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A Structural Theorem For Shortest Vertex-Disjoint Paths Computation in Planar Graphs
Given k terminal pairs (s₁,t₁),(s₂,t₂),..., (s[subscript k],t[subscript k]) in an edge-weighted graph G, the k Shortest Vertex-Disjoint Paths problem is to find a collection P₁, P₂,..., P[subscript k] of vertex-disjoint paths with minimum total length, where P[subscript i] is an s[subscript i]-to-t[subscript i] path. As a special case of the multi-commodity flow problem, computing vertex disjoint paths has found several applications, for example in VLSI design, or network routing. In this thesis we describe a Structural Theorem for a special case of the Shortest Vertex-Disjoint Paths problem in undirected planar graphs where the terminal vertices are on the boundary of the outer face. At a high level, our Structural Theorem guarantees that the i[superscript th] path of the k Shortest Vertex-Disjoint paths does not cross j[superscript th] (j ≠ i) path of the k-1 Vertex-Disjoint Paths problem
Tight Approximation and Kernelization Bounds for Vertex-Disjoint Shortest Paths
We examine the possibility of approximating Maximum Vertex-Disjoint Shortest
Paths. In this problem, the input is an edge-weighted (directed or undirected)
-vertex graph along with terminal pairs
. The task is to connect as many terminal
pairs as possible by pairwise vertex-disjoint paths such that each path is a
shortest path between the respective terminals. Our work is anchored in the
recent breakthrough by Lochet [SODA '21], which demonstrates the
polynomial-time solvability of the problem for a fixed value of .
Lochet's result implies the existence of a polynomial-time -approximation
for Maximum Vertex-Disjoint Shortest Paths, where is a constant. Our
first result suggests that this approximation algorithm is, in a sense, the
best we can hope for. More precisely, assuming the gap-ETH, we exclude the
existence of an -approximations within poly() time for
any function that only depends on .
Our second result demonstrates the infeasibility of achieving an
approximation ratio of in polynomial time, unless
P = NP. It is not difficult to show that a greedy algorithm selecting a path
with the minimum number of arcs results in a
-approximation, where is the number of edges in
all the paths of an optimal solution. Since , this underscores the
tightness of the -inapproximability bound.
Additionally, we establish that Maximum Vertex-Disjoint Shortest Paths is
fixed-parameter tractable when parameterized by but does not admit a
polynomial kernel. Our hardness results hold for undirected graphs with unit
weights, while our positive results extend to scenarios where the input graph
is directed and features arbitrary (non-negative) edge weights
Parameterizing Path Partitions
We study the algorithmic complexity of partitioning the vertex set of a given
(di)graph into a small number of paths. The Path Partition problem (PP) has
been studied extensively, as it includes Hamiltonian Path as a special case.
The natural variants where the paths are required to be either \emph{induced}
(Induced Path Partition, IPP) or \emph{shortest} (Shortest Path Partition,
SPP), have received much less attention. Both problems are known to be
NP-complete on undirected graphs; we strengthen this by showing that they
remain so even on planar bipartite directed acyclic graphs (DAGs), and that SPP
remains \NP-hard on undirected bipartite graphs. When parameterized by the
natural parameter ``number of paths'', both SPP and IPP are shown to be
W{1}-hard on DAGs. We also show that SPP is in \XP both for DAGs and undirected
graphs for the same parameter, as well as for other special subclasses of
directed graphs (IPP is known to be NP-hard on undirected graphs, even for two
paths). On the positive side, we show that for undirected graphs, both problems
are in FPT, parameterized by neighborhood diversity. We also give an explicit
algorithm for the vertex cover parameterization of PP. When considering the
dual parameterization (graph order minus number of paths), all three variants,
IPP, SPP and PP, are shown to be in FPT for undirected graphs. We also lift the
mentioned neighborhood diversity and dual parameterization results to directed
graphs; here, we need to define a proper novel notion of directed neighborhood
diversity. As we also show, most of our results also transfer to the case of
covering by edge-disjoint paths, and purely covering.Comment: 27 pages, 8 figures. A short version appeared in the proceedings of
the CIAC 2023 conferenc
Walking Through Waypoints
We initiate the study of a fundamental combinatorial problem: Given a
capacitated graph , find a shortest walk ("route") from a source to a destination that includes all vertices specified by a set
: the \emph{waypoints}. This waypoint routing problem
finds immediate applications in the context of modern networked distributed
systems. Our main contribution is an exact polynomial-time algorithm for graphs
of bounded treewidth. We also show that if the number of waypoints is
logarithmically bounded, exact polynomial-time algorithms exist even for
general graphs. Our two algorithms provide an almost complete characterization
of what can be solved exactly in polynomial-time: we show that more general
problems (e.g., on grid graphs of maximum degree 3, with slightly more
waypoints) are computationally intractable
Counting Shortest Two Disjoint Paths in Cubic Planar Graphs with an NC Algorithm
Given an undirected graph and two disjoint vertex pairs and
, the Shortest two disjoint paths problem (S2DP) asks for the minimum
total length of two vertex disjoint paths connecting with , and
with , respectively.
We show that for cubic planar graphs there are NC algorithms, uniform
circuits of polynomial size and polylogarithmic depth, that compute the S2DP
and moreover also output the number of such minimum length path pairs.
Previously, to the best of our knowledge, no deterministic polynomial time
algorithm was known for S2DP in cubic planar graphs with arbitrary placement of
the terminals. In contrast, the randomized polynomial time algorithm by
Bj\"orklund and Husfeldt, ICALP 2014, for general graphs is much slower, is
serial in nature, and cannot count the solutions.
Our results are built on an approach by Hirai and Namba, Algorithmica 2017,
for a generalisation of S2DP, and fast algorithms for counting perfect
matchings in planar graphs
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