprlxq

The Complexity of Decision Versus Search

Mihir Bellare, Shafi Goldwasser
1994 SIAM journal on computing (Print)  
A basic question about NP is whether or not search reduces in polynomial time to decision.  ...  reduce to decision.  ...  We thank Oded Goldreich for many helpful comments on the paper. We thank Satish Thate for  ... 
doi:10.1137/s0097539792228289 fatcat:5h53wc3gsbeqjaneurwasrqa5a

Computational complexity of vacua and near-vacua in field and string theory

James Halverson, Fabian Ruehle
2019 Physical Review D  
Multiple problems relevant for computing effective potential contributions from string theory are shown to be instances of NP-hard problems.  ...  We demonstrate that the problems of finding stable or metastable vacua in a low energy effective field theory requires solving nested NP-hard and co-NP-hard problems, while the problem of finding near-vacua  ...  Except for the first problem, which is NP-hard, the other problems below are all known to be NP-complete; see [40] for further discussion.  ... 
doi:10.1103/physrevd.99.046015 fatcat:uvtgwn4dwzavfj3tei26w2vf5y

Duality Gap, Computational Complexity and NP Completeness: A Survey [article]

Prabhu Manyem
2011 arXiv   pre-print
We survey research that studies the connection between the computational complexity of optimization problems on the one hand, and the duality gap between the primal and dual optimization problems on the  ...  Ramana [22] exhibited strong duality for the SDP problem. However, the complexity of SDP is unknown; it was shown in [22] that the decision version of SDP is NP-complete if and only if NP = CoNP.  ...  However, this problem would be NP-complete, unless we can tell whether it has a saddle point by looking at the structure of the problem or by running a polynomial time algorithm.  ... 
arXiv:1012.5568v2 fatcat:sptvhuucijfwbdfrqjmab7zjde

Self-concordance is NP-hard [article]

Lek-Heng Lim
2013 arXiv   pre-print
We give an elementary proof of a somewhat curious result, namely, that deciding whether a convex function is self-concordant is in general an intractable problem.  ...  In particular, their NP-hardness serves as yet reminder of the complexity of tensor problems [7] . Acknowledgement To be included.  ...  use as barriers in cone programming -see "How to construct self-concordant barriers" in [20, Chapter 5] for an extensive discussion or "Self-concordant calculus" in [3, Section 9.6] for a summary.  ... 
arXiv:1303.7455v1 fatcat:jstv322uqfc4leippnipw2cqai

Self-concordance is NP-hard

Lek-Heng Lim
2016 Journal of Global Optimization  
We give an elementary proof of a somewhat curious result, namely, that deciding whether a convex function is self-concordant is in general an intractable problem.  ...  In particular, their NP-hardness serves as yet reminder of the complexity of tensor problems [7] . Acknowledgement To be included.  ...  use as barriers in cone programming -see "How to construct self-concordant barriers" in [20, Chapter 5] for an extensive discussion or "Self-concordant calculus" in [3, Section 9.6] for a summary.  ... 
doi:10.1007/s10898-016-0469-6 fatcat:zzkxptrzyvahpaa7mpdkwvx3fa

On the Complexity of Quadratization for Polynomial Differential Equations [article]

Mathieu Hemery
2020 arXiv   pre-print
We show that both problems of minimizing either the number of variables (i.e., molecular species) or the number of monomials (i.e. elementary reactions) in a quadratic transformation of a PIVP are NP-hard  ...  The proof of that result is based on a previous result of Bournez et al. on the Turing-completeness of polyno-mial ordinary differential equations with polynomial initial conditions (PIVP).  ...  Acknowledgements This work was jointly supported by ANR-MOST BIOPSY Biochemical Programming System grant ANR-16-CE18-0029 and ANR-DFG SYMBIONT Symbolic Methods for Biological Networks grant ANR-17-CE40  ... 
arXiv:2007.08910v2 fatcat:z5pj4dq2q5e7dhngt55za6up5i

On the theory of average case complexity

Shai Ben-David, Benny Chor, Oded Goldreich, Michel Luby
1992 Journal of computer and system sciences (Print)  
Levin's work can be viewed as the basic for a theory of average NP-completeness, much the same way as [2] (and [19]) are the basis for the theory of NP-completeness.  ...  Our results include: l the equivalence of search and decision problems in the context of average case complexity; 9 an initial analysis of the structure of distributional-NP (i.e., NP problems coupled  ...  ACKNOWLEDGMENTS We thank Shimon Even, Mauricio Karchmer, Hugo Krawczyk, Ronny Roth, and Avi Wigderson for helpful discussions. We are grateful to Leonid Levin for very interesting discussions.  ... 
doi:10.1016/0022-0000(92)90019-f fatcat:ngf4z4bn65binajjhc2zkwabd4

The Complexity of Optimizing Over a Simplex, Hypercube or Sphere: A Short Survey

Etienne de Klerk
2006 Social Science Research Network  
These relatively simple optimization problems arise naturally from diverse applications.  ...  This decision problem is clearly in NP for the real number model, but the situation is much more difficult for the bit model.  ...  The case of the simplex If K = n , then computing f is an NP-hard problem, already for quadratic polynomials, as it contains the maximum stable set problem as a special case.  ... 
doi:10.2139/ssrn.932525 fatcat:pbbyh6pjmngg7afefazjvgb7zm

The complexity of optimizing over a simplex, hypercube or sphere: a short survey

Etienne de Klerk
2007 Central European Journal of Operations Research  
These relatively simple optimization problems arise naturally from diverse applications.  ...  This decision problem is clearly in NP for the real number model, but the situation is much more difficult for the bit model.  ...  The case of the simplex If K = n , then computing f is an NP-hard problem, already for quadratic polynomials, as it contains the maximum stable set problem as a special case.  ... 
doi:10.1007/s10100-007-0052-9 fatcat:3iynpel655hohamavx6dlmcqim

NP-Complete decision problems for binary quadratics

Kenneth L. Manders, Leonard Adleman
1978 Journal of computer and system sciences (Print)  
We show that for the two-variable quadratics of the form c& + fly -y = 0; 01, fi, y E w, the problem is &Y-complete. Xl .*a x,J = 0 has a solution).  ...  Even for equations with two unknowns, the decision problem is tantalizingly difficult.  ...  OPEN PROBLEM 7 (classification of the set PY of prime numbers). PY E NP [17] , and PY E P(ERH) [15] .  ... 
doi:10.1016/0022-0000(78)90044-2 fatcat:bmhpbziwwvbgpgcslr66jv3pua
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