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1. Hardness of Approximation for Orienteering with Multiple Time Windows

   Garg, Naveen; Khanna, Sanjeev; Kumar, Amit (January 2021, Proceedings of the ACM-SIAM Symposium on Discrete Algorithms)

   null (Ed.)

   Vehicle routing problems are a broad class of combinatorial optimization problems that can be formulated as the problem of finding a tour in a weighted graph that optimizes some function of the visited vertices. For instance, a canonical and extensively studied vehicle routing problem is the orienteering problem where the goal is to find a tour that maximizes the number of vertices visited by a given deadline. In this paper, we consider the computational tractability of a well-known generalization of the orienteering problem called the Orient-MTW problem. The input to Orient-MTW consists of a weighted graph G(V, E) where for each vertex v ∊ V we are given a set of time instants Tv ⊆ [T], and a source vertex s. A tour starting at s is said to visit a vertex v if it transits through v at any time in the set Tv. The goal is to find a tour starting at the source vertex that maximizes the number of vertices visited. It is known that this problem admits a quasi-polynomial time O(log OPT)-approximation ratio where OPT is the optimal solution value but until now no hardness better than an APX-hardness was known for this problem. Our main result is an -hardness for this problem that holds even when the underlying graph G is an undirected tree. This is the first super-constant hardness result for the Orient-MTW problem. The starting point for our result is the hardness of the SetCover problem which is known to hold on instances with a special structure. We exploit this special structure of the hard SetCover instances to first obtain a new proof of the APX-hardness result for Orient-MTW that holds even on trees of depth 2. We then recursively amplify this constant factor hardness to an -hardness, while keeping the resulting topology to be a tree. Our amplified hardness proof crucially utilizes a delicate concavity property which shows that in our encoding of SetCover instances as instances of the Orient-MTW problem, whenever the optimal cost for SetCover instance is large, any tour, no matter how it allocates its time across different sub-trees, can not visit too many vertices overall. We believe that this reduction template may also prove useful in showing hardness of other vehicle routing problems. 

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2. Leibniz International Proceedings in Informatics (LIPIcs):15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

   https://doi.org/10.4230/LIPICS.ITCS.2024.62  

   Henzinger, Monika; Saha, Barna; Seybold, Martin P.; Ye, Christopher (January 2024, 15th Innovations in Theoretical Computer Science Conference, {ITCS})

   Guruswami, Venkatesan (Ed.)

   Algorithms with predictions is a new research direction that leverages machine learned predictions for algorithm design. So far a plethora of recent works have incorporated predictions to improve on worst-case bounds for online problems. In this paper, we initiate the study of complexity of dynamic data structures with predictions, including dynamic graph algorithms. Unlike online algorithms, the goal in dynamic data structures is to maintain the solution efficiently with every update. We investigate three natural models of prediction: (1) δ-accurate predictions where each predicted request matches the true request with probability δ, (2) list-accurate predictions where a true request comes from a list of possible requests, and (3) bounded delay predictions where the true requests are a permutation of the predicted requests. We give general reductions among the prediction models, showing that bounded delay is the strongest prediction model, followed by list-accurate, and δ-accurate. Further, we identify two broad problem classes based on lower bounds due to the Online Matrix Vector (OMv) conjecture. Specifically, we show that locally correctable dynamic problems have strong conditional lower bounds for list-accurate predictions that are equivalent to the non-prediction setting, unless list-accurate predictions are perfect. Moreover, we show that locally reducible dynamic problems have time complexity that degrades gracefully with the quality of bounded delay predictions. We categorize problems with known OMv lower bounds accordingly and give several upper bounds in the delay model that show that our lower bounds are almost tight. We note that concurrent work by v.d.Brand et al. [SODA '24] and Liu and Srinivas [arXiv:2307.08890] independently study dynamic graph algorithms with predictions, but their work is mostly focused on showing upper bounds. 

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3. Graph Exploration by Energy-Sharing Mobile Agents

   Czyzowicz, J.; Dobrev, S.; Killick, R.; Kranakis, E.; Krizanc, D.; Narayanan, L.; Opatrny, J; Pankratov, D.; Shende, S. (July 2021, 28th International Colloquium on Structural Information and Communication Complexity (SIROCCO))

   null (Ed.)

   We consider the problem of collective exploration of a known n- node edge-weighted graph by k mobile agents that have limited energy but are capable of energy transfers. The agents are initially placed at an arbitrary subset of nodes in the graph, and each agent has an initial, possibly different, amount of energy. The goal of the exploration problem is for every edge in the graph to be traversed by at least one agent. The amount of energy used by an agent to travel distance x is proportional to x. In our model, the agents can share energy when co-located: when two agents meet, one can transfer part of its energy to the other. For an n-node path, we give an O(n+k) time algorithm that either nds an exploration strategy, or reports that one does not exist. For an n-node tree with l leaves, we give an O(n+lk^2) algorithm that finds an exploration strategy if one exists. Finally, for the general graph case, we show that the problem of deciding if exploration is possible by energy-sharing agents is NP-hard, even for 3-regular graphs. In addition, we show that it is always possible to find an exploration strategy if the total energy of the agents is at least twice the total weight of the edges; moreover, this is asymptotically optimal. 

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4. Local Frechet Permutation

   Perry, Jonathan; Raichel, Benjamin (July 2024, Proceedings 36th Canadian Conference on Computational Geometry)

   Nishat, Rahnuma Islam (Ed.)

   In this paper we consider computing the Fréchet distance between two curves where we are allowed to locally permute the vertices. Specifically, we limit each vertex to move at most k positions from where it started, and give fixed parameter tractable algorithms in this parameter k, whose running times match the standard Fréchet distance computation running time when k is a constant. Furthermore we also show that computing such a local permutation Fréchet distance is NP-hard when considering the weak Fréchet distance. 

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5. Subcubic Equivalences between Graph Centrality Problems, APSP, and Diameter

   https://doi.org/10.1145/3563393  

   Abboud, Amir; Grandoni, Fabrizio; Vassilevska_Williams, Virginia (January 2023, ACM Transactions on Algorithms)

   Measuring the importance of a node in a network is a major goal in the analysis of social networks, biological systems, transportation networks, and so forth. Differentcentralitymeasures have been proposed to capture the notion of node importance. For example, thecenterof a graph is a node that minimizes the maximum distance to any other node (the latter distance is theradiusof the graph). Themedianof a graph is a node that minimizes the sum of the distances to all other nodes. Informally, thebetweenness centralityof a nodewmeasures the fraction of shortest paths that havewas an intermediate node. Finally, thereach centralityof a nodewis the smallest distancersuch that anys-tshortest path passing throughwhas eithersortin the ball of radiusraroundw. The fastest known algorithms to compute the center and the median of a graph and to compute the betweenness or reach centrality even of a single node take roughly cubic time in the numbernof nodes in the input graph. It is open whether these problems admit truly subcubic algorithms, i.e., algorithms with running time Õ(n^3-δ) for some constant δ > 0.¹ We relate the complexity of the mentioned centrality problems to two classical problems for which no truly subcubic algorithm is known, namely All Pairs Shortest Paths (APSP) and Diameter. We show that Radius, Median, and Betweenness Centrality areequivalent under subcubic reductionsto APSP, i.e., that a truly subcubic algorithm for any of these problems implies a truly subcubic algorithm for all of them. We then show that Reach Centrality is equivalent to Diameter under subcubic reductions. The same holds for the problem of approximating Betweenness Centrality within any finite factor. Thus, the latter two centrality problems could potentially be solved in truly subcubic time, even if APSP required essentially cubic time. On the positive side, our reductions for Reach Centrality imply an improved Õ(Mn^ω)-time algorithm for this problem in case of non-negative integer weights upper bounded byM, where ω is a fast matrix multiplication exponent. 

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