86 research outputs found
Fine-grained parameterized complexity analysis of graph coloring problems
The q-COLORING problem asks whether the vertices of a graph can be properly colored with q colors. In this paper we perform a fine-grained analysis of the complexity of q-COLORING with respect to a hierarchy of structural parameters. We show that unless the Exponential Time Hypothesis fails, there is no constant θ such that q-COLORING parameterized by the size k of a vertex cover can be solved in O ∗(θ k) time for all fixed q. We prove that there are O ∗((q−ɛ) k) time algorithms where k is the vertex deletion distance to several graph classes for which q-COLORING is known to be solvable in polynomial time, including all graph classes F whose (q+1)-colorable members have bounded treedepth. In contrast, we prove that if F is the class of paths – some of the simplest graphs of unbounded treedepth – then no such algorithm can exist unless the Strong Exponential Time Hypothesis fails.</p
Fine-grained parameterized complexity analysis of graph coloring problems
The q-COLORING problem asks whether the vertices of a graph can be properly colored with q colors. In this paper we perform a fine-grained analysis of the complexity of q-COLORING with respect to a hierarchy of structural parameters. We show that unless the Exponential Time Hypothesis fails, there is no constant θ such that q-COLORING parameterized by the size k of a vertex cover can be solved in O ∗(θ k) time for all fixed q. We prove that there are O ∗((q−ɛ) k) time algorithms where k is the vertex deletion distance to several graph classes for which q-COLORING is known to be solvable in polynomial time, including all graph classes F whose (q+1)-colorable members have bounded treedepth. In contrast, we prove that if F is the class of paths – some of the simplest graphs of unbounded treedepth – then no such algorithm can exist unless the Strong Exponential Time Hypothesis fails.</p
Improved Bounds for the Excluded-Minor Approximation of Treedepth
Treedepth, a more restrictive graph width parameter than treewidth and pathwidth, plays a major role in the theory of sparse graph classes. We show that there exists a constant C such that for every integers a,b >= 2 and a graph G, if the treedepth of G is at least Cab log a, then the treewidth of G is at least a or G contains a subcubic (i.e., of maximum degree at most 3) tree of treedepth at least b as a subgraph.
As a direct corollary, we obtain that every graph of treedepth Omega(k^3 log k) is either of treewidth at least k, contains a subdivision of full binary tree of depth k, or contains a path of length 2^k. This improves the bound of Omega(k^5 log^2 k) of Kawarabayashi and Rossman [SODA 2018].
We also show an application for approximation algorithms of treedepth: given a graph G of treedepth k and treewidth t, one can in polynomial time compute a treedepth decomposition of G of width O(kt log^{3/2} t). This improves upon a bound of O(kt^2 log t) stemming from a tradeoff between known results.
The main technical ingredient in our result is a proof that every tree of treedepth d contains a subcubic subtree of treedepth at least d * log_3 ((1+sqrt{5})/2)
Structural sparsity of complex networks: bounded expansion in random models and real-world graphs
This research establishes that many real-world networks exhibit bounded expansion, a strong notion of structural sparsity, and demonstrates that it can be leveraged to design efficient algorithms for network analysis. Specifically, we give a new linear-time fpt algorithm for motif counting and linear time algorithms to compute localized variants of several centrality measures. To establish structural sparsity in real-world networks, we analyze several common network models regarding their structural sparsity. We show that, with high probability, (1) graphs sampled with a prescribed sparse degree sequence; (2) perturbed bounded-degree graphs; (3) stochastic block models with small probabilities; result in graphs of bounded expansion. In contrast, we show that the Kleinberg and the Barabási–Albert model have unbounded expansion. We support our findings with empirical measurements on a corpus of real-world networks
Flexible List Colorings in Graphs with Special Degeneracy Conditions
For a given , we say that a graph is
-flexibly -choosable if the following holds: for any assignment
of color lists of size on , if a preferred color from a list is
requested at any set of vertices, then at least of these
requests are satisfied by some -coloring. We consider the question of
flexible choosability in several graph classes with certain degeneracy
conditions. We characterize the graphs of maximum degree that are
-flexibly -choosable for some , which answers a question of Dvo\v{r}\'ak, Norin, and
Postle [List coloring with requests, JGT 2019]. In particular, we show that for
any , any graph of maximum degree that is not isomorphic
to is -flexibly -choosable. Our
fraction of is within a constant factor of being the best
possible. We also show that graphs of treewidth are -flexibly
-choosable, answering a question of Choi et al.~[arXiv 2020], and we give
conditions for list assignments by which graphs of treewidth are
-flexibly -choosable. We show furthermore that graphs of
treedepth are -flexibly -choosable. Finally, we introduce a
notion of flexible degeneracy, which strengthens flexible choosability, and we
show that apart from a well-understood class of exceptions, 3-connected
non-regular graphs of maximum degree are flexibly -degenerate.Comment: 21 pages, 5 figure
- …