Thomas Dueholm Hansen

Strategy Iteration Is Strongly Polynomial for 2-Player Turn-Based Stochastic Games with a Constant Discount Factor

Ye [2011] showed recently that the simplex method with Dantzig’s pivoting rule, as well as Howard’s <i>policy iteration</i> algorithm, solve discounted Markov decision processes (MDPs), with a constant discount factor, in strongly polynomial time. More precisely, Ye showed that both algorithms terminate after at most <i>O</i>(<i>mn</i>1−<i>γ</i> log <i>n</i>1−<i>γ</i>) iterations, where <i>n</i> is the number of states, <i>m</i> is the total number of actions in the MDP, and 0 < <i>γ</i> < 1 is the discount factor. We improve Ye’s analysis in two respects. First, we improve the bound given by Ye and show that Howard’s policy iteration algorithm actually terminates after at most <i>O</i>(<i>m</i>1−<i>γ</i> log <i>n</i>1−<i>γ</i>) iterations. Second, and more importantly, we show that the same bound applies to the number of iterations performed by the <i>strategy iteration</i> (or <i>strategy improvement</i>) algorithm, a generalization of Howard’s policy iteration algorithm used for solving 2-player turn-based <i>stochastic games</i> with discounted zero-sum rewards. This provides the first strongly polynomial algorithm for solving these games, solving a long standing open problem. Combined with other recent results, this provides a complete characterization of the complexity the standard strategy iteration algorithm for 2-player turn-based stochastic games; it is strongly polynomial for a fixed discount factor, and exponential otherwise.

Genetic determinants of both ethanol and acetaldehyde metabolism influence alcohol hypersensitivity and drinking behaviour among Scandinavians

Background Although hypersensitivity reactions following intake of alcoholic drinks are common in Caucasians, the underlying mechanisms and clinical significance are not known. In contrast, in Asians, alcohol‐induced asthma and flushing have been shown to be because of a single nucleotide polymorphism (SNP), the acetaldehyde dehydrogenase 2 (ALDH2) 487lys, causing decreased acetaldehyde (the metabolite of ethanol) metabolism and high levels of histamine. However, the ALDH2 487lys is absent in Caucasians.

Subexponential lower bounds for randomized pivoting rules for the simplex algorithm

The simplex algorithm is among the most widely used algorithms for solving linear programs in practice. With essentially all deterministic pivoting rules it is known, however, to require an exponential number of steps to solve some linear programs. No non-polynomial lower bounds were known, prior to this work, for randomized pivoting rules. We provide the first subexponential (i.e., of the form 2Ω(nα), for some α>0) lower bounds for the two most natural, and most studied, randomized pivoting rules suggested to date. The first randomized pivoting rule considered is Random-Edge, which among all improving pivoting steps (or edges) from the current basic feasible solution (or vertex) chooses one uniformly at random. The second randomized pivoting rule considered is Random-Facet, a more complicated randomized pivoting rule suggested by Kalai and by Matousek, Sharir and Welzl. Our lower bound for the Random-Facet pivoting rule essentially matches the subexponential upper bounds given by Kalai and by Matousek et al Lower bounds for Random-Edge and Random-Facet were known before only in abstract settings, and not for concrete linear programs. Our lower bounds are obtained by utilizing connections between pivoting steps performed by simplex-based algorithms and improving switches performed by policy iteration algorithms for solving Markov Decision Processes (MDPs).

Development of a catalytic cycle for the generation of C1-glycosyl carbanions with Cp2TiCl2: application to glycal synthesis

Abstract A catalytic cycle has been developed for the conversion of glycosyl halides to their corresponding glycals using Cp2TiCl2. This process can be effectively used with only 30% of the in situ generated single electron reducing agent in contrast to the 2 equivalents normally employed.

Simulating branching programs with edit distance and friends: or: a polylog shaved is a lower bound made

A recent, active line of work achieves tight lower bounds for fundamental problems under the Strong Exponential Time Hypothesis (SETH). A celebrated result of Backurs and Indyk (STOC’15) proves that computing the Edit Distance of two sequences of length n in truly subquadratic O(n2−ε) time, for some ε>0, is impossible under SETH. The result was extended by follow-up works to simpler looking problems like finding the Longest Common Subsequence (LCS). SETH is a very strong assumption, asserting that even linear size CNF formulas cannot be analyzed for satisfiability with an exponential speedup over exhaustive search. We consider much safer assumptions, e.g. that such a speedup is impossible for SAT on more expressive representations, like subexponential-size NC circuits. Intuitively, this assumption is much more plausible: NC circuits can implement linear algebra and complex cryptographic primitives, while CNFs cannot even approximately compute an XOR of bits. Our main result is a surprising reduction from SAT on Branching Programs to fundamental problems in P like Edit Distance, LCS, and many others. Truly subquadratic algorithms for these problems therefore have far more remarkable consequences than merely faster CNF-SAT algorithms. For example, SAT on arbitrary o(n)-depth bounded fan-in circuits (and therefore also NC-Circuit-SAT) can be solved in (2−ε)n time. An interesting feature of our work is that we get major consequences even from mildly subquadratic algorithms for Edit Distance or LCS. For example, we show that if an arbitrarily large polylog factor is shaved from n2 for Edit Distance then NEXP does not have non-uniform NC1 circuits.

Approximability and Parameterized Complexity of Minmax Values

We consider approximating the minmax value of a multi-playergame in strategic form. Tightening recent bounds by Borgs et al.,we observe that approximating the value with a precision ofelogn digits (for any constant e> 0) isNP-hard, where n is the size of the game. On the other hand,approximating the value with a precision of c loglogn digits (forany constant c ≥ 1) can be done inquasi-polynomial time. We consider the parameterized complexity ofthe problem, with the parameter being the number of pure strategiesk of the player for which the minmax value is computed. We showthat if there are three players, k = 2 and there areonly two possible rational payoffs, the minmax value is a rationalnumber and can be computed exactly in linear time. In the generalcase, we show that the value can be approximated with anypolynomial number of digits of accuracy in time n O(k). On theother hand, we show that minmax value approximation is W[1]-hardand hence not likely to be fixed parameter tractable. Concretely,we show that if k-Clique requires time n Ω(k) then so doesminmax value computation.

A convenient synthesis of glycals employing in-situ generated Cp2TiCl

Abstract Reductive elimination of acetylated glycosyl bromides to the corresponding glycal is easily achieved by mixing the bromide with Cp 2 TiCl 2 and Mn in THF, and hence does not require the separate preparation of Cp 2 TiCl using glove-box techniques.

Lower Bounds for Howard’s Algorithm for Finding Minimum Mean-Cost Cycles

Howard’s policy iteration algorithm is one of the most widely used algorithms for finding optimal policies for controlling Markov Decision Processes (MDPs). When applied to weighted directed graphs, which may be viewed as Deterministic MDPs (DMDPs), Howard’s algorithm can be used to find Minimum Mean-Cost cycles (MMCC). Experimental studies suggest that Howard’s algorithm works extremely well in this context. The theoretical complexity of Howard’s algorithm for finding MMCCs is a mystery. No polynomial time bound is known on its running time. Prior to this work, there were only linear lower bounds on the number of iterations performed by Howard’s algorithm. We provide the first weighted graphs on which Howard’s algorithm performs Ω(n 2) iterations, where n is the number of vertices in the graph.

Elevated pressure selectively blunts flow-evoked vasodilatation in rat mesenteric small arteries

The present study investigated mechanisms underlying impaired endothelium‐dependent vasodilatation elicited by elevating the intraluminal pressure in rat mesenteric small arteries.

On acyclicity of games with cycles

We study restricted improvement cycles (ri-cycles) in finite positional n-person games with perfect information modeled by directed graphs (di-graphs) that may contain directed cycles (di-cycles). We assume that all these di-cycles form one outcome c, for example, a draw. We obtain criteria of restricted improvement acyclicity (ri-acyclicity) in two cases: for n=2 and for acyclic di-graphs. We provide several examples that outline the limits of these criteria and show that, essentially, there are no other ri-acyclic cases. We also discuss connections between ri-acyclicity and some open problems related to Nash-solvability.