421 research outputs found
A Local-to-Global Theorem for Congested Shortest Paths
Amiri and Wargalla (2020) proved the following local-to-global theorem in
directed acyclic graphs (DAGs): if is a weighted DAG such that for each
subset of 3 nodes there is a shortest path containing every node in ,
then there exists a pair of nodes such that there is a shortest
-path containing every node in .
We extend this theorem to general graphs. For undirected graphs, we prove
that the same theorem holds (up to a difference in the constant 3). For
directed graphs, we provide a counterexample to the theorem (for any constant),
and prove a roundtrip analogue of the theorem which shows there exists a pair
of nodes such that every node in is contained in the union of a
shortest -path and a shortest -path.
The original theorem for DAGs has an application to the -Shortest Paths
with Congestion (()-SPC) problem. In this problem, we are given a
weighted graph , together with node pairs ,
and a positive integer . We are tasked with finding paths such that each is a shortest path from to , and every
node in the graph is on at most paths , or reporting that no such
collection of paths exists.
When the problem is easily solved by finding shortest paths for each
pair independently. When , the -SPC problem recovers
the -Disjoint Shortest Paths (-DSP) problem, where the collection of
shortest paths must be node-disjoint. For fixed , -DSP can be solved in
polynomial time on DAGs and undirected graphs. Previous work shows that the
local-to-global theorem for DAGs implies that -SPC on DAGs whenever
is constant. In the same way, our work implies that -SPC can be
solved in polynomial time on undirected graphs whenever is constant.Comment: Updated to reflect reviewer comment
How finance shapes careers of highly skilled individuals
This dissertation analyzes how finance shapes careers of highly skilled individuals. In three chapters, I analyze the role of financial education, labor regulation, and angel investments on choices and careers of individuals
Planar Disjoint Paths, Treewidth, and Kernels
In the Planar Disjoint Paths problem, one is given an undirected planar graph
with a set of vertex pairs and the task is to find pairwise
vertex-disjoint paths such that the -th path connects to . We
study the problem through the lens of kernelization, aiming at efficiently
reducing the input size in terms of a parameter. We show that Planar Disjoint
Paths does not admit a polynomial kernel when parameterized by unless coNP
NP/poly, resolving an open problem by [Bodlaender, Thomass{\'e},
Yeo, ESA'09]. Moreover, we rule out the existence of a polynomial Turing kernel
unless the WK-hierarchy collapses. Our reduction carries over to the setting of
edge-disjoint paths, where the kernelization status remained open even in
general graphs.
On the positive side, we present a polynomial kernel for Planar Disjoint
Paths parameterized by , where denotes the treewidth of the input
graph. As a consequence of both our results, we rule out the possibility of a
polynomial-time (Turing) treewidth reduction to under the same
assumptions. To the best of our knowledge, this is the first hardness result of
this kind. Finally, combining our kernel with the known techniques [Adler,
Kolliopoulos, Krause, Lokshtanov, Saurabh, Thilikos, JCTB'17; Schrijver,
SICOMP'94] yields an alternative (and arguably simpler) proof that Planar
Disjoint Paths can be solved in time , matching the
result of [Lokshtanov, Misra, Pilipczuk, Saurabh, Zehavi, STOC'20].Comment: To appear at FOCS'23, 82 pages, 30 figure
Online Learning of Energy Consumption for Navigation of Electric Vehicles
Energy efficient navigation constitutes an important challenge in electric vehicles, due to their limited battery capacity. We employ a Bayesian approach to model the energy consumption at road segments for efficient navigation. In order to learn the model parameters, we develop an online learning framework and investigate several exploration strategies such as Thompson Sampling and Upper Confidence Bound. We then extend our online learning framework to the multi-agent setting, where multiple vehicles adaptively navigate and learn the parameters of the energy model. We analyze Thompson Sampling and establish rigorous regret bounds on its performance in the single-agent and multi-agent settings, through an analysis of the algorithm under batched feedback. Finally, we demonstrate the performance of our methods via experiments on several real-world city road networks
Longest Path and Cycle Transversal and Gallai Families
A longest path transversal in a graph G is a set of vertices S of G such that every longest path in G has a vertex in S. The longest path transversal number of a graph G is the size of a smallest longest path transversal in G and is denoted lpt(G). Similarly, a longest cycle transversal is a set of vertices S in a graph G such that every longest cycle in G has a vertex in S. The longest cycle transversal number of a graph G is the size of a smallest longest cycle transversal in G and is denoted lct(G). A Gallai family is a family of graphs whose connected members have longest path transversal number 1. In this paper we find several Gallai families and give upper bounds on lpt(G) and lct(G) for general graphs and chordal graphs in terms of |V(G)|
Designing Equitable Transit Networks
Public transit is an essential infrastructure enabling access to employment,
healthcare, education, and recreational facilities. While accessibility to
transit is important in general, some sections of the population depend
critically on transit. However, existing public transit is often not designed
equitably, and often, equity is only considered as an additional objective post
hoc, which hampers systemic changes. We present a formulation for transit
network design that considers different notions of equity and welfare
explicitly. We study the interaction between network design and various
concepts of equity and present trade-offs and results based on real-world data
from a large metropolitan area in the United States of America.Comment: Accepted in the non-archival track at the ACM Conference on Equity
and Access in Algorithms, Mechanisms, and Optimization (EAAMO), 202
End-of-Horizon Load Balancing Problems: Algorithms and Insights
Effective load balancing is at the heart of many applications in operations.
Often tackled via the balls-into-bins paradigm, seminal results have shown that
a limited amount of flexibility goes a long way in order to maintain
(approximately) balanced loads throughout the decision-making horizon. This
paper is motivated by the fact that balance across time is too stringent a
requirement for some applications; rather, the only desideratum is approximate
balance at the end of the horizon. In this work we design
``limited-flexibility'' algorithms for three instantiations of the
end-of-horizon balance problem: the balls-into-bins problem, opaque selling
strategies for inventory management, and parcel delivery for e-commerce
fulfillment. For the balls-into-bins model, we show that a simple policy which
begins exerting flexibility toward the end of the time horizon (i.e., when
periods remain), suffices to achieve an
approximately balanced load (i.e., a maximum load within of the
average load). Moreover, with just a small amount of adaptivity, a threshold
policy achieves the same result, while only exerting flexibility in
periods, matching a natural lower bound. We then
adapt these algorithms to develop order-wise optimal policies for the opaque
selling problem. Finally, we show via a data-driven case study that the
adaptive policy designed for the balls-into-bins model can be modified to (i)
achieve approximate balance at the end of the horizon and (ii) yield
significant cost savings relative to policies which either never exert
flexibility, or exert flexibility aggressively enough to achieve anytime
balance. The unifying motivation behind our algorithms is the observation that
exerting flexibility at the beginning of the horizon is likely wasted when
system balance is only evaluated at the end
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Odd Paths, Cycles and -joins: Connections and Algorithms
Minimizing the weight of an edge set satisfying parity constraints is a
challenging branch of combinatorial optimization as witnessed by the binary
hypergraph chapter of Alexander Schrijver's book ``Combinatorial Optimization''
(Chapter 80). This area contains relevant graph theory problems including open
cases of the Max Cut problem, or some multiflow problems. We clarify the
interconnections of some problems and establish three levels of difficulties.
On the one hand, we prove that the Shortest Odd Path problem in an undirected
graph without cycles of negative total weight and several related problems are
NP-hard, settling a long-standing open question asked by Lov\'asz (Open Problem
27 in Schrijver's book ``Combinatorial Optimization''. On the other hand, we
provide a polynomial-time algorithm to the closely related and well-studied
Minimum-weight Odd -Join problem for non-negative weights, whose
complexity, however, was not known; more generally, we solve the Minimum-weight
Odd -Join problem in FPT time when parameterized by . If negative
weights are also allowed, then finding a minimum-weight odd -join is
equivalent to the Minimum-weight Odd -Join problem for arbitrary weights,
whose complexity is only conjectured to be polynomially solvable. The analogous
problems for digraphs are also considered.Comment: 24 pages, 2 figure
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