10 research outputs found
Multi-Clique-Width
Multi-clique-width is obtained by a simple modification in the definition of clique-width. It has the advantage of providing a natural extension of tree-width. Unlike clique-width, it does not explode exponentially compared to tree-width. Efficient algorithms based on multi-clique-width are still possible for interesting tasks like computing the independent set polynomial or testing c-colorability. In particular, c-colorability can be tested in time linear in n and singly exponential in c and the width k of a given multi-k-expression. For these tasks, the running time as a function of the multi-clique-width is the same as the running time of the fastest known algorithm as a function of the clique-width. This results in an exponential speed-up for some graphs, if the corresponding graph generating expressions are given. The reason is that the multi-clique-width is never bigger, but is exponentially smaller than the clique-width for many graphs. This gap shows up when the tree-width is basically equal to the multi-clique width as well as when the tree-width is not bounded by any function of the clique-width
Space-Efficient Parameterized Algorithms on Graphs of Low Shrubdepth
Dynamic programming on various graph decompositions is one of the most
fundamental techniques used in parameterized complexity. Unfortunately, even if
we consider concepts as simple as path or tree decompositions, such dynamic
programming uses space that is exponential in the decomposition's width, and
there are good reasons to believe that this is necessary. However, it has been
shown that in graphs of low treedepth it is possible to design algorithms which
achieve polynomial space complexity without requiring worse time complexity
than their counterparts working on tree decompositions of bounded width. Here,
treedepth is a graph parameter that, intuitively speaking, takes into account
both the depth and the width of a tree decomposition of the graph, rather than
the width alone.
Motivated by the above, we consider graphs that admit clique expressions with
bounded depth and label count, or equivalently, graphs of low shrubdepth (sd).
Here, sd is a bounded-depth analogue of cliquewidth, in the same way as td is a
bounded-depth analogue of treewidth. We show that also in this setting,
bounding the depth of the decomposition is a deciding factor for improving the
space complexity. Precisely, we prove that on -vertex graphs equipped with a
tree-model (a decomposition notion underlying sd) of depth and using
labels, we can solve
- Independent Set in time using
space;
- Max Cut in time using space; and
- Dominating Set in time using space via
a randomized algorithm.
We also establish a lower bound, conditional on a certain assumption about
the complexity of Longest Common Subsequence, which shows that at least in the
case of IS the exponent of the parametric factor in the time complexity has to
grow with if one wishes to keep the space complexity polynomial.Comment: Conference version to appear at the European Symposium on Algorithms
(ESA 2023
Space-Efficient Parameterized Algorithms on Graphs of Low Shrubdepth
Dynamic programming on various graph decompositions is one of the most fundamental techniques used in parameterized complexity. Unfortunately, even if we consider concepts as simple as path or tree decompositions, such dynamic programming uses space that is exponential in the decomposition’s width, and there are good reasons to believe that this is necessary. However, it has been shown that in graphs of low treedepth it is possible to design algorithms which achieve polynomial space complexity without requiring worse time complexity than their counterparts working on tree decompositions of bounded width. Here, treedepth is a graph parameter that, intuitively speaking, takes into account both the depth and the width of a tree decomposition of the graph, rather than the width alone. Motivated by the above, we consider graphs that admit clique expressions with bounded depth and label count, or equivalently, graphs of low shrubdepth. Here, shrubdepth is a bounded-depth analogue of cliquewidth, in the same way as treedepth is a bounded-depth analogue of treewidth. We show that also in this setting, bounding the depth of the decomposition is a deciding factor for improving the space complexity. More precisely, we prove that on n-vertex graphs equipped with a tree-model (a decomposition notion underlying shrubdepth) of depth d and using k labels, - Independent Set can be solved in time 2^(dk) ⋅ n^(1) using (dk²log n) space; - Max Cut can be solved in time n^(dk) using (dk log n) space; and - Dominating Set can be solved in time 2^(dk) ⋅ n^(1) using n^(1) space via a randomized algorithm. We also establish a lower bound, conditional on a certain assumption about the complexity of Longest Common Subsequence, which shows that at least in the case of Independent Set the exponent of the parametric factor in the time complexity has to grow with d if one wishes to keep the space complexity polynomial
Space-Efficient Parameterized Algorithms on Graphs of Low Shrubdepth
Dynamic programming on various graph decompositions is one of the most fundamental techniques used in parameterized complexity. Unfortunately, even if we consider concepts as simple as path or tree decompositions, such dynamic programming uses space that is exponential in the decomposition's width, and there are good reasons to believe that this is necessary. However, it has been shown that in graphs of low treedepth it is possible to design algorithms which achieve polynomial space complexity without requiring worse time complexity than their counterparts working on tree decompositions of bounded width. Here, treedepth is a graph parameter that, intuitively speaking, takes into account both the depth and the width of a tree decomposition of the graph, rather than the width alone. Motivated by the above, we consider graphs that admit clique expressions with bounded depth and label count, or equivalently, graphs of low shrubdepth (sd). Here, sd is a bounded-depth analogue of cliquewidth, in the same way as td is a bounded-depth analogue of treewidth. We show that also in this setting, bounding the depth of the decomposition is a deciding factor for improving the space complexity. Precisely, we prove that on n-vertex graphs equipped with a tree-model (a decomposition notion underlying sd) of depth d and using k labels, we can solve - Independent Set in time 2O(dk)⋅nO(1) using O(dk2logn) space; - Max Cut in time nO(dk) using O(dklogn) space; and - Dominating Set in time 2O(dk)⋅nO(1) using nO(1) space via a randomized algorithm. We also establish a lower bound, conditional on a certain assumption about the complexity of Longest Common Subsequence, which shows that at least in the case of IS the exponent of the parametric factor in the time complexity has to grow with d if one wishes to keep the space complexity polynomial
LIPIcs, Volume 274, ESA 2023, Complete Volume
LIPIcs, Volume 274, ESA 2023, Complete Volum
Multi-Clique-Width, a Powerful New Width Parameter
Multi-clique-width is obtained by a simple modication in the denition of clique-width. It has
the advantage of providing a natural extension of tree-width. Unlike clique-width, it does not explode
exponentially compared to tree-width. Ecient algorithms based on multi-clique-width are still possible
for interesting tasks like computing the independent set polynomial or testing c-colorability. In particular,
c-colorability can be tested in time linear in n and singly exponential in c and the width k of a given
multi-k-expression. For these tasks, the running time as a function of the multi-clique-width is the same
as the running time of the fastest known algorithm as a function of the clique-width. This results in an
exponential speed-up for some graphs, if the corresponding graph generating expressions are given. The
reason is that the multi-clique-width is never bigger, but is exponentially smaller than the clique-width for
many graphs.Non UBCUnreviewedAuthor affiliation: The Pennsylvania State UniversityFacult