On Space Efficiency of Algorithms Working on Structural Decompositions of Graphs

Pilipczuk, Michał; Wrochna, Marcin

doi:10.1145/3154856

Cited by 29 publications

(51 citation statements)

References 61 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Typically dynamic programming algorithms on tree decompositions use space exponential in the width of the decomposition and there are complexity-theoretical reasons to believe that without significant increase in time complexity, this cannot be avoided. On the other hand treedepth decompositions, sometimes called elimination trees, allow to devise algorithms using only polynomial space in the height of the decomposition, see a thorough study of this phenomenon by Pilipczuk and Wrochna [22].…”

Section: Algorithmic Applicationsmentioning

confidence: 99%

Improved Bounds for Centered Colorings

Dębski

Felsner

Micek

et al. 2021

AiC

View full text Add to dashboard Cite

A vertex coloring $\phi$ of a graph $G$ is $p$-centered if for every connected subgraph $H$ of $G$ either $\phi$ uses more than $p$ colors on $H$ or there is a color that appears exactly once on $H$. Centered colorings form one of the families of parameters that allow to capture notions of sparsity of graphs: A class of graphs has bounded expansion if and only if there is a function $f$ such that for every $p\geq1$, every graph in the class admits a $p$-centered coloring using at most $f(p)$ colors. In this paper, we give upper bounds for the maximum number of colors needed in a $p$-centered coloring of graphs from several widely studied graph classes. We show that: (1) planar graphs admit $p$-centered colorings with $O(p^3\log p)$ colors where the previous bound was $O(p^{19})$; (2) bounded degree graphs admit $p$-centered colorings with $O(p)$ colors while it was conjectured that they may require exponential number of colors. All these upper bounds imply polynomial algorithms for computing the colorings. Prior to this work there were no non-trivial lower bounds known. We show that: (4) there are graphs of treewidth $t$ that require $\binom{p+t}{t}$ colors in any $p$-centered coloring. This bound matches the upper bound; (5) there are planar graphs that require $\Omega(p^2\log p)$ colors in any $p$-centered coloring. We also give asymptotically tight bounds for outerplanar graphs and planar graphs of treewidth $3$. We prove our results with various proof techniques. The upper bound for planar graphs involves an application of a recent structure theorem while the upper bound for bounded degree graphs comes from the entropy compression method. We lift the result for bounded degree graphs to graphs avoiding a fixed topological minor using the Grohe-Marx structure theorem.

show abstract

Section: Algorithmic Applicationsmentioning

confidence: 99%

Improved Bounds for Centered Colorings

Dębski

Felsner

Micek

et al. 2021

AiC

View full text Add to dashboard Cite

show abstract

“…A problem is in XNLP, if it can be solved with a non-deterministic algorithm that uses f (k) log n space and f (k)n c time, where f is a computable function, n the input size, k the parameter and c a constant. XNLP-completeness has a number of consequences: it implies hardness for all classes W [t] for all positive integers t, and a conjecture from Pilipczuk and Wrochna [17] implies that it is unlikely that the problem has an XP algorithm that uses f (k)n c space (f , k, n and c as above). (See the discussion in Section 8.…”

Section: Introductionmentioning

confidence: 99%

XNLP-completeness for Parameterized Problems on Graphs with a Linear Structure

Bodlaender¹,

Groenland²,

Jacob³

2022

Preprint

View full text Add to dashboard Cite

In this paper, we show several parameterized problems to be complete for the class XNLP: parameterized problems that can be solved with a non-deterministic algorithm that uses f (k) log n space and f (k)n c time, with f a computable function, n the input size, k the parameter and c a constant. The problems include Maximum Regular Induced Subgraph and Max Cut parameterized by linear clique-width, Capacitated (Red-Blue) Dominating Set parameterized by pathwidth, Odd Cycle Transversal parameterized by a new parameter we call logarithmic linear clique-width (defined as k/ log n for an n-vertex graph of linear clique-width k), and Bipartite Bandwidth.

show abstract

“…Typically dynamic programming algorithms on tree decompositions use space exponential in the width of the decomposition and there are complexity-theoretical reasons to believe that without significant loss on time complexity, this cannot be avoided. On the other hand treedepth decompositions, sometimes called elimination trees, allow to device algorithms using only polynomial space in the height of the decomposition, see a thorough study of this phenomenon by Pilipczuk and Wrochna in [15].…”

Section: Introductionmentioning

confidence: 99%

Improved bounds for centered colorings

Dębski

Felsner

Micek

et al. 2020

Proceedings of the Fourteenth Annual ACM-SIAM Symposium on Discrete Algorithms

View full text Add to dashboard Cite

A vertex coloring φ of a graph G is p-centered if for every connected subgraph H of G either φ uses more than p colors on H or there is a color that appears exactly once on H. Centered colorings form one of the families of parameters that allow to capture notions of sparsity of graphs: A class of graphs has bounded expansion if and only if there is a function f such that for every p 1, every graph in the class admits a p-centered coloring using at most f (p) colors.In this paper, we give upper bounds for the maximum number of colors needed in a p-centered coloring of graphs from several widely studied graph classes. We show that:(1) planar graphs admit p-centered colorings with O(p 3 log p) colors where the previous bound was O(p 19 ); (2) bounded degree graphs admit p-centered colorings with O(p) colors while it was conjectured that they may require exponential number of colors in p;(3) graphs avoiding a fixed graph as a topological minor admit p-centered colorings with a polynomial in p number of colors. All these upper bounds imply polynomial algorithms for computing the colorings. Prior to this work there were no non-trivial lower bounds known. We show that: (4) there are graphs of treewidth t that require p+t t colors in any p-centered coloring and this bound matches the upper bound; (5) there are planar graphs that require Ω(p 2 log p) colors in any p-centered coloring. We also give asymptotically tight bounds for outerplanar graphs and planar graphs of treewidth 3. We prove our results with various proof techniques. The upper bound for planar graphs involves an application of a recent structure theorem while the upper bound for bounded degree graphs comes from the entropy compression method. We lift the result for bounded degree graphs to graphs avoiding a fixed topological minor using the Grohe-Marx structure theorem.

show abstract

On Space Efficiency of Algorithms Working on Structural Decompositions of Graphs

Cited by 29 publications

References 61 publications

Improved Bounds for Centered Colorings

Improved Bounds for Centered Colorings

XNLP-completeness for Parameterized Problems on Graphs with a Linear Structure

Improved bounds for centered colorings

Contact Info

Product

Resources

About