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1. Tractable Orders for Direct Access to Ranked Answers of Conjunctive Queries

   https://doi.org/10.1145/3578517  

   Carmeli, Nofar; Tziavelis, Nikolaos; Gatterbauer, Wolfgang; Kimelfeld, Benny; Riedewald, Mirek (March 2023, ACM Transactions on Database Systems)

   We study the question of when we can provide direct access to the k-th answer to a Conjunctive Query (CQ) according to a specified order over the answers in time logarithmic in the size of the database, following a preprocessing step that constructs a data structure in time quasilinear in database size. Specifically, we embark on the challenge of identifying the tractable answer orderings , that is, those orders that allow for such complexity guarantees. To better understand the computational challenge at hand, we also investigate the more modest task of providing access to only a single answer (i.e., finding the answer at a given position), a task that we refer to as the selection problem , and ask when it can be performed in quasilinear time. We also explore the question of when selection is indeed easier than ranked direct access. We begin with lexicographic orders . For each of the two problems, we give a decidable characterization (under conventional complexity assumptions) of the class of tractable lexicographic orders for every CQ without self-joins. We then continue to the more general orders by the sum of attribute weights and establish the corresponding decidable characterizations, for each of the two problems, of the tractable CQs without self-joins. Finally, we explore the question of when the satisfaction of Functional Dependencies (FDs) can be utilized for tractability and establish the corresponding generalizations of our characterizations for every set of unary FDs. 

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2. Waring’s Problem for Rational Functions in One Variable

   https://doi.org/10.1093/qmathj/haz052  

   Im, Bo-Hae; Larsen, Michael (February 2020, The quarterly journal of mathematics)

   Let f∈ℚ(x) be a non-constant rational function. We consider ‘Waring’s problem for f(x), i.e., whether every element of ℚ can be written as a bounded sum of elements of {f(a)∣a∈ℚ}. For rational functions of degree 2, we give necessary and sufficient conditions. For higher degrees, we prove that every polynomial of odd degree and every odd Laurent polynomial satisfies Waring’s problem. We also consider the 'easier Waring’s problem': whether every element of ℚ can be represented as a bounded sum of elements of {±f(a)∣a∈ℚ}. ⁠. 

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3. Algorithms for Automatic Ranking of Participants and Tasks in an Anonymized Contest

   https://doi.org/10.1007/978-3-319-53925-6_26  

   Jiao, Yang; Ravi, R; Gatterbauer, Wolfgang (January 2017, Walcom)

   We introduce a new set of problems based on the Chain Editing problem. In our version of Chain Editing, we are given a set of anonymous participants and a set of undisclosed tasks that every participant attempts. For each participant-task pair, we know whether the participant has succeeded at the task or not. We assume that participants vary in their ability to solve tasks, and that tasks vary in their difficulty to be solved. In an ideal world, stronger participants should succeed at a superset of tasks that weaker participants succeed at. Similarly, easier tasks should be completed successfully by a superset of participants who succeed at harder tasks. In reality, it can happen that a stronger participant fails at a task that a weaker participants succeeds at. Our goal is to find a perfect nesting of the participant-task relations by flipping a minimum number of participant-task relations, implying such a “nearest perfect ordering” to be the one that is closest to the truth of participant strengths and task difficulties. Many variants of the problem are known to be NP-hard. We propose six natural k-near versions of the Chain Editing problem and classify their complexity. The input to a k-near Chain Editing problem includes an initial ordering of the participants (or tasks) that we are required to respect by moving each participant (or task) at most k positions from the initial ordering. We obtain surprising results on the complexity of the six k-near problems: Five of the problems are polynomial-time solvable using dynamic programming, but one of them is NP-hard. 

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4. The Computational Complexity of Factored Graphs

   https://doi.org/10.4230/LIPICS.ITCS.2025.58  

   Gupta, Shreya; Huang, Boyang; Impagliazzo, Russell; Woo, Stanley; Ye, Christopher (January 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Meka, Raghu (Ed.)

   While graphs and abstract data structures can be large and complex, practical instances are often regular or highly structured. If the instance has sufficient structure, we might hope to compress the object into a more succinct representation. An efficient algorithm (with respect to the compressed input size) could then lead to more efficient computations than algorithms taking the explicit, uncompressed object as input. This leads to a natural question: when does knowing the input instance has a more succinct representation make computation easier? We initiate the study of the computational complexity of problems on factored graphs: graphs that are given as a formula of products and unions on smaller graphs. For any graph problem, we define a parameterized version that takes factored graphs as input, parameterized by the number of (smaller) ordinary graphs used to construct the factored graph. In this setting, we characterize the parameterized complexity of several natural graph problems, exhibiting a variety of complexities. We show that a decision version of lexicographically first maximal independent set is XP-complete, and therefore unconditionally not fixed-parameter tractable (FPT). On the other hand, we show that clique counting is FPT. Finally, we show that reachability is XNL-complete. Moreover, XNL is contained in FPT if and only if NL is contained in some fixed polynomial time. 

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5. On Super Strong ETH

   https://doi.org/10.1613/jair.1.11859  

   Vyas, Nikhil; Williams, Ryan (January 2021, Journal of Artificial Intelligence Research)

   Multiple known algorithmic paradigms (backtracking, local search and the polynomial method) only yield a 2n(1-1/O(k)) time algorithm for k-SAT in the worst case. For this reason, it has been hypothesized that the worst-case k-SAT problem cannot be solved in 2n(1-f(k)/k) time for any unbounded function f. This hypothesis has been called the "Super-Strong ETH", modelled after the ETH and the Strong ETH. It has also been hypothesized that k-SAT is hard to solve for randomly chosen instances near the "critical threshold", where the clause-to-variable ratio is such that randomly chosen instances are satisfiable with probability 1/2. We give a randomized algorithm which refutes the Super-Strong ETH for the case of random k-SAT and planted k-SAT for any clause-to-variable ratio. For example, given any random k-SAT instance F with n variables and m clauses, our algorithm decides satisfiability for F in  2n(1-c*log(k)/k) time with high probability (over the choice of the formula and the randomness of the algorithm). It turns out that a well-known algorithm from the literature on SAT algorithms does the job: the PPZ algorithm of Paturi, Pudlak, and Zane (1999).   The Unique k-SAT problem is the special case where there is at most one satisfying assignment. Improving prior reductions, we show that the Super-Strong ETHs for Unique k-SAT and k-SAT are equivalent. More precisely, we show the time complexities of Unique k-SAT and k-SAT are very tightly correlated: if Unique k-SAT is in  2n(1-f(k)/k) time for an unbounded f, then k-SAT is in 2n(1-f(k)/(2k)) time. 

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