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1. Approximation Algorithms for 𝓁_p-Shortest Path and 𝓁_p-Group Steiner Tree

   https://doi.org/10.4230/LIPIcs.ICALP.2024.111  

   Makarychev, Yury; Ovsiankin, Max; Tani, Erasmo (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Bringmann, Karl; Grohe, Martin; Puppis, Gabriele; Svensson, Ola (Ed.)

   We present polylogarithmic approximation algorithms for variants of the Shortest Path, Group Steiner Tree, and Group ATSP problems with vector costs. In these problems, each edge e has a vector cost c_e ∈ ℝ_{≥0}^𝓁;. For a feasible solution - a path, subtree, or tour (respectively) - we find the total vector cost of all the edges in the solution and then compute the 𝓁_p-norm of the obtained cost vector (we assume that p ≥ 1 is an integer). Our algorithms for series-parallel graphs run in polynomial time and those for arbitrary graphs run in quasi-polynomial time. To obtain our results, we introduce and use new flow-based Sum-of-Squares relaxations. We also obtain a number of hardness results. 

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2. Local Computation Algorithms for Graphs of Non-constant Degrees

   https://doi.org/10.1007/s00453-016-0126-y  

   Levi, Reut; Rubinfeld, Ronitt; Yodpinyanee, Anak (April 2017, Algorithmica)

   In the model of local computation algorithms (LCAs), we aim to compute the queried part of the output by examining only a small (sublinear) portion of the input. Many recently developed LCAs on graph problems achieve time and space complexities with very low dependence on n, the number of vertices. Nonetheless, these complexities are generally at least exponential in d, the upper bound on the degree of the input graph. Instead, we consider the case where parameter d can be moderately dependent on n, and aim for complexities with subexponential dependence on d, while maintaining polylogarithmic dependence on n. We present: -a randomized LCA for computing maximal independent sets whose time and space complexities are quasi-polynomial in d and polylogarithmic in n; -for constant ε>0, a randomized LCA that provides a (1−ε)-approximation to maximum matching with high probability, whose time and space complexities are polynomial in d and polylogarithmic in n. 

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3. Time-Dependent Shortest Path Problems with Penalties and Limits on Waiting

   https://doi.org/10.1287/ijoc.2020.0985  

   He, Edward; Boland, Natashia; Nemhauser, George; Savelsbergh, Martin (January 2020, INFORMS Journal on Computing)

   null (Ed.)

   Waiting at the right location at the right time can be critically important in certain variants of time-dependent shortest path problems. We investigate the computational complexity of time-dependent shortest path problems in which there is either a penalty on waiting or a limit on the total time spent waiting at a given subset of the nodes. We show that some cases are nondeterministic polynomial-time hard, and others can be solved in polynomial time, depending on the choice of the subset of nodes, on whether waiting is penalized or constrained, and on the magnitude of the penalty/waiting limit parameter. Summary of Contributions: This paper addresses simple yet relevant extensions of a fundamental problem in Operations Research: the Shortest Path Problem (SPP). It considers time-dependent variants of SPP, which can account for changing traffic and/or weather conditions. The first variant that is tackled allows for waiting at certain nodes but at a cost. The second variant instead places a limit on the total waiting. Both variants have applications in transportation, e.g., when it is possible to wait at certain locations if the benefits outweigh the costs. The paper investigates these problems using complexity analysis and algorithm design, both tools from the field of computing. Different cases are considered depending on which of the nodes contribute to the waiting cost or waiting limit (all nodes, all nodes except the origin, a subset of nodes…). The computational complexity of all cases is determined, providing complexity proofs for the variants that are NP-Hard and polynomial time algorithms for the variants that are in P. 

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4. Privately Estimating Graph Parameters in Sublinear Time

   https://doi.org/10.4230/LIPIcs.ICALP.2022.82  

   Blocki, J; Grigorescu, E; Mukherjee, T. (January 2022, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022).)

   Mikołaj Boja´nczyk, Emanuela Merelli (Ed.)

   We initiate a systematic study of algorithms that are both differentially-private and run in sublinear time for several problems in which the goal is to estimate natural graph parameters. Our main result is a differentially-private $$(1+\rho)$$-approximation algorithm for the problem of computing the average degree of a graph, for every $$\rho>0$$. The running time of the algorithm is roughly the same (for sparse graphs) as its non-private version proposed by Goldreich and Ron (Sublinear Algorithms, 2005). We also obtain the first differentially-private sublinear-time approximation algorithms for the maximum matching size and the minimum vertex cover size of a graph. An overarching technique we employ is the notion of \emph{coupled global sensitivity} of randomized algorithms. Related variants of this notion of sensitivity have been used in the literature in ad-hoc ways. Here we formalize the notion and develop it as a unifying framework for privacy analysis of randomized approximation algorithms. 

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5. The best ways to slice a polytope

   https://doi.org/10.1090/mcom/4006  

   Brandenburg, Marie-Charlotte; De_Loera, Jesus; Meroni, Chiara (July 2024, American Mathematical Society)

   We study the structure of the set of all possible affine hyperplane sections of a convex polytope. We present two different cell decompositions of this set, induced by hyperplane arrangements. Using our decomposition, we bound the number of possible combinatorial types of sections and craft algorithms that compute optimal sections of the polytope according to various combinatorial and metric criteria, including sections that maximize the number of -dimensional faces, maximize the volume, and maximize the integral of a polynomial. Our optimization algorithms run in polynomial time in fixed dimension, but the same problems show computational complexity hardness otherwise. Our tools can be extended to intersection with halfspaces and projections onto hyperplanes. Finally, we present several experiments illustrating our theorems and algorithms on famous polytopes. 

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