45 research outputs found
Improved Dynamic Graph Coloring
This paper studies the fundamental problem of graph coloring in fully dynamic graphs. Since the problem of computing an optimal coloring, or even approximating it to within n^{1-epsilon} for any epsilon > 0, is NP-hard in static graphs, there is no hope to achieve any meaningful computational results for general graphs in the dynamic setting. It is therefore only natural to consider the combinatorial aspects of dynamic coloring, or alternatively, study restricted families of graphs.
Towards understanding the combinatorial aspects of this problem, one may assume a black-box access to a static algorithm for C-coloring any subgraph of the dynamic graph, and investigate the trade-off between the number of colors and the number of recolorings per update step. Optimizing the number of recolorings, sometimes referred to as the recourse bound, is important for various practical applications. In WADS\u2717, Barba et al. devised two complementary algorithms: For any beta > 0, the first (respectively, second) maintains an O(C beta n^{1/beta}) (resp., O(C beta))-coloring while recoloring O(beta) (resp., O(beta n^{1/beta})) vertices per update. Barba et al. also showed that the second trade-off appears to exhibit the right behavior, at least for beta = O(1): Any algorithm that maintains a c-coloring of an n-vertex dynamic forest must recolor Omega(n^{2/(c(c-1))}) vertices per update, for any constant c >= 2. Our contribution is two-fold:
- We devise a new algorithm for general graphs that improves significantly upon the first trade-off in a wide range of parameters: For any beta > 0, we get a O~(C/(beta)log^2 n)-coloring with O(beta) recolorings per update, where the O~ notation supresses polyloglog(n) factors. In particular, for beta = O(1) we get constant recolorings with polylog(n) colors; not only is this an exponential improvement over the previous bound, but it also unveils a rather surprising phenomenon: The trade-off between the number of colors and recolorings is highly non-symmetric.
- For uniformly sparse graphs, we use low out-degree orientations to strengthen the above result by bounding the update time of the algorithm rather than the number of recolorings. Then, we further improve this result by introducing a new data structure that refines bounded out-degree edge orientations and is of independent interest
A Local-To-Global Theorem for Congested Shortest Paths
P_k such that each P_i is a shortest path from s_i to t_i, and every node in the graph is on at most c paths P_i, or reporting that no such collection of paths exists. When c = k, there are no congestion constraints, and the problem can be solved easily by running a shortest path algorithm for each pair (s_i,t_i) independently. At the other extreme, when c = 1, the (k,c)-SPC problem is equivalent to the k-Disjoint Shortest Paths (k-DSP) problem, where the collection of shortest paths must be node-disjoint. For fixed k, k-DSP is polynomial-time solvable on DAGs and undirected graphs. Amiri and Wargalla interpolated between these two extreme values of c, to obtain an algorithm for (k,c)-SPC on DAGs that runs in polynomial time when k-c is constant.
In the same way, we prove that (k,c)-SPC can be solved in polynomial time on undirected graphs whenever k-c is constant. For directed graphs, because of our counterexample to the original theorem statement, our roundtrip local-to-global result does not imply such an algorithm (k,c)-SPC. Even without an algorithmic application, our proof for directed graphs may be of broader interest because it characterizes intriguing structural properties of shortest paths in directed graphs
Closing the Gap Between Directed Hopsets and Shortcut Sets
For an n-vertex directed graph , a -\emph{shortcut set}
is a set of additional edges such that has
the same transitive closure as , and for every pair , there is a
-path in with at most edges. A natural generalization of
shortcut sets to distances is a -\emph{hopset} , where the requirement is that and have the same
shortest-path distances, and for every , there is a
-approximate shortest path in with at most
edges.
There is a large literature on the tradeoff between the size of a shortcut
set / hopset and the value of . We highlight the most natural point on
this tradeoff: what is the minimum value of , such that for any graph
, there exists a -shortcut set (or a -hopset) with
edges? Not only is this a natural structural question in its own right,
but shortcuts sets / hopsets form the core of many distributed, parallel, and
dynamic algorithms for reachability / shortest paths. Until very recently the
best known upper bound was a folklore construction showing , but in a breakthrough result Kogan and Parter [SODA 2022] improve
this to for shortcut sets and
for hopsets.
Our result is to close the gap between shortcut sets and hopsets. That is, we
show that for any graph and any fixed there is a
hopset with edges. More generally, we
achieve a smooth tradeoff between hopset size and which exactly matches
the tradeoff of Kogan and Parter for shortcut sets (up to polylog factors).
Using a very recent black-box reduction of Kogan and Parter, our new hopset
implies improved bounds for approximate distance preservers.Comment: Abstract shortened to meet arXiv requirements, v2: fixed a typ
A Local-to-Global Theorem for Congested Shortest Paths
Amiri and Wargalla (2020) proved the following local-to-global theorem in
directed acyclic graphs (DAGs): if is a weighted DAG such that for each
subset of 3 nodes there is a shortest path containing every node in ,
then there exists a pair of nodes such that there is a shortest
-path containing every node in .
We extend this theorem to general graphs. For undirected graphs, we prove
that the same theorem holds (up to a difference in the constant 3). For
directed graphs, we provide a counterexample to the theorem (for any constant),
and prove a roundtrip analogue of the theorem which shows there exists a pair
of nodes such that every node in is contained in the union of a
shortest -path and a shortest -path.
The original theorem for DAGs has an application to the -Shortest Paths
with Congestion (()-SPC) problem. In this problem, we are given a
weighted graph , together with node pairs ,
and a positive integer . We are tasked with finding paths such that each is a shortest path from to , and every
node in the graph is on at most paths , or reporting that no such
collection of paths exists.
When the problem is easily solved by finding shortest paths for each
pair independently. When , the -SPC problem recovers
the -Disjoint Shortest Paths (-DSP) problem, where the collection of
shortest paths must be node-disjoint. For fixed , -DSP can be solved in
polynomial time on DAGs and undirected graphs. Previous work shows that the
local-to-global theorem for DAGs implies that -SPC on DAGs whenever
is constant. In the same way, our work implies that -SPC can be
solved in polynomial time on undirected graphs whenever is constant.Comment: Updated to reflect reviewer comment
Lower Bounds for Dynamic Distributed Task Allocation
We study the problem of distributed task allocation in multi-agent systems. Suppose there is a collection of agents, a collection of tasks, and a demand vector, which specifies the number of agents required to perform each task. The goal of the agents is to cooperatively allocate themselves to the tasks to satisfy the demand vector. We study the dynamic version of the problem where the demand vector changes over time. Here, the goal is to minimize the switching cost, which is the number of agents that change tasks in response to a change in the demand vector. The switching cost is an important metric since changing tasks may incur significant overhead.
We study a mathematical formalization of the above problem introduced by Su, Su, Dornhaus, and Lynch [Su et al., 2017], which can be reformulated as a question of finding a low distortion embedding from symmetric difference to Hamming distance. In this model it is trivial to prove that the switching cost is at least 2. We present the first non-trivial lower bounds for the switching cost, by giving lower bounds of 3 and 4 for different ranges of the parameters
Are there graphs whose shortest path structure requires large edge weights?
The aspect ratio of a weighted graph is the ratio of its maximum edge
weight to its minimum edge weight. Aspect ratio commonly arises as a complexity
measure in graph algorithms, especially related to the computation of shortest
paths. Popular paradigms are to interpolate between the settings of weighted
and unweighted input graphs by incurring a dependence on aspect ratio, or by
simply restricting attention to input graphs of low aspect ratio.
This paper studies the effects of these paradigms, investigating whether
graphs of low aspect ratio have more structured shortest paths than graphs in
general. In particular, we raise the question of whether one can generally take
a graph of large aspect ratio and \emph{reweight} its edges, to obtain a graph
with bounded aspect ratio while preserving the structure of its shortest paths.
Our findings are:
- Every weighted DAG on nodes has a shortest-paths preserving graph of
aspect ratio . A simple lower bound shows that this is tight.
- The previous result does not extend to general directed or undirected
graphs; in fact, the answer turns out to be exponential in these settings. In
particular, we construct directed and undirected -node graphs for which any
shortest-paths preserving graph has aspect ratio .
We also consider the approximate version of this problem, where the goal is
for shortest paths in to correspond to approximate shortest paths in .
We show that our exponential lower bounds extend even to this setting. We also
show that in a closely related model, where approximate shortest paths in
must also correspond to approximate shortest paths in , even DAGs require
exponential aspect ratio
Algorithms for the Minimum Dominating Set Problem in Bounded Arboricity Graphs: Simpler, Faster, and Combinatorial
We revisit the minimum dominating set problem on graphs with arboricity
bounded by . In the (standard) centralized setting, Bansal and Umboh
[BU17] gave an -approximation LP rounding algorithm. Moreover,
[BU17] showed that it is NP-hard to achieve an asymptotic improvement. On the
other hand, the previous two non-LP-based algorithms, by Lenzen and Wattenhofer
[LW10], and Jones et al. [JLR+13], achieve an approximation factor of
in linear time.
There is a similar situation in the distributed setting: While there are
-round LP-based -approximation algorithms [KMW06,
DKM19], the best non-LP-based algorithm by Lenzen and Wattenhofer [LW10] is an
implementation of their centralized algorithm, providing an
-approximation within rounds with high probability.
We address the question of whether one can achieve a simple, elementary
-approximation algorithm not based on any LP-based methods, either
in the centralized setting or in the distributed setting. We resolve these
questions in the affirmative. More specifically, our contribution is two-fold:
1. In the centralized setting, we provide a surprisingly simple combinatorial
algorithm that is asymptotically optimal in terms of both approximation factor
and running time: an -approximation in linear time.
2. Based on our centralized algorithm, we design a distributed combinatorial
-approximation algorithm in the model that runs
in rounds with high probability. Our round complexity
outperforms the best LP-based distributed algorithm for a wide range of
parameters