2,448 research outputs found
List Colouring Trees in Logarithmic Space
We show that List Colouring can be solved on n-vertex trees by a deterministic Turing machine using O(log n) bits on the worktape. Given an n-vertex graph G = (V,E) and a list L(v) ⊆ {1, . . . , n} of available colours for each v ∈ V , a list colouring for G is a proper colouring c such that c(v) ∈ L(v) for all v
On the Parameterized Complexity of Computing Tree-Partitions
We study the parameterized complexity of computing the tree-partition-width,
a graph parameter equivalent to treewidth on graphs of bounded maximum degree.
On one hand, we can obtain approximations of the tree-partition-width
efficiently: we show that there is an algorithm that, given an -vertex graph
and an integer , constructs a tree-partition of width for
or reports that has tree-partition width more than , in time
. We can improve on the approximation factor or the dependence on
by sacrificing the dependence on .
On the other hand, we show the problem of computing tree-partition-width
exactly is XALP-complete, which implies that it is -hard for all . We
deduce XALP-completeness of the problem of computing the domino treewidth.
Finally, we adapt some known results on the parameter tree-partition-width and
the topological minor relation, and use them to compare tree-partition-width to
tree-cut width
On the parameterized complexity of computing tree-partitions
We study the parameterized complexity of computing the tree-partition-width,
a graph parameter equivalent to treewidth on graphs of bounded maximum degree.
On one hand, we can obtain approximations of the tree-partition-width
efficiently: we show that there is an algorithm that, given an -vertex graph
and an integer , constructs a tree-partition of width for
or reports that has tree-partition width more than , in time
. We can improve on the approximation factor or the dependence on
by sacrificing the dependence on .
On the other hand, we show the problem of computing tree-partition-width
exactly is XALP-complete, which implies that it is -hard for all . We
deduce XALP-completeness of the problem of computing the domino treewidth.
Finally, we adapt some known results on the parameter tree-partition-width and
the topological minor relation, and use them to compare tree-partition-width to
tree-cut width
Close Relatives (Of Feedback Vertex Set), Revisited
At IPEC 2020, Bergougnoux, Bonnet, Brettell, and Kwon (Close Relatives of Feedback Vertex Set Without Single-Exponential Algorithms Parameterized by Treewidth, IPEC 2020, LIPIcs vol. 180, pp. 3:1-3:17) showed that a number of problems related to the classic Feedback Vertex Set (FVS) problem do not admit a 2^{o(k log k)} ? n^{?(1)}-time algorithm on graphs of treewidth at most k, assuming the Exponential Time Hypothesis. This contrasts with the 3^{k} ? k^{?(1)} ? n-time algorithm for FVS using the Cut&Count technique.
During their live talk at IPEC 2020, Bergougnoux et al. posed a number of open questions, which we answer in this work.
- Subset Even Cycle Transversal, Subset Odd Cycle Transversal, Subset Feedback Vertex Set can be solved in time 2^{?(k log k)} ? n in graphs of treewidth at most k. This matches a lower bound for Even Cycle Transversal of Bergougnoux et al. and improves the polynomial factor in some of their upper bounds.
- Subset Feedback Vertex Set and Node Multiway Cut can be solved in time 2^{?(k log k)} ? n, if the input graph is given as a cliquewidth expression of size n and width k.
- Odd Cycle Transversal can be solved in time 4^k ? k^{?(1)} ? n if the input graph is given as a cliquewidth expression of size n and width k. Furthermore, the existence of a constant ? > 0 and an algorithm performing this task in time (4-?)^k ? n^{?(1)} would contradict the Strong Exponential Time Hypothesis. A common theme of the first two algorithmic results is to represent connectivity properties of the current graph in a state of a dynamic programming algorithm as an auxiliary forest with ?(k) nodes. This results in a 2^{?(k log k)} bound on the number of states for one node of the tree decomposition or cliquewidth expression and allows to compare two states in k^{?(1)} time, resulting in linear time dependency on the size of the graph or the input cliquewidth expression
Exact antichain saturation numbers via a generalisation of a result of Lehman-Ron
For given positive integers and , a family of subsets of
is -antichain saturated if it does not contain an antichain
of size , but adding any set to creates an antichain of size
. We use sat to denote the smallest size of such a family. For all
and sufficiently large , we determine the exact value of sat.
Our result implies that sat, which confirms
several conjectures on antichain saturation. Previously, exact values for
sat were only known for up to .
We also prove a generalisation of a result of Lehman-Ron which may be of
independent interest. We show that given disjoint chains in the Boolean
lattice, we can create disjoint skipless chains that cover the same
elements (where we call a chain skipless if any two consecutive elements differ
in size by exactly one).Comment: 29 pages, 3 figure
A spot-size transformer for fiber-chip coupling in sensor applications at 633 nm in silicon oxynitride
A mode-size adapter was designed, fabricated in SiON/SiO2 and tested. It consists of a laterally tapered SiON waveguide having a step-wise decrease in thickness towards the taper point which may have up to 0.5 ¿m residual widt
On the Complexity of Problems on Tree-structured Graphs
In this paper, we introduce a new class of parameterized problems, which we
call XALP: the class of all parameterized problems that can be solved in
time and space on a non-deterministic Turing
Machine with access to an auxiliary stack (with only top element lookup
allowed). Various natural problems on `tree-structured graphs' are complete for
this class: we show that List Coloring and All-or-Nothing Flow parameterized by
treewidth are XALP-complete. Moreover, Independent Set and Dominating Set
parameterized by treewidth divided by , and Max Cut parameterized by
cliquewidth are also XALP-complete. Besides finding a `natural home' for these
problems, we also pave the road for future reductions. We give a number of
equivalent characterisations of the class XALP, e.g., XALP is the class of
problems solvable by an Alternating Turing Machine whose runs have tree size at
most and use space. Moreover, we introduce
`tree-shaped' variants of Weighted CNF-Satisfiability and Multicolor Clique
that are XALP-complete
On the Complexity of Problems on Tree-Structured Graphs
In this paper, we introduce a new class of parameterized problems, which we call XALP: the class of all parameterized problems that can be solved in f(k)n^O(1) time and f(k)log n space on a non-deterministic Turing Machine with access to an auxiliary stack (with only top element lookup allowed). Various natural problems on "tree-structured graphs" are complete for this class: we show that List Coloring and All-or-Nothing Flow parameterized by treewidth are XALP-complete. Moreover, Independent Set and Dominating Set parameterized by treewidth divided by log n, and Max Cut parameterized by cliquewidth are also XALP-complete.
Besides finding a "natural home" for these problems, we also pave the road for future reductions. We give a number of equivalent characterisations of the class XALP, e.g., XALP is the class of problems solvable by an Alternating Turing Machine whose runs have tree size at most f(k)n^O(1) and use f(k)log n space. Moreover, we introduce "tree-shaped" variants of Weighted CNF-Satisfiability and Multicolor Clique that are XALP-complete
XNLP-Completeness for Parameterized Problems on Graphs with a Linear Structure
In this paper, we showcase the class XNLP as a natural place for many hard problems parameterized by linear width measures. This strengthens existing W[1]-hardness proofs for these problems, since XNLP-hardness implies W[t]-hardness for all t. It also indicates, via a conjecture by Pilipczuk and Wrochna [ToCT 2018], that any XP algorithm for such problems is likely to require XP space.
In particular, we show XNLP-completeness for natural problems parameterized by pathwidth, linear clique-width, and linear mim-width. The problems we consider are Independent Set, Dominating Set, Odd Cycle Transversal, (q-)Coloring, Max Cut, Maximum Regular Induced Subgraph, Feedback Vertex Set, Capacitated (Red-Blue) Dominating Set, and Bipartite Bandwidth
- …