74 research outputs found
On Directed Feedback Vertex Set parameterized by treewidth
We study the Directed Feedback Vertex Set problem parameterized by the
treewidth of the input graph. We prove that unless the Exponential Time
Hypothesis fails, the problem cannot be solved in time on general directed graphs, where is the treewidth of
the underlying undirected graph. This is matched by a dynamic programming
algorithm with running time .
On the other hand, we show that if the input digraph is planar, then the
running time can be improved to .Comment: 20
An O(log k)-Approximation for Directed Steiner Tree in Planar Graphs
We present an O(log k)-approximation for both the edge-weighted and node-weighted versions of Directed Steiner Tree in planar graphs where k is the number of terminals. We extend our approach to Multi-Rooted Directed Steiner Tree, in which we get a O(R+log k)-approximation for planar graphs for where R is the number of roots
On Temporal Graph Exploration
A temporal graph is a graph in which the edge set can change from step to
step. The temporal graph exploration problem TEXP is the problem of computing a
foremost exploration schedule for a temporal graph, i.e., a temporal walk that
starts at a given start node, visits all nodes of the graph, and has the
smallest arrival time. In the first part of the paper, we consider only
temporal graphs that are connected at each step. For such temporal graphs with
nodes, we show that it is NP-hard to approximate TEXP with ratio
for any . We also provide an explicit
construction of temporal graphs that require steps to be
explored. We then consider TEXP under the assumption that the underlying graph
(i.e. the graph that contains all edges that are present in the temporal graph
in at least one step) belongs to a specific class of graphs. Among other
results, we show that temporal graphs can be explored in steps if the underlying graph has treewidth and in
steps if the underlying graph is a grid. In the second part of the
paper, we replace the connectedness assumption by a weaker assumption and show
that -edge temporal graphs with regularly present edges and with random
edges can always be explored in steps and steps with high
probability, respectively. We finally show that the latter result can be used
to obtain a distributed algorithm for the gossiping problem.Comment: This is an extended version of an ICALP 2015 pape
An -Approximation for Directed Steiner Tree in Planar Graphs
We present an -approximation for both the edge-weighted and
node-weighted versions of \DST in planar graphs where is the number of
terminals. We extend our approach to \MDST (in general graphs \MDST and \DST
are easily seen to be equivalent but in planar graphs this is not the case
necessarily) in which we get an -approximation for planar graphs
for where is the number of roots
Decremental Strongly-Connected Components and Single-Source Reachability in Near-Linear Time
Computing the Strongly-Connected Components (SCCs) in a graph is
known to take only time using an algorithm by Tarjan from
1972[SICOMP 72] where , . For fully-dynamic graphs, conditional
lower bounds provide evidence that the update time cannot be improved by
polynomial factors over recomputing the SCCs from scratch after every update.
Nevertheless, substantial progress has been made to find algorithms with fast
update time for \emph{decremental} graphs, i.e. graphs that undergo edge
deletions.
In this paper, we present the first algorithm for general decremental graphs
that maintains the SCCs in total update time , thus only a
polylogarithmic factor from the optimal running time. Previously such a result
was only known for the special case of planar graphs [Italiano et al, STOC
2017]. Our result should be compared to the formerly best algorithm for general
graphs achieving total update time by Chechik et.al.
[FOCS 16] which improved upon a breakthrough result leading to total update time by Henzinger, Krinninger and Nanongkai [STOC 14,
ICALP 15]; these results in turn improved upon the longstanding bound of
by Roditty and Zwick [STOC 04].
All of the above results also apply to the decremental Single-Source
Reachability (SSR) problem, which can be reduced to decrementally maintaining
SCCs. A bound of total update time for decremental SSR was established
already in 1981 by Even and Shiloach [JACM 1981].
Using a well known reduction, we can maintain the reachability of pairs , in fully-dynamic graphs with update time
and query time for all ; this
generalizes an earlier All-Pairs Reachability where [{\L}\k{a}cki, TALG
2013].Comment: Accepted to STOC 1
On the Power of Tree-Depth for Fully Polynomial FPT Algorithms
There are many classical problems in P whose time complexities have not been improved over the past decades.
Recent studies of "Hardness in P" have revealed that, for several of such problems, the current fastest algorithm is the best possible under some complexity assumptions.
To bypass this difficulty, the concept of "FPT inside P" has been introduced.
For a problem with the current best time complexity O(n^c), the goal is to design an algorithm running in k^{O(1)}n^{c\u27} time for a parameter k and a constant c\u27<c.
In this paper, we investigate the complexity of graph problems in P parameterized by tree-depth, a graph parameter related to tree-width.
We show that a simple divide-and-conquer method can solve many graph problems, including
Weighted Matching, Negative Cycle Detection, Minimum Weight Cycle, Replacement Paths, and 2-hop Cover,
in O(td m) time or O(td (m+nlog n)) time, where td is the tree-depth of the input graph.
Because any graph of tree-width tw has tree-depth at most (tw+1)log_2 n, our algorithms also run in O(tw mlog n) time or O(tw (m+nlog n)log n) time.
These results match or improve the previous best algorithms parameterized by tree-width.
Especially, we solve an open problem of fully polynomial FPT algorithm for Weighted Matching parameterized by tree-width posed by Fomin et al. (SODA 2017)
- …