65 research outputs found
Relations between automata and the simple k-path problem
Let be a directed graph on vertices. Given an integer , the
SIMPLE -PATH problem asks whether there exists a simple -path in . In
case is weighted, the MIN-WT SIMPLE -PATH problem asks for a simple
-path in of minimal weight. The fastest currently known deterministic
algorithm for MIN-WT SIMPLE -PATH by Fomin, Lokshtanov and Saurabh runs in
time for graphs with integer weights in
the range . This is also the best currently known deterministic
algorithm for SIMPLE k-PATH- where the running time is the same without the
factor. We define to be the set of words of
length whose symbols are all distinct. We show that an explicit
construction of a non-deterministic automaton (NFA) of size for implies an algorithm of running time for MIN-WT SIMPLE -PATH when the weights are
non-negative or the constructed NFA is acyclic as a directed graph. We show
that the algorithm of Kneis et al. and its derandomization by Chen et al. for
SIMPLE -PATH can be used to construct an acylic NFA for of size
.
We show, on the other hand, that any NFA for must be size at least
. We thus propose closing this gap and determining the smallest NFA for
as an interesting open problem that might lead to faster algorithms
for MIN-WT SIMPLE -PATH.
We use a relation between SIMPLE -PATH and non-deterministic xor automata
(NXA) to give another direction for a deterministic algorithm with running time
for SIMPLE -PATH
Capacitated Vehicle Routing with Non-Uniform Speeds
The capacitated vehicle routing problem (CVRP) involves distributing
(identical) items from a depot to a set of demand locations, using a single
capacitated vehicle. We study a generalization of this problem to the setting
of multiple vehicles having non-uniform speeds (that we call Heterogenous
CVRP), and present a constant-factor approximation algorithm.
The technical heart of our result lies in achieving a constant approximation
to the following TSP variant (called Heterogenous TSP). Given a metric denoting
distances between vertices, a depot r containing k vehicles with possibly
different speeds, the goal is to find a tour for each vehicle (starting and
ending at r), so that every vertex is covered in some tour and the maximum
completion time is minimized. This problem is precisely Heterogenous CVRP when
vehicles are uncapacitated.
The presence of non-uniform speeds introduces difficulties for employing
standard tour-splitting techniques. In order to get a better understanding of
this technique in our context, we appeal to ideas from the 2-approximation for
scheduling in parallel machine of Lenstra et al.. This motivates the
introduction of a new approximate MST construction called Level-Prim, which is
related to Light Approximate Shortest-path Trees. The last component of our
algorithm involves partitioning the Level-Prim tree and matching the resulting
parts to vehicles. This decomposition is more subtle than usual since now we
need to enforce correlation between the size of the parts and their distances
to the depot
The robot routing problem for collecting aggregate stochastic rewards
We propose a new model for formalizing reward collection problems on graphs with dynamically generated rewards which may appear and disappear based on a stochastic model. The robot routing problem is modeled as a graph whose nodes are stochastic processes generating potential rewards over discrete time. The rewards are generated according to the stochastic process, but at each step, an existing reward disappears with a given probability. The edges in the graph encode the (unit-distance) paths between the rewards' locations. On visiting a node, the robot collects the accumulated reward at the node at that time, but traveling between the nodes takes time. The optimization question asks to compute an optimal (or epsilon-optimal) path that maximizes the expected collected rewards. We consider the finite and infinite-horizon robot routing problems. For finite-horizon, the goal is to maximize the total expected reward, while for infinite horizon we consider limit-average objectives. We study the computational and strategy complexity of these problems, establish NP-lower bounds and show that optimal strategies require memory in general. We also provide an algorithm for computing epsilon-optimal infinite paths for arbitrary epsilon > 0
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