On the Complexity of Quadratization for Polynomial Differential Equations

Chemical reaction networks (CRNs) are a standard formalism used in chemistry and biology to reason about the dynamics of molecular interaction networks. In their interpretation by ordinary differential equations, CRNs provide a Turing-complete model of analog computattion, in the sense that any computable function over the reals can be computed by a finite number of molecular species with a continuous CRN which approximates the result of that function in one of its components in arbitrary precision. 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). It uses an encoding of real variables by two non-negative variables for concentrations, and a transformation to an equivalent quadratic PIVP (i.e. with degrees at most 2) for restricting ourselves to at most bimolecular reactions. In this paper, we study the theoretical and practical complexities of the quadratic transformation. 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. We present an encoding of those problems in MAX-SAT and show the practical complexity of this algorithm on a benchmark of quadratization problems inspired from CRN design problems

Holographic renormalization as a canonical transformation

The gauge/string dualities have drawn attention to a class of variational problems on a boundary at infinity, which are not well defined unless a certain boundary term is added to the classical action. In the context of supergravity in asymptotically AdS spaces these problems are systematically addressed by the method of holographic renormalization. We argue that this class of a priori ill defined variational problems extends far beyond the realm of holographic dualities. As we show, exactly the same issues arise in gravity in non asymptotically AdS spaces, in point particles with certain unbounded from below potentials, and even fundamental strings in flat or AdS backgrounds. We show that the variational problem in all such cases can be made well defined by the following procedure, which is intrinsic to the system in question and does not rely on the existence of a holographically dual theory: (i) The first step is the construction of the space of the most general asymptotic solutions of the classical equations of motion that inherits a well defined symplectic form from that on phase space. The requirement of a well defined symplectic form is essential and often leads to a necessary repackaging of the degrees of freedom. (ii) Once the space of asymptotic solutions has been constructed in terms of the correct degrees of freedom, then there exists a boundary term that is obtained as a certain solution of the Hamilton-Jacobi equation which simultaneously makes the variational problem well defined and preserves the symplectic form. This procedure is identical to holographic renormalization in the case of asymptotically AdS gravity, but it is applicable to any Hamiltonian system.Comment: 37 pages; v2 minor corrections in section 2, 2 references and a footnote on Palatini gravity added. Version to appear in JHE

Most vital segment barriers

We study continuous analogues of "vitality" for discrete network flows/paths, and consider problems related to placing segment barriers that have highest impact on a flow/path in a polygonal domain. This extends the graph-theoretic notion of "most vital arcs" for flows/paths to geometric environments. We give hardness results and efficient algorithms for various versions of the problem, (almost) completely separating hard and polynomially-solvable cases

Using Sat solvers for synchronization issues in partial deterministic automata

We approach the task of computing a carefully synchronizing word of minimum length for a given partial deterministic automaton, encoding the problem as an instance of SAT and invoking a SAT solver. Our experimental results demonstrate that this approach gives satisfactory results for automata with up to 100 states even if very modest computational resources are used.Comment: 15 pages, 3 figure

Travelling on Graphs with Small Highway Dimension

We study the Travelling Salesperson (TSP) and the Steiner Tree problem (STP) in graphs of low highway dimension. This graph parameter was introduced by Abraham et al. [SODA 2010] as a model for transportation networks, on which TSP and STP naturally occur for various applications in logistics. It was previously shown [Feldmann et al. ICALP 2015] that these problems admit a quasi-polynomial time approximation scheme (QPTAS) on graphs of constant highway dimension. We demonstrate that a significant improvement is possible in the special case when the highway dimension is 1, for which we present a fully-polynomial time approximation scheme (FPTAS). We also prove that STP is weakly NP-hard for these restricted graphs. For TSP we show NP-hardness for graphs of highway dimension 6, which answers an open problem posed in [Feldmann et al. ICALP 2015]

Stochastic quantization and holographic Wilsonian renormalization group

We study relation between stochastic quantization and holographic Wilsonian renormalization group flow. Considering stochastic quantization of the boundary on-shell actions with the Dirichlet boundary condition for certain $AdS$ bulk gravity theories, we find that the radial flows of double trace deformations in the boundary effective actions are completely captured by stochastic time evolution with identification of the $AdS$ radial coordinate `$r$' with the stochastic time '$t$' as $r=t$. More precisely, we investigate Langevin dynamics and find an exact relation between radial flow of the double trace couplings and 2-point correlation functions in stochastic quantization. We also show that the radial evolution of double trace deformations in the boundary effective action and the stochastic time evolution of the Fokker-Planck action are the same. We demonstrate this relation with a couple of examples: (minimally coupled)massless scalar fields in $AdS_2$ and U(1) vector fields in $AdS_4$.Comment: 1+30 pages, a new subsection is added, references are adde

Complexity of Discrete Energy Minimization Problems

Discrete energy minimization is widely-used in computer vision and machine learning for problems such as MAP inference in graphical models. The problem, in general, is notoriously intractable, and finding the global optimal solution is known to be NP-hard. However, is it possible to approximate this problem with a reasonable ratio bound on the solution quality in polynomial time? We show in this paper that the answer is no. Specifically, we show that general energy minimization, even in the 2-label pairwise case, and planar energy minimization with three or more labels are exp-APX-complete. This finding rules out the existence of any approximation algorithm with a sub-exponential approximation ratio in the input size for these two problems, including constant factor approximations. Moreover, we collect and review the computational complexity of several subclass problems and arrange them on a complexity scale consisting of three major complexity classes -- PO, APX, and exp-APX, corresponding to problems that are solvable, approximable, and inapproximable in polynomial time. Problems in the first two complexity classes can serve as alternative tractable formulations to the inapproximable ones. This paper can help vision researchers to select an appropriate model for an application or guide them in designing new algorithms.Comment: ECCV'16 accepte

An umbrella review of the evidence associating diet and cancer risk at 11 anatomical sites

There is evidence that diet and nutrition are modifiable risk factors for several cancers, but associations may be flawed due to inherent biases. Nutritional epidemiology studies have largely relied on a single assessment of diet using food frequency questionnaires. We conduct an umbrella review of meta-analyses of observational studies to evaluate the strength and validity of the evidence for the association between food/nutrient intake and risk of developing or dying from 11 primary cancers. It is estimated that only few single food/nutrient and cancer associations are supported by strong or highly suggestive meta-analytic evidence, and future similar research is unlikely to change this evidence. Alcohol consumption is positively associated with risk of postmenopausal breast, colorectal, esophageal, head & neck and liver cancer. Consumption of dairy products, milk, calcium and wholegrains are inversely associated with colorectal cancer risk. Coffee consumption is inversely associated with risk of liver cancer and skin basal cell carcinoma

A Better-response Strategy for Self-interested Planning Agents

[EN] When self-interested agents plan individually, interactions that prevent them from executing their actions as planned may arise. In these coordination problems, game-theoretic planning can be used to enhance the agents¿ strategic behavior considering the interactions as part of the agents¿ utility. In this work, we define a general-sum game in which interactions such as conflicts and congestions are reflected in the agents¿ utility. We propose a better-response planning strategy that guarantees convergence to an equilibrium joint plan by imposing a tax to agents involved in conflicts. We apply our approach to a real-world problem in which agents are Electric Autonomous Vehicles (EAVs). The EAVs intend to find a joint plan that ensures their individual goals are achievable in a transportation scenario where congestion and conflicting situations may arise. Although the task is computationally hard, as we theoretically prove, the experimental results show that our approach outperforms similar approaches in both performance and solution quality.This work is supported by the GLASS project TIN2014-55637-C2-2-R of the Spanish MINECO and the Prometeo project II/2013/019 funded by the Valencian Government.Jordán, J.; Torreño Lerma, A.; De Weerdt, M.; Onaindia De La Rivaherrera, E. (2018). A Better-response Strategy for Self-interested Planning Agents. Applied Intelligence. 48(4):1020-1040. https://doi.org/10.1007/s10489-017-1046-5S10201040484Aghighi M, Bäckström C (2016) A multi-parameter complexity analysis of cost-optimal and net-benefit planning. In: Proceedings of the Twenty-Sixth International Conference on International Conference on Automated Planning and Scheduling. AAAI Press, London, pp 2–10Bercher P, Mattmüller R (2008) A planning graph heuristic for forward-chaining adversarial planning. In: ECAI, vol 8, pp 921–922Brafman RI, Domshlak C, Engel Y, Tennenholtz M (2009) Planning games. In: IJCAI 2009, Proceedings of the 21st international joint conference on artificial intelligence, pp 73–78Bylander T (1994) The computational complexity of propositional strips planning. Artif Intell 69(1):165–204Chen X, Deng X (2006) Settling the complexity of two-player nash equilibrium. In: 47th annual IEEE symposium on foundations of computer science, 2006. FOCS’06. IEEE, pp 261–272Chien S, Sinclair A (2011) Convergence to approximate nash equilibria in congestion games. Games and Economic Behavior 71(2):315–327de Cote EM, Chapman A, Sykulski AM, Jennings N (2010) Automated planning in repeated adversarial games. In: 26th conference on uncertainty in artificial intelligence (UAI 2010), pp 376–383Dunne PE, Kraus S, Manisterski E, Wooldridge M (2010) Solving coalitional resource games. Artif Intell 174(1):20–50Fabrikant A, Papadimitriou C, Talwar K (2004) The complexity of pure nash equilibria. In: Proceedings of the thirty-sixth annual ACM symposium on theory of computing, STOC ’04, pp 604–612Friedman JW, Mezzetti C (2001) Learning in games by random sampling. J Econ Theory 98(1):55–84Ghallab M, Nau D, Traverso P (2004) Automated planning: theory & practice. ElsevierGoemans M, Mirrokni V, Vetta A (2005) Sink equilibria and convergence. In: Proceedings of the 46th annual IEEE symposium on foundations of computer science, FOCS ’05, pp 142–154Hadad M, Kraus S, Hartman IBA, Rosenfeld A (2013) Group planning with time constraints. Ann Math Artif Intell 69(3):243–291Hart S, Mansour Y (2010) How long to equilibrium? the communication complexity of uncoupled equilibrium procedures. Games and Economic Behavior 69(1):107–126Helmert M (2003) Complexity results for standard benchmark domains in planning. Artif Intell 143(2):219–262Helmert M (2006) The fast downward planning system. J Artif Intell Res 26(1):191–246Jennings N, Faratin P, Lomuscio A, Parsons S, Wooldrige M, Sierra C (2001) Automated negotiation: prospects, methods and challenges. Group Decis Negot 10(2):199–215Johnson DS, Papadimtriou CH, Yannakakis M (1988) How easy is local search? J Comput Syst Sci 37 (1):79–100Jonsson A, Rovatsos M (2011) Scaling up multiagent planning: a best-response approach. In: Proceedings of the 21st international conference on automated planning and scheduling, ICAPSJordán J, Onaindía E (2015) Game-theoretic approach for non-cooperative planning. In: 29th AAAI conference on artificial intelligence (AAAI-15), pp 1357–1363McDermott D, Ghallab M, Howe A, Knoblock C, Ram A, Veloso M, Weld D, Wilkins D (1998) PDDL: the planning domain definition language. Yale Center for Computational Vision and Control, New HavenMilchtaich I (1996) Congestion games with player-specific payoff functions. Games and Economic Behavior 13(1):111–124Monderer D, Shapley LS (1996) Potential games. Games and Economic Behavior 14(1):124–143Nigro N, Welch D, Peace J (2015) Strategic planning to implement publicly available ev charching stations: a guide for business and policy makers. Tech rep, Center for Climate and Energy SolutionsNisan N, Ronen A (2007) Computationally feasible vcg mechanisms. J Artif Intell Res 29(1):19–47Nisan N, Roughgarden T, Tardos E, Vazirani VV (2007) Algorithmic game theory. Cambridge University Press, New YorkPapadimitriou CH (1994) On the complexity of the parity argument and other inefficient proofs of existence. J Comput Syst Sci 48(3):498–532Richter S, Westphal M (2010) The LAMA planner: guiding cost-based anytime planning with landmarks. J Artif Intell Res 39(1):127–177Rosenthal RW (1973) A class of games possessing pure-strategy nash equilibria. Int J Game Theory 2(1):65–67Shoham Y, Leyton-Brown K (2009) Multiagent systems: algorithmic, game-theoretic, and logical foundations. Cambridge University PressTorreño A, Onaindia E, Sapena Ó (2014) A flexible coupling approach to multi-agent planning under incomplete information. Knowl Inf Syst 38(1):141–178Torreño A, Onaindia E, Sapena Ó (2014) FMAP: distributed cooperative multi-agent planning. Appl Intell 41(2):606– 626Torreño A, Sapena Ó, Onaindia E (2015) Global heuristics for distributed cooperative multi-agent planning. In: ICAPS 2015. 25th international conference on automated planning and scheduling. AAAI Press, pp 225–233Von Neumann J, Morgenstern O (2007) Theory of games and economic behavior. Princeton University Pressde Weerdt M, Bos A, Tonino H, Witteveen C (2003) A resource logic for multi-agent plan merging. Ann Math Artif Intell 37(1):93–130Wooldridge M, Endriss U, Kraus S, Lang J (2013) Incentive engineering for boolean games. Artif Intell 195:418–43

Holographic Renormalization for Asymptotically Lifshitz Spacetimes

A variational formulation is given for a theory of gravity coupled to a massive vector in four dimensions, with Asymptotically Lifshitz boundary conditions on the fields. For theories with critical exponent z=2 we obtain a well-defined variational principle by explicitly constructing two actions with local boundary counterterms. As part of our analysis we obtain solutions of these theories on a neighborhood of spatial infinity, study the asymptotic symmetries, and consider different definitions of the boundary stress tensor and associated charges. A constraint on the boundary data for the fields figures prominently in one of our formulations, and in that case the only suitable definition of the boundary stress tensor is due to Hollands, Ishibashi, and Marolf. Their definition naturally emerges from our requirement of finiteness of the action under Hamilton-Jacobi variations of the fields. A second, more general variational principle also allows the Brown-York definition of a boundary stress tensor.Comment: 34 pages, Added Reference