Unique End of Potential Line

This paper studies the complexity of problems in PPAD $\cap$ PLS that have unique solutions. Three well-known examples of such problems are the problem of finding a fixpoint of a contraction map, finding the unique sink of a Unique Sink Orientation (USO), and solving the P-matrix Linear Complementarity Problem (P-LCP). Each of these are promise-problems, and when the promise holds, they always possess unique solutions. We define the complexity class UEOPL to capture problems of this type. We first define a class that we call EOPL, which consists of all problems that can be reduced to End-of-Potential-Line. This problem merges the canonical PPAD-complete problem End-of-Line, with the canonical PLS-complete problem Sink-of-Dag, and so EOPL captures problems that can be solved by a line-following algorithm that also simultaneously decreases a potential function. Promise-UEOPL is a promise-subclass of EOPL in which the line in the End-of-Potential-Line instance is guaranteed to be unique via a promise. We turn this into a non-promise class UEOPL, by adding an extra solution type to EOPL that captures any pair of points that are provably on two different lines. We show that UEOPL $\subseteq$ EOPL $\subseteq$ CLS, and that all of our motivating problems are contained in UEOPL: specifically USO, P-LCP, and finding a fixpoint of a Piecewise-Linear Contraction under an $\ell_p$-norm all lie in UEOPL. Our results also imply that parity games, mean-payoff games, discounted games, and simple-stochastic games lie in UEOPL. All of our containment results are proved via a reduction to a problem that we call One-Permutation Discrete Contraction (OPDC). This problem is motivated by a discretized version of contraction, but it is also closely related to the USO problem. We show that OPDC lies in UEOPL, and we are also able to show that OPDC is UEOPL-complete.Comment: This paper substantially revises and extends the work described in our previous preprint "End of Potential Line'' (arXiv:1804.03450). The abstract has been shortened to meet the arXiv character limi

End of Potential Line

We introduce the problem EndOfPotentialLine and the corresponding complexity class EOPL of all problems that can be reduced to it in polynomial time. This class captures problems that admit a single combinatorial proof of their joint membership in the complexity classes PPAD of fixpoint problems and PLS of local search problems. EOPL is a combinatorially-defined alternative to the class CLS (for Continuous Local Search), which was introduced in with the goal of capturing the complexity of some well-known problems in PPAD $\cap$ PLS that have resisted, in some cases for decades, attempts to put them in polynomial time. Two of these are Contraction, the problem of finding a fixpoint of a contraction map, and P-LCP, the problem of solving a P-matrix Linear Complementarity Problem. We show that EndOfPotentialLine is in CLS via a two-way reduction to EndOfMeteredLine. The latter was defined in to show query and cryptographic lower bounds for CLS. Our two main results are to show that both PL-Contraction (Piecewise-Linear Contraction) and P-LCP are in EOPL. Our reductions imply that the promise versions of PL-Contraction and P-LCP are in the promise class UniqueEOPL, which corresponds to the case of a single potential line. This also shows that simple-stochastic, discounted, mean-payoff, and parity games are in EOPL. Using the insights from our reduction for PL-Contraction, we obtain the first polynomial-time algorithms for finding fixed points of contraction maps in fixed dimension for any $\ell_p$ norm, where previously such algorithms were only known for the $\ell_2$ and $\ell_\infty$ norms. Our reduction from P-LCP to EndOfPotentialLine allows a technique of Aldous to be applied, which in turn gives the fastest-known randomized algorithm for the P-LCP

Unique End of Potential Line

This paper studies the complexity of problems in PPAD $\cap$ PLS that have unique solutions. Three well-known examples of such problems are the problem of finding a fixpoint of a contraction map, finding the unique sink of a Unique Sink Orientation (USO), and solving the P-matrix Linear Complementarity Problem (P-LCP). Each of these are promise-problems, and when the promise holds, they always possess unique solutions. We define the complexity class UEOPL to capture problems of this type. We first define a class that we call EOPL, which consists of all problems that can be reduced to End-of-Potential-Line. This problem merges the canonical PPAD-complete problem End-of-Line, with the canonical PLS-complete problem Sink-of-Dag, and so EOPL captures problems that can be solved by a line-following algorithm that also simultaneously decreases a potential function. Promise-UEOPL is a promise-subclass of EOPL in which the line in the End-of-Potential-Line instance is guaranteed to be unique via a promise. We turn this into a non-promise class UEOPL, by adding an extra solution type to EOPL that captures any pair of points that are provably on two different lines. We show that UEOPL $\subseteq$ EOPL $\subseteq$ CLS, and that all of our motivating problems are contained in UEOPL: specifically USO, P-LCP, and finding a fixpoint of a Piecewise-Linear Contraction under an $\ell_p$-norm all lie in UEOPL. Our results also imply that parity games, mean-payoff games, discounted games, and simple-stochastic games lie in UEOPL. All of our containment results are proved via a reduction to a problem that we call One-Permutation Discrete Contraction (OPDC). This problem is motivated by a discretized version of contraction, but it is also closely related to the USO problem. We show that OPDC lies in UEOPL, and we are also able to show that OPDC is UEOPL-complete

Evidence of a higher-order singularity in dense short-ranged attractive colloids

We study a model in which particles interact through a hard-core repulsion complemented by a short-ranged attractive potential, of the kind found in colloidal suspensions. Combining theoretical and numerical work we locate the line of higher-order glass transition singularities and its end-point -- named $A_4$ -- on the fluid-glass line. Close to the $A_4$ point, we detect logarithmic decay of density correlations and sub linear power-law increase of the mean square displacement, for time intervals up to four order of magnitudes. We establish the presence of the $A_4$ singularity by studying how the range of the potential affects the time-window where anomalous dynamics is observed.Comment: 4 pages, 4 figures, REVTE

An assessment of the Irish speciality food enterprises’ use of the internet as a marketing tool

End of Project ReportThis study set out to explore the role of the Internet as a marketing tool for Irish speciality food producers and to research on-line speciality food sales as a business opportunity. The project achieved this through a combination of consumer focus groups, a producer web audit, producer depth interviews and an e-mailed on-line producer survey. Irish consumers acknowledged potential for on-line sales of Irish speciality food products to export and gift markets; however they could not see significant advantages for on-line sales in the domestic market. Experience with the product (and consequent importance of the purchase experience), the high delivery cost of an already premium priced product and difficulties associated with receipt of deliveries were identified as the main reasons for not purchasing on-line

Lattice QCD at Finite Density

I discuss different approaches to finite density lattice QCD. In particular, I focus on the structure of the phase diagram and discuss attempts to determine the location of the critical end-point. Recent results on the transiton line as function of the chemical potential ($T_c(\mu_q)$) are reviewed. Along the transition line, hadronic fluctuations have been calculated, which can be used to characterize properties of the Quark Gluon plasma and eventually can also help to identify the location of the critical end-point in the QCD phase diagram on the lattice and in heavy ion experiments. Furthermore, I comment on the structure of the phase diagram at large $\mu_q$.Comment: Plenary talk at XXIVth International Symposium on Lattice Field Theory (Lattice 2006). Version to appear as PoS (LAT2006) 021; references adde

Universality, the QCD critical/tricritical point and the quark number susceptibility

The quark number susceptibility near the QCD critical end-point (CEP), the tricritical point (TCP) and the O(4) critical line at finite temperature and quark chemical potential is investigated. Based on the universality argument and numerical model calculations we propose a possibility that the hidden tricritical point strongly affects the critical phenomena around the critical end-point. We made a semi-quantitative study of the quark number susceptibility near CEP/TCP for several quark masses on the basis of the Cornwall-Jackiw-Tomboulis (CJT) potential for QCD in the improved-ladder approximation. The results show that the susceptibility is enhanced in a wide region around CEP inside which the critical exponent gradually changes from that of CEP to that of TCP, indicating a crossover of different universality classes.Comment: 18 pages, 10 figure