Periodic Reporting for period 4 - TUgbOAT (Towards Unification of Algorithmic Tools)
Periodo di rendicontazione: 2023-03-01 al 2024-08-31
Over last 50 years, extensive algorithmic research gave rise to a plethora of fundamental results. These results equipped us with increasingly better solutions to a number of core problems. However, many of these solutions are incomparable. The main reason for that is the fact that many cutting-edge algorithmic results are very specialized in their applicability. Often, they are limited to particular parameter range or require different assumptions.
A natural question arises: is it possible to get “one to rule them all” algorithm for some core problems such as matchings and maximum flow? In other words, can we unify our algorithms? That is, can we develop an algorithmic framework that enables us to combine a number of existing, only “conditionally” optimal, algorithms into a single all-around optimal solution? Such results would unify the landscape of algorithmic theory but would also greatly enhance the impact of these cutting-edge developments on the real world. After all, algorithms and data structures are the basic building blocks of every computer program. However, currently using cutting-edge algorithms in an optimal way requires extensive expertise and thorough understanding of both the underlying implementation and the characteristics of the input data.
Hence, the need for such unified solutions seems to be critical from both theoretical and practical perspective. However, obtaining such algorithmic unification poses serious theoretical challenges. We believe that some of the recent advances in algorithms provide us with an opportunity to make serious progress towards solving these challenges in the context of several fundamental algorithmic problems. This project should be seen as the start of such a systematic study of unification of algorithmic tools with the aim to remove the need to “under the hood” while still guaranteeing an optimal performance independently of the particular usage case.
During the project we have successfully made progress to towards the stated objectives by delivering several state-of-art algorithms
that have been obtained using unified framework. This includes:
- [1] that introduces a general framework for sampling random walks in graphs in Massive Parallel Computing (MPC) model and then using this new faster algorithms reduces several graphs problems (including PageRank) to random walk sampling.
- [2] includes a series of reductions of more general problems to less general ones. The reduction starts from submodular decomposable flows, goes through parametrized from, and ends with standard min-cost flow problem, thus efficiently solving these problems in the same complexity as min-cost flow.
- in [3] we introduce almost optimal dynamic algorithm for the Frobenious normal form and show how using it solve several dynamic shortest paths problems efficently.
- [4] gives first known subquadratic algorithms for reporting paths in dynamic problems against adaptive adversary.
- [5,6] - the set of these two papers finalize the study of f-factors in general graphs giving the general theory for this generalization
of matchings as well as unification of algorithms solving these problems.
[1] Jakub Lacki, Slobodan Mitrovic, Krzysztof Onak, Piotr Sankowski: Walking randomly, massively, and efficiently. STOC 2020: 364-377
[2] yriakos Axiotis, Adam Karczmarz, Anish Mukherjee, Piotr Sankowski, Adrian Vladu: Decomposable Submodular Function Minimization via Maximum Flow. ICML 2021: 446-456
[3] Adam Karczmarz, Piotr Sankowski: Sensitivity and Dynamic Distance Oracles via Generic Matrices and Frobenius Form. FOCS 2023: 1745-1756
[4] Adam Karczmarz, Anish Mukherjee, Piotr Sankowski: Subquadratic dynamic path reporting in directed graphs against an adaptive adversary. STOC 2022: 1643-1656
[5] Harold N. Gabow, Piotr Sankowski: Algorithms for Weighted Matching Generalizations I: Bipartite Graphs, b-matching, and Unweighted f-factors. SIAM J. Comput. 50(2): 440-486 (2021)
[6] Harold N. Gabow, Piotr Sankowski: Algorithms for Weighted Matching Generalizations II: f-factors and the Special Case of Shortest Paths. SIAM J. Comput. 50(2): 555-601 (2021)
A natural question arises: is it possible to get “one to rule them all” algorithm for some core problems such as matchings and maximum flow? In other words, can we unify our algorithms? That is, can we develop an algorithmic framework that enables us to combine a number of existing, only “conditionally” optimal, algorithms into a single all-around optimal solution? Such results would unify the landscape of algorithmic theory but would also greatly enhance the impact of these cutting-edge developments on the real world. After all, algorithms and data structures are the basic building blocks of every computer program. However, currently using cutting-edge algorithms in an optimal way requires extensive expertise and thorough understanding of both the underlying implementation and the characteristics of the input data.
Hence, the need for such unified solutions seems to be critical from both theoretical and practical perspective. However, obtaining such algorithmic unification poses serious theoretical challenges. We believe that some of the recent advances in algorithms provide us with an opportunity to make serious progress towards solving these challenges in the context of several fundamental algorithmic problems. This project should be seen as the start of such a systematic study of unification of algorithmic tools with the aim to remove the need to “under the hood” while still guaranteeing an optimal performance independently of the particular usage case.
During the project we have successfully made progress to towards the stated objectives by delivering several state-of-art algorithms
that have been obtained using unified framework. This includes:
- [1] that introduces a general framework for sampling random walks in graphs in Massive Parallel Computing (MPC) model and then using this new faster algorithms reduces several graphs problems (including PageRank) to random walk sampling.
- [2] includes a series of reductions of more general problems to less general ones. The reduction starts from submodular decomposable flows, goes through parametrized from, and ends with standard min-cost flow problem, thus efficiently solving these problems in the same complexity as min-cost flow.
- in [3] we introduce almost optimal dynamic algorithm for the Frobenious normal form and show how using it solve several dynamic shortest paths problems efficently.
- [4] gives first known subquadratic algorithms for reporting paths in dynamic problems against adaptive adversary.
- [5,6] - the set of these two papers finalize the study of f-factors in general graphs giving the general theory for this generalization
of matchings as well as unification of algorithms solving these problems.
[1] Jakub Lacki, Slobodan Mitrovic, Krzysztof Onak, Piotr Sankowski: Walking randomly, massively, and efficiently. STOC 2020: 364-377
[2] yriakos Axiotis, Adam Karczmarz, Anish Mukherjee, Piotr Sankowski, Adrian Vladu: Decomposable Submodular Function Minimization via Maximum Flow. ICML 2021: 446-456
[3] Adam Karczmarz, Piotr Sankowski: Sensitivity and Dynamic Distance Oracles via Generic Matrices and Frobenius Form. FOCS 2023: 1745-1756
[4] Adam Karczmarz, Anish Mukherjee, Piotr Sankowski: Subquadratic dynamic path reporting in directed graphs against an adaptive adversary. STOC 2022: 1643-1656
[5] Harold N. Gabow, Piotr Sankowski: Algorithms for Weighted Matching Generalizations I: Bipartite Graphs, b-matching, and Unweighted f-factors. SIAM J. Comput. 50(2): 440-486 (2021)
[6] Harold N. Gabow, Piotr Sankowski: Algorithms for Weighted Matching Generalizations II: f-factors and the Special Case of Shortest Paths. SIAM J. Comput. 50(2): 555-601 (2021)
Task A. This was the most challenging task of the project, which aimed to deliver methodologically new unified algorithms or deliver black-box reductions between algorithmic problems, including:
- unified view on how to use shortest path computations to speed-up min-cost flows in planar graphs,
- general algorithms and reductions for non-bipartite matching-type problems.
- general reductions that reduce decomposable submodular function minimization to parameterized flow problem and then to the classical maximum-flow problems.
- computing all pairs shortest paths and related problems in the case of parallel algorithms.
- improved oracles for the max s,t-flow problem in the case of planar graphs.
Task B. In this task the we were working on two seemingly unrelated algorithmic setting, e.g. dynamic graph algorithms as well as fast algorithms for data science. However, as exemplified by some papers, we report here, these two regimes often are related by algorithmic tools used. As for dynamic algorithms the results of the project include:
- new partially-dynamic algorithms for the all-pairs shortest paths problem in weighted directed graphs.
- novel constant time, and thus optimal, approximate algorithms for incremental matchings in general graphs.
- the first known polylogarithmic dynamic algorithms for directed reachability problems.
- almost optimal algorithms for sensitivity and distance oracles using reduction to dynamic Frobenious form computation.
Furthermore, we have approached the two strongly interconnected problems, i.e. randomized algorithms against adaptive adversary, and deterministic algorithms:
- the first known subquadratic algorithms for shortest path reporting against adaptive adversary.
- the first non-trivial fully dynamic algorithm maintaining exact single-source distances in unweighted graphs.
- deterministic incremental algorithms for the maximum approximate matching in general graphs.
The final goal of this task was to work on efficient algorithms for data science:
- exponential time improvement for the implementation of PageRank algorithm in Massive Parallel Computation model.
- new improved algorithms for computing Shapley and Banzhaf values that are used for explanations of decision trees.
- matching upper and lower bounds for the problem of maintaining graph PageRank dynamically.
Task C. In this task we have been styling additional properties of data which allow to develop faster algorithms:
- improved NC algorithms for constructing perfect matching in general single crossing minor free graphs.
- new parameterized approximation scheme for the geometric knapsack problem with wide items.
- improved algorithm for the natural combinatorial pricing problem for sequentially arriving buyers with equal budgets.
- the first known fixed-parameter tractable algorithm for finding shortest k-disjoint paths on rectangular grids.
Finally, we have been studying stochastic assumptions by:
- showing that natural greedy algorithm is constant-competitive for the stochastic problem of min-cost perfect matchings with delays.
- giving a deterministic online algorithm which achieves a constant ratio of expectations for multi-level aggregation with delays and stochastic arrival times.
- unified view on how to use shortest path computations to speed-up min-cost flows in planar graphs,
- general algorithms and reductions for non-bipartite matching-type problems.
- general reductions that reduce decomposable submodular function minimization to parameterized flow problem and then to the classical maximum-flow problems.
- computing all pairs shortest paths and related problems in the case of parallel algorithms.
- improved oracles for the max s,t-flow problem in the case of planar graphs.
Task B. In this task the we were working on two seemingly unrelated algorithmic setting, e.g. dynamic graph algorithms as well as fast algorithms for data science. However, as exemplified by some papers, we report here, these two regimes often are related by algorithmic tools used. As for dynamic algorithms the results of the project include:
- new partially-dynamic algorithms for the all-pairs shortest paths problem in weighted directed graphs.
- novel constant time, and thus optimal, approximate algorithms for incremental matchings in general graphs.
- the first known polylogarithmic dynamic algorithms for directed reachability problems.
- almost optimal algorithms for sensitivity and distance oracles using reduction to dynamic Frobenious form computation.
Furthermore, we have approached the two strongly interconnected problems, i.e. randomized algorithms against adaptive adversary, and deterministic algorithms:
- the first known subquadratic algorithms for shortest path reporting against adaptive adversary.
- the first non-trivial fully dynamic algorithm maintaining exact single-source distances in unweighted graphs.
- deterministic incremental algorithms for the maximum approximate matching in general graphs.
The final goal of this task was to work on efficient algorithms for data science:
- exponential time improvement for the implementation of PageRank algorithm in Massive Parallel Computation model.
- new improved algorithms for computing Shapley and Banzhaf values that are used for explanations of decision trees.
- matching upper and lower bounds for the problem of maintaining graph PageRank dynamically.
Task C. In this task we have been styling additional properties of data which allow to develop faster algorithms:
- improved NC algorithms for constructing perfect matching in general single crossing minor free graphs.
- new parameterized approximation scheme for the geometric knapsack problem with wide items.
- improved algorithm for the natural combinatorial pricing problem for sequentially arriving buyers with equal budgets.
- the first known fixed-parameter tractable algorithm for finding shortest k-disjoint paths on rectangular grids.
Finally, we have been studying stochastic assumptions by:
- showing that natural greedy algorithm is constant-competitive for the stochastic problem of min-cost perfect matchings with delays.
- giving a deterministic online algorithm which achieves a constant ratio of expectations for multi-level aggregation with delays and stochastic arrival times.
All the results described above go beyond the state of art.