23 research outputs found
Finding detours is fixed-parameter tractable
We consider the following natural "above guarantee" parameterization of the
classical Longest Path problem: For given vertices s and t of a graph G, and an
integer k, the problem Longest Detour asks for an (s,t)-path in G that is at
least k longer than a shortest (s,t)-path. Using insights into structural graph
theory, we prove that Longest Detour is fixed-parameter tractable (FPT) on
undirected graphs and actually even admits a single-exponential algorithm, that
is, one of running time exp(O(k)) poly(n). This matches (up to the base of the
exponential) the best algorithms for finding a path of length at least k.
Furthermore, we study the related problem Exact Detour that asks whether a
graph G contains an (s,t)-path that is exactly k longer than a shortest
(s,t)-path. For this problem, we obtain a randomized algorithm with running
time about 2.746^k, and a deterministic algorithm with running time about
6.745^k, showing that this problem is FPT as well. Our algorithms for Exact
Detour apply to both undirected and directed graphs.Comment: Extended abstract appears at ICALP 201
Computing and Counting Longest Paths on Circular-Arc Graphs in Polynomial Time
The longest path problem asks for a path with the largest number of vertices in a given graph. The first polynomial time algorithm (with running time O(n4)) has been recently developed for interval graphs. Even though interval and circular-arc graphs look superficially similar, they differ substantially, as circular-arc graphs are not perfect. In this paper, we prove that for every path P of a circular-arc graph G, we can appropriately “cut” the circle, such that the obtained (not induced) interval subgraph G′ of G admits a path P′ on the same vertices as P. This non-trivial result is of independent interest, as it suggests a generic reduction of a number of path problems on circular-arc graphs to the case of interval graphs with a multiplicative linear time overhead of O(n). As an application of this reduction, we present the first polynomial algorithm for the longest path problem on circular-arc graphs, which turns out to have the same running time O(n4) with the one on interval graphs, as we manage to get rid of the linear overhead of the reduction. This algorithm computes in the same time an n-approximation of the number of different vertex sets that provide a longest path; in the case where G is an interval graph, we compute the exact number. Moreover, our algorithm can be directly extended with the same running time to the case where every vertex has an arbitrary positive weight
An EPTAS for Scheduling on Unrelated Machines of Few Different Types
In the classical problem of scheduling on unrelated parallel machines, a set
of jobs has to be assigned to a set of machines. The jobs have a processing
time depending on the machine and the goal is to minimize the makespan, that is
the maximum machine load. It is well known that this problem is NP-hard and
does not allow polynomial time approximation algorithms with approximation
guarantees smaller than unless PNP. We consider the case that there
are only a constant number of machine types. Two machines have the same
type if all jobs have the same processing time for them. This variant of the
problem is strongly NP-hard already for . We present an efficient
polynomial time approximation scheme (EPTAS) for the problem, that is, for any
an assignment with makespan of length at most
times the optimum can be found in polynomial time in the
input length and the exponent is independent of . In particular
we achieve a running time of , where
denotes the input length. Furthermore, we study three other problem
variants and present an EPTAS for each of them: The Santa Claus problem, where
the minimum machine load has to be maximized; the case of scheduling on
unrelated parallel machines with a constant number of uniform types, where
machines of the same type behave like uniformly related machines; and the
multidimensional vector scheduling variant of the problem where both the
dimension and the number of machine types are constant. For the Santa Claus
problem we achieve the same running time. The results are achieved, using mixed
integer linear programming and rounding techniques
Hitting Time of Quantum Walks with Perturbation
The hitting time is the required minimum time for a Markov chain-based walk
(classical or quantum) to reach a target state in the state space. We
investigate the effect of the perturbation on the hitting time of a quantum
walk. We obtain an upper bound for the perturbed quantum walk hitting time by
applying Szegedy's work and the perturbation bounds with Weyl's perturbation
theorem on classical matrix. Based on the definition of quantum hitting time
given in MNRS algorithm, we further compute the delayed perturbed hitting time
(DPHT) and delayed perturbed quantum hitting time (DPQHT). We show that the
upper bound for DPQHT is actually greater than the difference between the
square root of the upper bound for a perturbed random walk and the square root
of the lower bound for a random walk.Comment: 9 page
Network Archaeology: Uncovering Ancient Networks from Present-day Interactions
Often questions arise about old or extinct networks. What proteins interacted
in a long-extinct ancestor species of yeast? Who were the central players in
the Last.fm social network 3 years ago? Our ability to answer such questions
has been limited by the unavailability of past versions of networks. To
overcome these limitations, we propose several algorithms for reconstructing a
network's history of growth given only the network as it exists today and a
generative model by which the network is believed to have evolved. Our
likelihood-based method finds a probable previous state of the network by
reversing the forward growth model. This approach retains node identities so
that the history of individual nodes can be tracked. We apply these algorithms
to uncover older, non-extant biological and social networks believed to have
grown via several models, including duplication-mutation with complementarity,
forest fire, and preferential attachment. Through experiments on both synthetic
and real-world data, we find that our algorithms can estimate node arrival
times, identify anchor nodes from which new nodes copy links, and can reveal
significant features of networks that have long since disappeared.Comment: 16 pages, 10 figure
Computing and counting longest paths on circular-arc graphs in polynomial time
The longest path problem asks for a path with the largest number of vertices in a given graph. In contrast to the Hamiltonian path problem, until recently polynomial algorithms for the longest path problem were known only for small graph classes, such as trees. Recently, a polynomial algorithm for this problem on interval graphs has been presented in Ioannidou et al. (2011) [19] with running time O(n4) on a graph with n vertices, thus answering the open question posed in Uehara and Uno (2004) [32]. Even though interval and circular-arc graphs look superficially similar, they differ substantially, as circular-arc graphs are not perfect; for instance, several problems– e.g. coloring –are NP-hard on circular-arc graphs, although they can be efficiently solved on interval graphs. In this paper, we prove that for every path P of a circular-arc graph G, we can appropriately “cut” the circle, such that the obtained (not induced) interval subgraph G′ of G admits a path P′ on the same vertices as P. This non-trivial result is of independent interest, as it suggests a generic reduction of a number of path problems on circular-arc graphs to the case of interval graphs with a multiplicative linear time overhead of O(n). As an application of this reduction, we present the first polynomial algorithm for the longest path problem on circular-arc graphs. In addition, by exploiting deeper the structure of circular-arc graphs, we manage to get rid of the linear time overhead of the reduction, and thus this algorithm turns out to have the same running time O(n4) as the one on interval graphs. Our algorithm, which significantly simplifies the approach of Ioannidou et al. (2011) [19], computes in the same time an n-approximation of the (exponentially large in worst case) number of different vertex sets that provide a longest path; in the case where G is an interval graph, we compute the exact number. Moreover, in contrast to Ioannidou et al. (2011) [19], this algorithm can be directly extended with the same running time to the case where every vertex has an arbitrary positive weight