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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Cumulative memory - the sum of space used per step over the duration of a computation - is a fine-grained measure of time-space complexity that was introduced to analyze cryptographic applications like password hashing. It is a more accurate cost measure for algorithms that have infrequent spikes in memory usage and are run in environments such as cloud computing that allow dynamic allocation and de-allocation of resources during execution, or when many multiple instances of an algorithm are interleaved in parallel.
We prove the first lower bounds on cumulative memory complexity for both sequential classical computation and quantum circuits. Moreover, we develop general paradigms for bounding cumulative memory complexity inspired by the standard paradigms for proving time-space tradeoff lower bounds that can only lower bound the maximum space used during an execution. The resulting lower bounds on cumulative memory that we obtain are just as strong as the best time-space tradeoff lower bounds, which are very often known to be tight.
Although previous results for pebbling and random oracle models have yielded time-space tradeoff lower bounds larger than the cumulative memory complexity, our results show that in general computational models such separations cannot follow from known lower bound techniques and are not true for many functions.
Among many possible applications of our general methods, we show that any classical sorting algorithm with success probability at least 1/poly(n) requires cumulative memory ̃ Ω(n²), any classical matrix multiplication algorithm requires cumulative memory Ω(n⁶/T), any quantum sorting circuit requires cumulative memory Ω(n³/T), and any quantum circuit that finds k disjoint collisions in a random function requires cumulative memory Ω(k³n/T²).

Paul Beame and Niels Kornerup. Cumulative Memory Lower Bounds for Randomized and Quantum Computation. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 17:1-17:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{beame_et_al:LIPIcs.ICALP.2023.17, author = {Beame, Paul and Kornerup, Niels}, title = {{Cumulative Memory Lower Bounds for Randomized and Quantum Computation}}, booktitle = {50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)}, pages = {17:1--17:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-278-5}, ISSN = {1868-8969}, year = {2023}, volume = {261}, editor = {Etessami, Kousha and Feige, Uriel and Puppis, Gabriele}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2023.17}, URN = {urn:nbn:de:0030-drops-180694}, doi = {10.4230/LIPIcs.ICALP.2023.17}, annote = {Keywords: Cumulative memory complexity, time-space tradeoffs, branching programs, quantum lower bounds} }

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**Published in:** LIPIcs, Volume 251, 14th Innovations in Theoretical Computer Science Conference (ITCS 2023)

Query-to-communication lifting theorems, which connect the query complexity of a Boolean function to the communication complexity of an associated "lifted" function obtained by composing the function with many copies of another function known as a gadget, have been instrumental in resolving many open questions in computational complexity. A number of important complexity questions could be resolved if we could make substantial improvements in the input size required for lifting with the Index function, which is a universal gadget for lifting, from its current near-linear size down to polylogarithmic in the number of inputs N of the original function or, ideally, constant. The near-linear size bound was recently shown by Lovett, Meka, Mertz, Pitassi and Zhang [Shachar Lovett et al., 2022] using a recent breakthrough improvement on the Sunflower Lemma to show that a certain graph associated with an Index function of that size is a disperser. They also stated a conjecture about the Index function that is essential for further improvements in the size required for lifting with Index using current techniques. In this paper we prove the following;
- The conjecture of Lovett et al. is false when the size of the Index gadget is less than logarithmic in N.
- The same limitation applies to the Inner-Product function. More precisely, the Inner-Product function, which is known to satisfy the disperser property at size O(log N), also does not have this property when its size is less than log N.
- Notwithstanding the above, we prove a lifting theorem that applies to Index gadgets of any size at least 4 and yields lower bounds for a restricted class of communication protocols in which one of the players is limited to sending parities of its inputs.
- Using a modification of the same idea with improved lifting parameters we derive a strong lifting theorem from decision tree size to parity decision tree size. We use this, in turn, to derive a general lifting theorem in proof complexity from tree-resolution size to tree-like Res(⊕) refutation size, which yields many new exponential lower bounds on such proofs.

Paul Beame and Sajin Koroth. On Disperser/Lifting Properties of the Index and Inner-Product Functions. In 14th Innovations in Theoretical Computer Science Conference (ITCS 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 251, pp. 14:1-14:17, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)

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@InProceedings{beame_et_al:LIPIcs.ITCS.2023.14, author = {Beame, Paul and Koroth, Sajin}, title = {{On Disperser/Lifting Properties of the Index and Inner-Product Functions}}, booktitle = {14th Innovations in Theoretical Computer Science Conference (ITCS 2023)}, pages = {14:1--14:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-263-1}, ISSN = {1868-8969}, year = {2023}, volume = {251}, editor = {Tauman Kalai, Yael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2023.14}, URN = {urn:nbn:de:0030-drops-175172}, doi = {10.4230/LIPIcs.ITCS.2023.14}, annote = {Keywords: Decision trees, communication complexity, lifting theorems, proof complexity} }

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**Published in:** LIPIcs, Volume 94, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)

We introduce and develop a new semi-algebraic proof system, called Stabbing Planes that is in the style of DPLL-based modern SAT solvers. As with DPLL, there is only one rule: the current polytope can be subdivided by branching on an inequality and its "integer negation." That is, we can (nondeterministically choose) a hyperplane a x >= b with integer coefficients, which partitions the polytope into three pieces: the points in the polytope satisfying a x >= b, the points satisfying a x <= b-1, and the middle slab b-1 < a x < b. Since the middle slab contains no integer points it can be safely discarded, and the algorithm proceeds recursively on the other two branches. Each path terminates when the current polytope is empty, which is polynomial-time checkable. Among our results, we show somewhat surprisingly that Stabbing Planes can efficiently simulate Cutting Planes, and moreover, is strictly stronger than Cutting Planes under a reasonable conjecture. We prove linear lower bounds on the rank of Stabbing Planes refutations, by adapting
a lifting argument in communication complexity.

Paul Beame, Noah Fleming, Russell Impagliazzo, Antonina Kolokolova, Denis Pankratov, Toniann Pitassi, and Robert Robere. Stabbing Planes. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 10:1-10:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{beame_et_al:LIPIcs.ITCS.2018.10, author = {Beame, Paul and Fleming, Noah and Impagliazzo, Russell and Kolokolova, Antonina and Pankratov, Denis and Pitassi, Toniann and Robere, Robert}, title = {{Stabbing Planes}}, booktitle = {9th Innovations in Theoretical Computer Science Conference (ITCS 2018)}, pages = {10:1--10:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-060-6}, ISSN = {1868-8969}, year = {2018}, volume = {94}, editor = {Karlin, Anna R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.10}, URN = {urn:nbn:de:0030-drops-83418}, doi = {10.4230/LIPIcs.ITCS.2018.10}, annote = {Keywords: Complexity Theory, Proof Complexity, Communication Complexity, Cutting Planes, Semi-Algebraic Proof Systems, Pseudo Boolean Solvers, SAT solvers, Inte} }

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**Published in:** LIPIcs, Volume 94, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)

We study the problem of estimating the number of edges in a graph with access to only an independent set oracle. Independent set queries draw motivation from group testing and have applications to the complexity of decision versus counting problems. We give two algorithms to estimate the number of edges in an n-vertex graph: one that uses only polylog(n) bipartite independent set queries, and another one that uses n^{2/3} polylog(n) independent set queries.

Paul Beame, Sariel Har-Peled, Sivaramakrishnan Natarajan Ramamoorthy, Cyrus Rashtchian, and Makrand Sinha. Edge Estimation with Independent Set Oracles. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 38:1-38:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{beame_et_al:LIPIcs.ITCS.2018.38, author = {Beame, Paul and Har-Peled, Sariel and Natarajan Ramamoorthy, Sivaramakrishnan and Rashtchian, Cyrus and Sinha, Makrand}, title = {{Edge Estimation with Independent Set Oracles}}, booktitle = {9th Innovations in Theoretical Computer Science Conference (ITCS 2018)}, pages = {38:1--38:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-060-6}, ISSN = {1868-8969}, year = {2018}, volume = {94}, editor = {Karlin, Anna R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.38}, URN = {urn:nbn:de:0030-drops-83552}, doi = {10.4230/LIPIcs.ITCS.2018.38}, annote = {Keywords: Approximate Counting, Independent Set Queries, Sparsification, Importance Sampling} }

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**Published in:** LIPIcs, Volume 48, 19th International Conference on Database Theory (ICDT 2016)

In this paper, we study the communication complexity for the problem of computing a conjunctive query on a large database in a parallel setting with p servers. In contrast to previous work, where upper and lower bounds on the communication were specified for particular structures of data (either data without skew, or data with specific types of skew), in this work we focus on worst-case analysis of the communication cost. The goal is to find worst-case optimal parallel algorithms, similar to the work of (Ngo et al. 2012) for sequential algorithms.
We first show that for a single round we can obtain an optimal worst-case algorithm. The optimal load for a conjunctive query q when all relations have size equal to M is O(M/p^{1/psi^*}), where psi^* is a new query-related quantity called the edge quasi-packing number, which is different from both the edge packing number and edge cover number of the query hypergraph. For multiple rounds, we present algorithms that are optimal for several classes of queries. Finally, we show a surprising connection to the external memory model, which allows us to translate parallel algorithms to external memory algorithms. This technique allows us to recover (within a polylogarithmic factor) several recent results on the I/O complexity for computing join queries, and also obtain optimal algorithms for other classes of queries.

Paraschos Koutris, Paul Beame, and Dan Suciu. Worst-Case Optimal Algorithms for Parallel Query Processing. In 19th International Conference on Database Theory (ICDT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 48, pp. 8:1-8:18, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{koutris_et_al:LIPIcs.ICDT.2016.8, author = {Koutris, Paraschos and Beame, Paul and Suciu, Dan}, title = {{Worst-Case Optimal Algorithms for Parallel Query Processing}}, booktitle = {19th International Conference on Database Theory (ICDT 2016)}, pages = {8:1--8:18}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-002-6}, ISSN = {1868-8969}, year = {2016}, volume = {48}, editor = {Martens, Wim and Zeume, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2016.8}, URN = {urn:nbn:de:0030-drops-57771}, doi = {10.4230/LIPIcs.ICDT.2016.8}, annote = {Keywords: conjunctive query, parallel computation, worst-case bounds} }

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