Periodic Reporting for period 2 - BOBR (Decomposition methods for discrete problems)
Reporting period: 2022-10-01 to 2024-03-31
The goal of the project is to advance the understanding of decomposition techniques within the field of algorithmics, with a particular focus on the design of efficient parameterized, approximation, and dynamic algorithms. We address four major directions of research that are interlinked by common concepts and techniques centered around various forms of graph decompositions.
1) Construct a sound mathematical theory of classes of well-structured graphs, centered around notions borrowed from model theory: first-order interpretations, stability, and dependence. Use this theory to design efficient parameterized algorithms for first-order definable problems on well-structured graphs.
2) Develop fundamental techniques for the design of dynamic data structures for parameterized problems, with a particular focus on dynamic maintenance of decompositions of tree-like graphs and of sparse graphs.
3) Address parameterized approximability of selected geometric and planar problems, in hope of introducing new decomposition tools for corresponding graph classes.
4) Explore the complexity of Maximum Independent Set and related problems in various hereditary graph classes, particularly in graphs excluding an induced path or an induced subdivided claw. This algorithmic goal should be treated as an excuse for obtaining a deep structural understanding of such graphs.
1) Construct a sound mathematical theory of classes of well-structured graphs, centered around notions borrowed from model theory: first-order interpretations, stability, and dependence. Use this theory to design efficient parameterized algorithms for first-order definable problems on well-structured graphs.
2) Develop fundamental techniques for the design of dynamic data structures for parameterized problems, with a particular focus on dynamic maintenance of decompositions of tree-like graphs and of sparse graphs.
3) Address parameterized approximability of selected geometric and planar problems, in hope of introducing new decomposition tools for corresponding graph classes.
4) Explore the complexity of Maximum Independent Set and related problems in various hereditary graph classes, particularly in graphs excluding an induced path or an induced subdivided claw. This algorithmic goal should be treated as an excuse for obtaining a deep structural understanding of such graphs.
We have achieved breakthroughs in all the four major directions covered by the project.
1) For the logical theory of classes of well-structured graphs, we proved that the every problem definable in first-order logic can be solved in time O(n^6) on every class of graphs that is monadically stable. This concept is a key notion borrowed from model theory that essentially says that graphs from the considered classes cannot be ordered by means of logical formulas. This achievement involved the introduction of a whole new structural theory of monadically stable classes, including novel decomposition tools: flip-flatness and the Flipper game.
2) We designed a dynamic data structure that maintains, for a given graph of bounded treewidth modified over time by edge insertions and deletions, an approximate tree decomposition with subpolynomial amortized update time. Here, treewidth is a fundamental notion of tree-likeness considered in structural graph theory. This result opens doors for multiple applications, both within the setting of dynamic and of static parameterized algorithms.
3) We gave an embedding algorithm that maps any metric originating from an edge-weighted planar graph to a very simple metric --- of polylogarithmic treewidth --- with only marginal distortion of the metric. This theorem provides a uniform explanation for the existence of quasi-polynomial time approximation schemes for a host of metric problems in planar graphs.
4) We proved that the Maximum Independent Set can be solved in quasi-polynomial time on every graph that excludes a fixed subdivided claw as an induced subgraph. This falls short of solving the main conjecture of the area, postulating that in fact polynomial-time solvability can be achieved.
1) For the logical theory of classes of well-structured graphs, we proved that the every problem definable in first-order logic can be solved in time O(n^6) on every class of graphs that is monadically stable. This concept is a key notion borrowed from model theory that essentially says that graphs from the considered classes cannot be ordered by means of logical formulas. This achievement involved the introduction of a whole new structural theory of monadically stable classes, including novel decomposition tools: flip-flatness and the Flipper game.
2) We designed a dynamic data structure that maintains, for a given graph of bounded treewidth modified over time by edge insertions and deletions, an approximate tree decomposition with subpolynomial amortized update time. Here, treewidth is a fundamental notion of tree-likeness considered in structural graph theory. This result opens doors for multiple applications, both within the setting of dynamic and of static parameterized algorithms.
3) We gave an embedding algorithm that maps any metric originating from an edge-weighted planar graph to a very simple metric --- of polylogarithmic treewidth --- with only marginal distortion of the metric. This theorem provides a uniform explanation for the existence of quasi-polynomial time approximation schemes for a host of metric problems in planar graphs.
4) We proved that the Maximum Independent Set can be solved in quasi-polynomial time on every graph that excludes a fixed subdivided claw as an induced subgraph. This falls short of solving the main conjecture of the area, postulating that in fact polynomial-time solvability can be achieved.
The project has already achieved many of its major goals and is now at the stage where we take a look further, to understand far-reaching extensions and applications of the introduced tools. Of particular significance are the new techniques introduced in the logic-based theory of well-structured graphs, which so far enabled us to obtain a good understanding of monadically stable classes of graphs. We wish to expand this understanding to monadically dependent classes, which we believe are the ultimate limit of tractable classes as far as model-theoretic notions are concerned. Within the topic of dynamic algorithms, we believe that the data structure for maintaining approximate tree decompositions of bounded width is a game-changer in the area, and we plan to explore its applications for both static and dynamic algorithms.