7 research outputs found
Simple dynamic algorithms for Maximal Independent Set, Maximum Flow and Maximum Matching
Peer reviewe
Dynamic Maxflow via Dynamic Interior Point Methods
In this paper we provide an algorithm for maintaining a
-approximate maximum flow in a dynamic, capacitated graph
undergoing edge additions. Over a sequence of -additions to an -node
graph where every edge has capacity our algorithm runs in
time . To obtain this result we
design dynamic data structures for the more general problem of detecting when
the value of the minimum cost circulation in a dynamic graph undergoing edge
additions obtains value at most (exactly) for a given threshold . Over a
sequence -additions to an -node graph where every edge has capacity
and cost we solve this thresholded
minimum cost flow problem in . Both of our algorithms
succeed with high probability against an adaptive adversary. We obtain these
results by dynamizing the recent interior point method used to obtain an almost
linear time algorithm for minimum cost flow (Chen, Kyng, Liu, Peng, Probst
Gutenberg, Sachdeva 2022), and introducing a new dynamic data structure for
maintaining minimum ratio cycles in an undirected graph that succeeds with high
probability against adaptive adversaries.Comment: 30 page
Recent Advances in Fully Dynamic Graph Algorithms
In recent years, significant advances have been made in the design and
analysis of fully dynamic algorithms. However, these theoretical results have
received very little attention from the practical perspective. Few of the
algorithms are implemented and tested on real datasets, and their practical
potential is far from understood. Here, we present a quick reference guide to
recent engineering and theory results in the area of fully dynamic graph
algorithms
Deepening the (Parameterized) Complexity Analysis of Incremental Stable Matching Problems
When computing stable matchings, it is usually assumed that the preferences
of the agents in the matching market are fixed. However, in many realistic
scenarios, preferences change over time. Consequently, an initially stable
matching may become unstable. Then, a natural goal is to find a matching which
is stable with respect to the modified preferences and as close as possible to
the initial one. For Stable Marriage/Roommates, this problem was formally
defined as Incremental Stable Marriage/Roommates by Bredereck et al. [AAAI
'20]. As they showed that Incremental Stable Roommates and Incremental Stable
Marriage with Ties are NP-hard, we focus on the parameterized complexity of
these problems. We answer two open questions of Bredereck et al. [AAAI '20]: We
show that Incremental Stable Roommates is W[1]-hard parameterized by the number
of changes in the preferences, yet admits an intricate XP-algorithm, and we
show that Incremental Stable Marriage with Ties is W[1]-hard parameterized by
the number of ties. Furthermore, we analyze the influence of the degree of
"similarity" between the agents' preference lists, identifying several
polynomial-time solvable and fixed-parameter tractable cases, but also proving
that Incremental Stable Roommates and Incremental Stable Marriage with Ties
parameterized by the number of different preference lists are W[1]-hard.Comment: Accepted to MFCS'2
Dense subgraph maintenance under streaming edge weight updates for real-time story identification
Recent years have witnessed an unprecedented proliferation of social media. People around the globe author, everyday, millions of blog posts, social network status updates, etc. This rich stream of information can be used to identify, on an ongoing basis, emerging stories, and events that capture popular attention. Stories can be identified via groups of tightly coupled real-world entities, namely the people, locations, products, etc, that are involved in the story. The sheer scale and rapid evolution of the data involved necessitate highly efficient techniques for identifying important stories at every point of time. The main challenge in real-time story identification is the maintenance of dense subgraphs (corresponding to groups of tightly coupled entities) under streaming edge weight updates (resulting from a stream of user-generated content). This is the first work to study the efficient maintenance of dense subgraphs under such streaming edge weight updates. For a wide range of definitions of density, we derive theoretical results regarding the magnitude of change that a single edge weight update can cause. Based on these, we propose a novel algorithm, DynDens, which outperforms adaptations of existing techniques to this setting and yields meaningful, intuitive results. Our approach is validated by a thorough experimental evaluation on large-scale real and synthetic datasets