"Tetravex is a widely played one person computer game in which you are given $n^2$ unit tiles, each edge of which is labelled with a number. The objective is to place each tile within a $n$ by $n$ square such that all neighbouring edges are labelled with an identical number. Unfortunately, playing Tetravex is computationally hard. More precisely, we prove that deciding if there is a tiling of the Tetravex board is NP-complete. Deciding where to place the tiles is therefore NP-hard. This may help to explain why Tetravex is a good puzzle. This result compliments a number of similar results for one person games involving tiling. For example, NP-completeness results have been shown for: the offline version of Tetris, KPlumber (which involves rotating tiles containing drawings of pipes to make a connected network), and shortest sliding puzzle problems. It raises a number of open questions. For example, is the infinite version Turing-complete? How do we generate Tetravex problems which are truly puzzling as random NP-complete problems are often surprising easy to solve? Can we observe phase transition behaviour? What about the complexity of the problem when it is guaranteed to have an unique solution? How do we generate puzzles with unique solutions?"

"Determining whether an unordered collection of overlapping substrings (called shingles) can be uniquely decoded into a consistent string is a problem that lies within the foundation of a broad assortment of disciplines ranging from networking and information theory through cryptography and even genetic engineering and linguistics. We present three perspectives on this problem: a graph theoretic framework due to Pevzner, an automata theoretic approach from our previous work, and a new insight that yields a time-optimal streaming algorithm for determining whether a string of $n$ characters over the alphabet $Sigma$ can be uniquely decoded from its two-character shingles. Our algorithm achieves an overall time complexity $Theta(n)$ and space complexity $O(|Sigma|)$. As an application, we demonstrate how this algorithm can be extended to larger shingles for efficient string reconciliation."

"This paper presents a study on two data structures that have been used to model several problems in computer science: and/or graphs and x-y graphs. An and/or graph is an acyclic digraph containing a source, such that every vertex v has a label f(v) in {and,or} and edges represent dependency relations between vertices: a vertex labeled and depends on all of its out-neighbors, while a vertex labeled or depends on only one of its out-neighbors. X-y graphs are defined as a natural generalization of and/or graphs: every vertex vi of an x-y graph has a label xi-yi to mean that vi depends on xi of its yi out-neighbors. We analyze the complexity of the optimization problems Min-and/or and Min-x-y, which consist of finding solution subgraphs of optimal weight for and/or and x-y graphs, respectively. Motivated by the large applicability as well as the hardness of Min-and/or and Min-x-y, we study new complexity aspects of such problems, both from a classical and a parameterized point of view. …"

"In this paper, first we give a sequential linear-time algorithm for the longest path problem in meshes. This algorithm can be considered as an improvement of [13]. Then based on this sequential algorithm, we present a constant-time parallel algorithm for the problem which can be run on every parallel machine."

"Computational complexity is one of the most beautiful fields of modern mathematics, and it is increasingly relevant to other sciences ranging from physics to biology. This book gives a lucid and playful explanation of the field, starting with P and NP-completeness. The authors explain why the P vs. NP problem is so fundamental, and why it is so hard to resolve. They then lead the reader through the complexity of mazes and games; optimization in theory and practice; randomized algorithms, interactive proofs, and pseudorandomness; Markov chains and phase transitions; and the outer reaches of quantum computing. At every turn, they use a minimum of formalism, providing explanations that are both deep and accessible. The book is intended for graduates and undergraduates, scientists from other areas who have long wanted to understand this subject, and experts who want to fall in love with this field all over again."

"The main source of this speed penalty is an insistence that the result of a matrix multiply should follow R’s rules for handling infinity, NaN (not-a-number), and NA.  These rules correspond to what happens with ordinary arithmetic operations on modern computers, which follow a standard for floating-point arithmetic in which, for example, 0/0 is NaN.  You might therefore think that nothing special is needed to arrange for matrix multiplies to produce NaNs as required.  However, R does matrix multiplications using the BLAS library, which comes in many versions, some of which may try to speed things up by avoiding “unnecessary” operations such as multiplication by zero — assuming that that will always result in zero.  However, zero times NaN or infinity is supposed to be NaN, not zero."

"We give a space-optimal algorithm with update time O(log^2(1/eps)loglog(1/eps)) for (1+eps)-approximating the pth frequency moment, 0 < p < 2, of a length-n vector updated in a data stream. This provides a nearly exponential improvement in the update time complexity over the previous space-optimal algorithm of [Kane-Nelson-Woodruff, SODA 2010], which had update time Omega(1/eps^2)."

"We provide several basic results. We obtain an efficient algorithm for determining the heapa- bility of a sequence, and also prove that the question of whether a sequence can be arranged in a complete binary heap is NP-hard. Regarding subsequences we show that, with high probability, the longest heapable subsequence of a random permutation of n numbers has length (1 − o(1))n, and a subsequence of length (1 − o(1))n can in fact be found online with high probability. We similarly show that for a random permutation a subsequence that yields a complete heap of size αn for a constant α can be found with high probability. Our work highlights the interesting structure underlying this class of subsequence problems, and we leave many further interesting variations open for future work."

"We study several problems related to finding reset words in deterministic finite automata. In particular, we establish that the problem of deciding whether a shortest reset word has length k is complete for the complexity class DP. This result answers a question posed by Volkov. For the search problems of finding a shortest reset word and the length of a shortest reset word, we establish membership in the complexity classes FP^NP and FP^NP[log], respectively. Moreover, we show that both these problems are hard for FP^NP[log]. Finally, we observe that computing a reset word of a given length is FNP-complete."

"The Lin-Kernighan heuristic is known to be one of the most successful heuristics for the Traveling Salesman Problem (TSP). It has also proven its efficiency in application to some other problems. In this paper we discuss possible adaptations of TSP heuristics for the Generalized Traveling Salesman Problem (GTSP) and focus on the case of the Lin-Kernighan algorithm. At first, we provide an easy-to-understand description of the original Lin-Kernighan heuristic. Then we propose several adaptations, both trivial and complicated. Finally, we conduct a fair competition between all the variations of the Lin-Kernighan adaptation and some other GTSP heuristics. It appears that our adaptation of the Lin-Kernighan algorithm for the GTSP reproduces the success of the original heuristic. Different variations of our adaptation outperform all other heuristics in a wide range of trade-offs between solution quality and running time, making Lin-Kernighan the state-of-the-art GTSP local search."

"The main goal of the paper is to introduce methods which compute B\'ezier curves faster than Casteljau's method does. These methods are based on the spectral factorization of a $n\times n$ Bernstein matrix, $B^e_n(s)= P_nG_n(s)P_n^{-1}$, where $P_n$ is the $n\times n$ lower triangular Pascal matrix. So we first calculate the exact optimum positive value $t$ in order to transform $P_n$ in a scaled Toeplitz matrix, which is a problem that was partially solved by X. Wang and J. Zhou (2006). Then fast Pascal matrix-vector multiplications and strategies of polynomial evaluation are put together to compute B\'ezier curves. Nevertheless, when $n$ increases, more precise Pascal matrix-vector multiplications allied to affine transformations of the vectors of coordinates of the control points of the curve are then necessary to stabilize all the computation."

I wonder to what extent this sort of thing might be useful in heuristics for Pareto-ranking

"We formalize the procedure of adaptation of a TSP neighborhood for GTSP and propose efficient algorithms to explore the obtained neighborhoods. We also generalize all other existing and some new GTSP neighborhoods. Apart from these theoretical results, we also provide the results of a thorough experimental analysis to compare the proposed algorithms implementations and find out which neighborhoods are the most efficient in practice."

"We present a near-linear time algorithm that approximates the edit distance between two strings within a polylogarithmic factor; specifically, for strings of length n and every fixed epsilon>0, it can compute a (log n)^O(1/epsilon) approximation in n^(1+epsilon) time. This is an exponential improvement over the previously known factor, 2^(O (sqrt(log n))), with a comparable running time (Ostrovsky and Rabani J.ACM 2007; Andoni and Onak STOC 2009). Previously, no efficient polylogarithmic approximation algorithm was known for any computational task involving edit distance (e.g., nearest neighbor search or sketching). "

"We describe a variation of a technique by Chiba and Nishizeki [3], leading to a data structure for graph algorithmic problems, called the h-graph data structure. It supports operations of insertion and removal of vertices, as well as insertion and removal of edges. Although the data structure can be used for general purpose, it is particularly suitable for applications in dynamic graph algorithms."

"Computational Complexity studies the power and limitations of efficient computation. So is efficient computation purely an abstract mathematical object or is it trying to model a real world phenomenon? I would argue the latter. Efficient computation occurs not just in computers but in biological systems, physical systems, chemical systems, economic systems and much more. Physics focuses on the "what", computational complexity on the "how"."

"… The results imply that several families of constraint satisfaction problems exhibit a complexity dichotomy: the problems are in P or NP-hard, depending on the choice of the allowed relations. As concrete examples, we investigate fundamental algebraic constraint satisfaction problems. The first class consists of all first-order expansions of (Q;+). The second class is the affine variant of the first class. In both cases, we obtain full dichotomies by utilising our general methods."

"Computational complexity has in the past adapted well to new computation models from the PRAM to biological and quantum computers. But we are seeing new computing paradigms in multicore and cloud computing and theory seems late to the party. There was a nice SODA paper on MapReduce, the basic cloud computing operation, but for the most part theorists haven't tackled the cloud computing model and only a few have looked at multicore. Theory can say much about new computational methods, both in how we can take advantage of them and what they can't do, but only if we make the effort to develop the proper models to capture these new approaches."

"We consider the computation of the volume of the union of high-dimensional geometric objects. While showing that this problem is #P-hard already for very simple bodies (i.e., axis-parallel boxes), we give a fast FPRAS for all objects where one can: (1) test whether a given point lies inside the object, (2) sample a point uniformly, (3) calculate the volume of the object in polynomial time. All three oracles can be weak, that is, just approximate. This implies that Klee's measure problem and the hypervolume indicator can be approximated efficiently even though they are #P-hard and hence cannot be solved exactly in time polynomial in the number of dimensions unless P=NP. Our algorithm also allows to approximate efficiently the volume of the union of convex bodies given by weak membership oracles. "