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**Published in:** LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

We present a pseudopolynomial-time algorithm for the Knapsack problem that has running time Õ(n + t√{p_{max}}), where n is the number of items, t is the knapsack capacity, and p_{max} is the maximum item profit. This improves over the Õ(n + t p_{max})-time algorithm based on the convolution and prediction technique by Bateni et al. (STOC 2018). Moreover, we give some evidence, based on a strengthening of the Min-Plus Convolution Hypothesis, that our running time might be optimal.
Our algorithm uses two new technical tools, which might be of independent interest. First, we generalize the Õ(n^{1.5})-time algorithm for bounded monotone min-plus convolution by Chi et al. (STOC 2022) to the rectangular case where the range of entries can be different from the sequence length. Second, we give a reduction from general knapsack instances to balanced instances, where all items have nearly the same profit-to-weight ratio, up to a constant factor.
Using these techniques, we can also obtain algorithms that run in time Õ(n + OPT√{w_{max}}), Õ(n + (nw_{max}p_{max})^{1/3}t^{2/3}), and Õ(n + (nw_{max}p_{max})^{1/3} OPT^{2/3}), where OPT is the optimal total profit and w_{max} is the maximum item weight.

Karl Bringmann, Anita Dürr, and Adam Polak. Even Faster Knapsack via Rectangular Monotone Min-Plus Convolution and Balancing. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 33:1-33:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bringmann_et_al:LIPIcs.ESA.2024.33, author = {Bringmann, Karl and D\"{u}rr, Anita and Polak, Adam}, title = {{Even Faster Knapsack via Rectangular Monotone Min-Plus Convolution and Balancing}}, booktitle = {32nd Annual European Symposium on Algorithms (ESA 2024)}, pages = {33:1--33:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-338-6}, ISSN = {1868-8969}, year = {2024}, volume = {308}, editor = {Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.33}, URN = {urn:nbn:de:0030-drops-211047}, doi = {10.4230/LIPIcs.ESA.2024.33}, annote = {Keywords: 0-1-Knapsack problem, bounded monotone min-plus convolution, fine-grained complexity} }

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**Published in:** LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024)

The problem of designing connectivity oracles supporting vertex failures is one of the basic data structures problems for undirected graphs. It is already well understood: previous works [Duan-Pettie STOC'10; Long-Saranurak FOCS'22] achieve query time linear in the number of failed vertices, and it is conditionally optimal as long as we require preprocessing time polynomial in the size of the graph and update time polynomial in the number of failed vertices.
We revisit this problem in the paradigm of algorithms with predictions: we ask if the query time can be improved if the set of failed vertices can be predicted beforehand up to a small number of errors. More specifically, we design a data structure that, given a graph G = (V,E) and a set of vertices predicted to fail D̂ ⊆ V of size d = |D̂|, preprocesses it in time Õ(d|E|) and then can receive an update given as the symmetric difference between the predicted and the actual set of failed vertices D̂△D = (D̂ ⧵ D) ∪ (D ⧵ D̂) of size η = |D̂△D|, process it in time Õ(η⁴), and after that answer connectivity queries in G ⧵ D in time O(η). Viewed from another perspective, our data structure provides an improvement over the state of the art for the fully dynamic subgraph connectivity problem in the sensitivity setting [Henzinger-Neumann ESA'16].
We argue that the preprocessing time and query time of our data structure are conditionally optimal under standard fine-grained complexity assumptions.

Bingbing Hu, Evangelos Kosinas, and Adam Polak. Connectivity Oracles for Predictable Vertex Failures. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 72:1-72:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{hu_et_al:LIPIcs.ESA.2024.72, author = {Hu, Bingbing and Kosinas, Evangelos and Polak, Adam}, title = {{Connectivity Oracles for Predictable Vertex Failures}}, booktitle = {32nd Annual European Symposium on Algorithms (ESA 2024)}, pages = {72:1--72:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-338-6}, ISSN = {1868-8969}, year = {2024}, volume = {308}, editor = {Chan, Timothy and Fischer, Johannes and Iacono, John and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.72}, URN = {urn:nbn:de:0030-drops-211437}, doi = {10.4230/LIPIcs.ESA.2024.72}, annote = {Keywords: Data structures, graph connectivity, algorithms with predictions} }

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**Published in:** LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

As one of the three main pillars of fine-grained complexity theory, the 3SUM problem explains the hardness of many diverse polynomial-time problems via fine-grained reductions. Many of these reductions are either directly based on or heavily inspired by Pătraşcu’s framework involving additive hashing and are thus randomized. Some selected reductions were derandomized in previous work [Chan, He; SOSA'20], but the current techniques are limited and a major fraction of the reductions remains randomized.
In this work we gather a toolkit aimed to derandomize reductions based on additive hashing. Using this toolkit, we manage to derandomize almost all known 3SUM-hardness reductions. As technical highlights we derandomize the hardness reductions to (offline) Set Disjointness, (offline) Set Intersection and Triangle Listing - these questions were explicitly left open in previous work [Kopelowitz, Pettie, Porat; SODA'16]. The few exceptions to our work fall into a special category of recent reductions based on structure-versus-randomness dichotomies.
We expect that our toolkit can be readily applied to derandomize future reductions as well. As a conceptual innovation, our work thereby promotes the theory of deterministic 3SUM-hardness.
As our second contribution, we prove that there is a deterministic universe reduction for 3SUM. Specifically, using additive hashing it is a standard trick to assume that the numbers in 3SUM have size at most n³. We prove that this assumption is similarly valid for deterministic algorithms.

Nick Fischer, Piotr Kaliciak, and Adam Polak. Deterministic 3SUM-Hardness. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 49:1-49:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{fischer_et_al:LIPIcs.ITCS.2024.49, author = {Fischer, Nick and Kaliciak, Piotr and Polak, Adam}, title = {{Deterministic 3SUM-Hardness}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {49:1--49:24}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-309-6}, ISSN = {1868-8969}, year = {2024}, volume = {287}, editor = {Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.49}, URN = {urn:nbn:de:0030-drops-195772}, doi = {10.4230/LIPIcs.ITCS.2024.49}, annote = {Keywords: 3SUM, derandomization, fine-grained complexity} }

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**Published in:** LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

This paper is about the problem of finding a shortest s-t path using at most h edges in edge-weighted graphs. The Bellman-Ford algorithm solves this problem in O(hm) time, where m is the number of edges. We show that this running time is optimal, up to subpolynomial factors, under popular fine-grained complexity assumptions.
More specifically, we show that under the APSP Hypothesis the problem cannot be solved faster already in undirected graphs with nonnegative edge weights. This lower bound holds even restricted to graphs of arbitrary density and for arbitrary h ∈ O(√m). Moreover, under a stronger assumption, namely the Min-Plus Convolution Hypothesis, we can eliminate the restriction h ∈ O(√m). In other words, the O(hm) bound is tight for the entire space of parameters h, m, and n, where n is the number of nodes.
Our lower bounds can be contrasted with the recent near-linear time algorithm for the negative-weight Single-Source Shortest Paths problem, which is the textbook application of the Bellman-Ford algorithm.

Tomasz Kociumaka and Adam Polak. Bellman-Ford Is Optimal for Shortest Hop-Bounded Paths. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 72:1-72:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{kociumaka_et_al:LIPIcs.ESA.2023.72, author = {Kociumaka, Tomasz and Polak, Adam}, title = {{Bellman-Ford Is Optimal for Shortest Hop-Bounded Paths}}, booktitle = {31st Annual European Symposium on Algorithms (ESA 2023)}, pages = {72:1--72:10}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-295-2}, ISSN = {1868-8969}, year = {2023}, volume = {274}, editor = {G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2023.72}, URN = {urn:nbn:de:0030-drops-187257}, doi = {10.4230/LIPIcs.ESA.2023.72}, annote = {Keywords: Fine-grained complexity, graph algorithms, lower bounds, shortest paths} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

We study the basic problem of assigning memoryless workers to tasks with dynamically changing demands. Given a set of w workers and a multiset T ⊆ [t] of |T| = w tasks, a memoryless worker-task assignment function is any function ϕ that assigns the workers [w] to the tasks T based only on the current value of T. The assignment function ϕ is said to have switching cost at most k if, for every task multiset T, changing the contents of T by one task changes ϕ(T) by at most k worker assignments. The goal of memoryless worker task assignment is to construct an assignment function with the smallest possible switching cost.
In past work, the problem of determining the optimal switching cost has been posed as an open question. There are no known sub-linear upper bounds, and after considerable effort, the best known lower bound remains 4 (ICALP 2020).
We show that it is possible to achieve polylogarithmic switching cost. We give a construction via the probabilistic method that achieves switching cost O(log w log (wt)) and an explicit construction that achieves switching cost polylog (wt). We also prove a super-constant lower bound on switching cost: we show that for any value of w, there exists a value of t for which the optimal switching cost is w. Thus it is not possible to achieve a switching cost that is sublinear strictly as a function of w.
Finally, we present an application of the worker-task assignment problem to a metric embeddings problem. In particular, we use our results to give the first low-distortion embedding from sparse binary vectors into low-dimensional Hamming space.

Aaron Berger, William Kuszmaul, Adam Polak, Jonathan Tidor, and Nicole Wein. Memoryless Worker-Task Assignment with Polylogarithmic Switching Cost. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 19:1-19:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{berger_et_al:LIPIcs.ICALP.2022.19, author = {Berger, Aaron and Kuszmaul, William and Polak, Adam and Tidor, Jonathan and Wein, Nicole}, title = {{Memoryless Worker-Task Assignment with Polylogarithmic Switching Cost}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {19:1--19:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.19}, URN = {urn:nbn:de:0030-drops-163608}, doi = {10.4230/LIPIcs.ICALP.2022.19}, annote = {Keywords: Distributed Task Allocation, Metric Embeddings, Probabilistic Method} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 229, 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)

We solve the Bin Packing problem in O^*(2^k) time, where k is the number of items less or equal to one third of the bin capacity. This parameter measures the distance from the polynomially solvable case of only large (i.e., greater than one third) items. Our algorithm is actually designed to work for a more general Vector Bin Packing problem, in which items are multidimensional vectors. We improve over the previous fastest O^*(k! ⋅ 4^k) time algorithm.
Our algorithm works by reducing the problem to finding an exact weight perfect matching in a (multi-)graph with O^*(2^k) edges, whose weights are integers of the order of O^*(2^k). To solve the matching problem in the desired time, we give a variant of the classic Mulmuley-Vazirani-Vazirani algorithm with only a linear dependence on the edge weights and the number of edges - which may be of independent interest.
Moreover, we give a tight lower bound, under the Strong Exponential Time Hypothesis (SETH), showing that the constant 2 in the base of the exponent cannot be further improved for Vector Bin Packing.
Our techniques also lead to improved algorithms for Vector Multiple Knapsack, Vector Bin Covering, and Perfect Matching with Hitting Constraints.

Alexandra Lassota, Aleksander Łukasiewicz, and Adam Polak. Tight Vector Bin Packing with Few Small Items via Fast Exact Matching in Multigraphs. In 49th International Colloquium on Automata, Languages, and Programming (ICALP 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 229, pp. 87:1-87:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{lassota_et_al:LIPIcs.ICALP.2022.87, author = {Lassota, Alexandra and {\L}ukasiewicz, Aleksander and Polak, Adam}, title = {{Tight Vector Bin Packing with Few Small Items via Fast Exact Matching in Multigraphs}}, booktitle = {49th International Colloquium on Automata, Languages, and Programming (ICALP 2022)}, pages = {87:1--87:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-235-8}, ISSN = {1868-8969}, year = {2022}, volume = {229}, editor = {Boja\'{n}czyk, Miko{\l}aj and Merelli, Emanuela and Woodruff, David P.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2022.87}, URN = {urn:nbn:de:0030-drops-164286}, doi = {10.4230/LIPIcs.ICALP.2022.87}, annote = {Keywords: Bin Packing, Vector Bin Packing, Parameterized Complexity, Matching} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

One of the most basic graph problems, All-Pairs Shortest Paths (APSP) is known to be solvable in n^{3-o(1)} time, and it is widely open whether it has an O(n^{3-ε}) time algorithm for ε > 0. To better understand APSP, one often strives to obtain subcubic time algorithms for structured instances of APSP and problems equivalent to it, such as the Min-Plus matrix product.
A natural structured version of Min-Plus product is Monotone Min-Plus product which has been studied in the context of the Batch Range Mode [SODA'20] and Dynamic Range Mode [ICALP'20] problems. This paper improves the known algorithms for Monotone Min-Plus Product and for Batch and Dynamic Range Mode, and establishes a connection between Monotone Min-Plus Product and the Single Source Replacement Paths (SSRP) problem on an n-vertex graph with potentially negative edge weights in {-M, …, M}.
SSRP with positive integer edge weights bounded by M can be solved in Õ(Mn^ω) time, whereas the prior fastest algorithm for graphs with possibly negative weights [FOCS'12] runs in O(M^{0.7519} n^{2.5286}) time, the current best running time for directed APSP with small integer weights. Using Monotone Min-Plus Product, we obtain an improved O(M^{0.8043} n^{2.4957}) time SSRP algorithm, showing that SSRP with constant negative integer weights is likely easier than directed unweighted APSP, a problem that is believed to require n^{2.5-o(1)} time.
Complementing our algorithm for SSRP, we give a reduction from the Bounded-Difference Min-Plus Product problem studied by Bringmann et al. [FOCS'16] to negative weight SSRP. This reduction shows that it might be difficult to obtain an Õ(M n^{ω}) time algorithm for SSRP with negative weight edges, thus separating the problem from SSRP with only positive weight edges.

Yuzhou Gu, Adam Polak, Virginia Vassilevska Williams, and Yinzhan Xu. Faster Monotone Min-Plus Product, Range Mode, and Single Source Replacement Paths. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 75:1-75:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{gu_et_al:LIPIcs.ICALP.2021.75, author = {Gu, Yuzhou and Polak, Adam and Vassilevska Williams, Virginia and Xu, Yinzhan}, title = {{Faster Monotone Min-Plus Product, Range Mode, and Single Source Replacement Paths}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {75:1--75:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.75}, URN = {urn:nbn:de:0030-drops-141440}, doi = {10.4230/LIPIcs.ICALP.2021.75}, annote = {Keywords: APSP, Min-Plus Product, Range Mode, Single-Source Replacement Paths} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

Knapsack and Subset Sum are fundamental NP-hard problems in combinatorial optimization. Recently there has been a growing interest in understanding the best possible pseudopolynomial running times for these problems with respect to various parameters.
In this paper we focus on the maximum item size s and the maximum item value v. We give algorithms that run in time O(n + s³) and O(n + v³) for the Knapsack problem, and in time Õ(n + s^{5/3}) for the Subset Sum problem.
Our algorithms work for the more general problem variants with multiplicities, where each input item comes with a (binary encoded) multiplicity, which succinctly describes how many times the item appears in the instance. In these variants n denotes the (possibly much smaller) number of distinct items.
Our results follow from combining and optimizing several diverse lines of research, notably proximity arguments for integer programming due to Eisenbrand and Weismantel (TALG 2019), fast structured (min,+)-convolution by Kellerer and Pferschy (J. Comb. Optim. 2004), and additive combinatorics methods originating from Galil and Margalit (SICOMP 1991).

Adam Polak, Lars Rohwedder, and Karol Węgrzycki. Knapsack and Subset Sum with Small Items. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 106:1-106:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{polak_et_al:LIPIcs.ICALP.2021.106, author = {Polak, Adam and Rohwedder, Lars and W\k{e}grzycki, Karol}, title = {{Knapsack and Subset Sum with Small Items}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {106:1--106:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.106}, URN = {urn:nbn:de:0030-drops-141752}, doi = {10.4230/LIPIcs.ICALP.2021.106}, annote = {Keywords: Knapsack, Subset Sum, Proximity, Additive Combinatorics, Multiset} }

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APPROX

**Published in:** LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

We study the online graph coloring problem restricted to the intersection graphs of intervals with lengths in [1,σ]. For σ = 1 it is the class of unit interval graphs, and for σ = ∞ the class of all interval graphs. Our focus is on intermediary classes. We present a (1+σ)-competitive algorithm, which beats the state of the art for 1 < σ < 2, and proves that the problem we study can be strictly easier than online coloring of general interval graphs. On the lower bound side, we prove that no algorithm is better than 5/3-competitive for any σ > 1, nor better than 7/4-competitive for any σ > 2, and that no algorithm beats the 5/2 asymptotic competitive ratio for all, arbitrarily large, values of σ. That last result shows that the problem we study can be strictly harder than unit interval coloring. Our main technical contribution is a recursive composition of strategies, which seems essential to prove any lower bound higher than 2.

Joanna Chybowska-Sokół, Grzegorz Gutowski, Konstanty Junosza-Szaniawski, Patryk Mikos, and Adam Polak. Online Coloring of Short Intervals. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 52:1-52:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{chybowskasokol_et_al:LIPIcs.APPROX/RANDOM.2020.52, author = {Chybowska-Sok\'{o}{\l}, Joanna and Gutowski, Grzegorz and Junosza-Szaniawski, Konstanty and Mikos, Patryk and Polak, Adam}, title = {{Online Coloring of Short Intervals}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)}, pages = {52:1--52:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-164-1}, ISSN = {1868-8969}, year = {2020}, volume = {176}, editor = {Byrka, Jaros{\l}aw and Meka, Raghu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.52}, URN = {urn:nbn:de:0030-drops-126550}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2020.52}, annote = {Keywords: Online algorithms, graph coloring, interval graphs} }

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**Published in:** LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

The most studied linear algebraic operation, matrix multiplication, has surprisingly fast O(n^ω) time algorithms for ω < 2.373. On the other hand, the (min,+) matrix product which is at the heart of many fundamental graph problems such as All-Pairs Shortest Paths, has received only minor n^o(1) improvements over its brute-force cubic running time and is widely conjectured to require n^{3-o(1)} time. There is a plethora of matrix products and graph problems whose complexity seems to lie in the middle of these two problems. For instance, the Min-Max matrix product, the Minimum Witness matrix product, All-Pairs Shortest Paths in directed unweighted graphs and determining whether an edge-colored graph contains a monochromatic triangle, can all be solved in Õ(n^{(3+ω)/2}) time. While slight improvements are sometimes possible using rectangular matrix multiplication, if ω=2, the best runtimes for these "intermediate" problems are all Õ(n^2.5).
A similar phenomenon occurs for convolution problems. Here, using the FFT, the usual (+,×)-convolution of two n-length sequences can be solved in O(n log n) time, while the (min,+) convolution is conjectured to require n^{2-o(1)} time, the brute force running time for convolution problems. There are analogous intermediate problems that can be solved in O(n^1.5) time, but seemingly not much faster: Min-Max convolution, Minimum Witness convolution, etc.
Can one improve upon the running times for these intermediate problems, in either the matrix product or the convolution world? Or, alternatively, can one relate these problems to each other and to other key problems in a meaningful way?
This paper makes progress on these questions by providing a network of fine-grained reductions. We show for instance that APSP in directed unweighted graphs and Minimum Witness product can be reduced to both the Min-Max product and a variant of the monochromatic triangle problem, so that a significant improvement over n^{(3+ω)/2} time for any of the latter problems would result in a similar improvement for both of the former problems. We also show that a natural convolution variant of monochromatic triangle is fine-grained equivalent to the famous 3SUM problem. As this variant is solvable in O(n^1.5) time and 3SUM is in O(n^2) time (and is conjectured to require n^{2-o(1)} time), our result gives the first fine-grained equivalence between natural problems of different running times. We also relate 3SUM to monochromatic triangle, and a coin change problem to monochromatic convolution, and thus to 3SUM.

Andrea Lincoln, Adam Polak, and Virginia Vassilevska Williams. Monochromatic Triangles, Intermediate Matrix Products, and Convolutions. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 53:1-53:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{lincoln_et_al:LIPIcs.ITCS.2020.53, author = {Lincoln, Andrea and Polak, Adam and Vassilevska Williams, Virginia}, title = {{Monochromatic Triangles, Intermediate Matrix Products, and Convolutions}}, booktitle = {11th Innovations in Theoretical Computer Science Conference (ITCS 2020)}, pages = {53:1--53:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-134-4}, ISSN = {1868-8969}, year = {2020}, volume = {151}, editor = {Vidick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.53}, URN = {urn:nbn:de:0030-drops-117382}, doi = {10.4230/LIPIcs.ITCS.2020.53}, annote = {Keywords: 3SUM, fine-grained complexity, matrix multiplication, monochromatic triangle} }

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**Published in:** LIPIcs, Volume 89, 12th International Symposium on Parameterized and Exact Computation (IPEC 2017)

We consider the canonical generalization of the well-studied Longest Increasing Subsequence problem to multiple sequences, called k-LCIS: Given k integer sequences X_1,...,X_k of length at most n, the task is to determine the length of the longest common subsequence of X_1,...,X_k that is also strictly increasing. Especially for the case of k=2 (called LCIS for short), several algorithms have been proposed that require quadratic time in the worst case.
Assuming the Strong Exponential Time Hypothesis (SETH), we prove a tight lower bound, specifically, that no algorithm solves LCIS in (strongly) subquadratic time. Interestingly, the proof makes no use of normalization tricks common to hardness proofs for similar problems such as LCS. We further strengthen this lower bound to rule out O((nL)^{1-epsilon}) time algorithms for LCIS, where L denotes the solution size, and to rule out O(n^{k-epsilon}) time algorithms for k-LCIS. We obtain the same conditional lower bounds for the related Longest Common Weakly Increasing Subsequence problem.

Lech Duraj, Marvin Künnemann, and Adam Polak. Tight Conditional Lower Bounds for Longest Common Increasing Subsequence. In 12th International Symposium on Parameterized and Exact Computation (IPEC 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 89, pp. 15:1-15:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{duraj_et_al:LIPIcs.IPEC.2017.15, author = {Duraj, Lech and K\"{u}nnemann, Marvin and Polak, Adam}, title = {{Tight Conditional Lower Bounds for Longest Common Increasing Subsequence}}, booktitle = {12th International Symposium on Parameterized and Exact Computation (IPEC 2017)}, pages = {15:1--15:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-051-4}, ISSN = {1868-8969}, year = {2018}, volume = {89}, editor = {Lokshtanov, Daniel and Nishimura, Naomi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2017.15}, URN = {urn:nbn:de:0030-drops-85706}, doi = {10.4230/LIPIcs.IPEC.2017.15}, annote = {Keywords: fine-grained complexity, combinatorial pattern matching, sequence alignments, parameterized complexity, SETH} }

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