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1. Attacking Similarity-Based Link Prediction in Social Networks

   Zhou, Kai; Michalak, Tomasz; Waniek, Marcin; Rahwan, Talal; Vorobeychik, Yevgeniy (January 2019, International Conference on Autonomous Agents and Multiagent Systems)

   Link prediction is one of the fundamental problems in computational social science. A particularly common means to predict existence of unobserved links is via structural similarity metrics, such as the number of common neighbors; node pairs with higher similarity are thus deemed more likely to be linked. However, a number of applications of link prediction, such as predicting links in gang or terrorist networks, are adversarial, with another party incentivized to minimize its effectiveness by manipulating observed information about the network. We offer a comprehensive algorithmic investigation of the problem of attacking similarity-based link prediction through link deletion, focusing on two broad classes of such approaches, one which uses only local information about target links, and another which uses global network information. While we show several variations of the general problem to be NP-Hard for both local and global metrics, we exhibit a number of well-motivated special cases which are tractable. Additionally, we provide principled and empirically effective algorithms for the intractable cases, in some cases proving worst-case approximation guarantees. 

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2. On Efficiently Solvable Cases of Quantum k-SAT

   https://doi.org/10.4230/LIPIcs.MFCS.2018.38  

   Aldi, Marco; De Beaudrap, Marco; Gharibian, Sevag; Saeedi, Seyran (January 2018, International Symposium on Mathematical Foundations of Computer Science)

   The constraint satisfaction problems k-SAT and Quantum k-SAT (k-QSAT) are canonical NP-complete and QMA_1-complete problems (for k >= 3), respectively, where QMA_1 is a quantum generalization of NP with one-sided error. Whereas k-SAT has been well-studied for special tractable cases, as well as from a parameterized complexity perspective, much less is known in similar settings for k-QSAT. Here, we study the open problem of computing satisfying assignments to k-QSAT instances which have a "matching" or "dimer covering"; this is an NP problem whose decision variant is trivial, but whose search complexity remains open. Our results fall into three directions, all of which relate to the "matching" setting: (1) We give a polynomial-time classical algorithm for k-QSAT when all qubits occur in at most two clauses. (2) We give a parameterized algorithm for k-QSAT instances from a certain non-trivial class, which allows us to obtain exponential speedups over brute force methods in some cases by reducing the problem to solving for a single root of a single univariate polynomial. (3) We conduct a structural graph theoretic study of 3-QSAT interaction graphs which have a "matching". We remark that the results of (2), in particular, introduce a number of new tools to the study of Quantum SAT, including graph theoretic concepts such as transfer filtrations and blow-ups from algebraic geometry; we hope these prove useful elsewhere. 

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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. On the geometry of k-SAT solutions: what more can PPZ and Schöning’s algorithms do?

   Austrin, Per; Bercea, Ioana O; Goswami, Mayank; Limaye, Nutan; Srinivasan, Adarsh (July 2024, Arxiv)

   Given a k-CNF formula and an integer s, we study algorithms that obtain s solutions to the formula that are maximally dispersed. For s=2, the problem of computing the diameter of a k-CNF formula was initiated by Creszenzi and Rossi, who showed strong hardness results even for k=2. Assuming SETH, the current best upper bound [Angelsmark and Thapper '04] goes to 4n as k→∞. As our first result, we give exact algorithms for using the Fast Fourier Transform and clique-finding that run in O(2^((s−1)n)) and O(s^2|Ω_F|^(ω⌈s/3⌉)) respectively, where |Ω_F| is the size of the solution space of the formula F and ω is the matrix multiplication exponent. As our main result, we re-analyze the popular PPZ (Paturi, Pudlak, Zane '97) and Schöning's ('02) algorithms (which find one solution in time O∗(2^(ε_k n)) for εk≈1−Θ(1/k)), and show that in the same time, they can be used to approximate the diameter as well as the dispersion (s>2) problems. While we need to modify Schöning's original algorithm, we show that the PPZ algorithm, without any modification, samples solutions in a geometric sense. We believe that this property may be of independent interest. Finally, we present algorithms to output approximately diverse, approximately optimal solutions to NP-complete optimization problems running in time poly(s)O(^(2εn)) with ε<1 for several problems such as Minimum Hitting Set and Feedback Vertex Set. For these problems, all existing exact methods for finding optimal diverse solutions have a runtime with at least an exponential dependence on the number of solutions s. Our methods find bi-approximations with polynomial dependence on s. 

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5. Computing welfare-Maximizing fair allocations of indivisible goods

   https://doi.org/10.1016/j.ejor.2022.10.013  

   Aziz, Haris; Huang, Xin; Mattei, Nicholas; Segal-Halevi, Erel (January 2022, European journal of operational research)

   We analyze the run-time complexity of computing allocations that are both fair and maximize the utilitarian social welfare, defined as the sum of agents’ utilities. We focus on two tractable fairness concepts: envy-freeness up to one item (EF1) and proportionality up to one item (PROP1). We consider two computational problems: (1) Among the utilitarian-maximal allocations, decide whether there exists one that is also fair; (2) among the fair allocations, compute one that maximizes the utilitarian welfare. We show that both problems are strongly NP-hard when the number of agents is variable, and remain NP-hard for a fixed number of agents greater than two. For the special case of two agents, we find that problem (1) is polynomial-time solvable, while problem (2) remains NP-hard. Finally, with a fixed number of agents, we design pseudopolynomial-time algorithms for both problems. We extend our results to the stronger fairness notions envy-freeness up to any item (EFx) and proportionality up to any item (PROPx). 

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