59 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
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
Simplifying Activity-On-Edge Graphs
We formalize the simplification of activity-on-edge graphs used for
visualizing project schedules, where the vertices of the graphs represent
project milestones, and the edges represent either tasks of the project or
timing constraints between milestones. In this framework, a timeline of the
project can be constructed as a leveled drawing of the graph, where the levels
of the vertices represent the time at which each milestone is scheduled to
happen. We focus on the following problem: given an activity-on-edge graph
representing a project, find an equivalent activity-on-edge graph (one with the
same critical paths) that has the minimum possible number of milestone vertices
among all equivalent activity-on-edge graphs. We provide a polynomial-time
algorithm for solving this graph minimization problem
Shortest path and maximum flow problems in planar flow networks with additive gains and losses
In contrast to traditional flow networks, in additive flow networks, to every
edge e is assigned a gain factor g(e) which represents the loss or gain of the
flow while using edge e. Hence, if a flow f(e) enters the edge e and f(e) is
less than the designated capacity of e, then f(e) + g(e) = 0 units of flow
reach the end point of e, provided e is used, i.e., provided f(e) != 0. In this
report we study the maximum flow problem in additive flow networks, which we
prove to be NP-hard even when the underlying graphs of additive flow networks
are planar. We also investigate the shortest path problem, when to every edge e
is assigned a cost value for every unit flow entering edge e, which we show to
be NP-hard in the strong sense even when the additive flow networks are planar
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)
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
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