434 research outputs found
Prize-Collecting TSP with a Budget Constraint
We consider constrained versions of the prize-collecting traveling salesman and the minimum spanning tree problems. The goal is to maximize the number of vertices in the returned tour/tree subject to a bound on the tour/tree cost. We present a 2-approximation algorithm for these problems based on a primal-dual approach. The algorithm relies on finding a threshold value for the dual variable corresponding to the budget constraint in the primal and then carefully constructing a tour/tree that is just within budget. Thereby, we improve the best-known guarantees from 3+epsilon and 2+epsilon for the tree and the tour version, respectively. Our analysis extends to the setting with weighted vertices, in which we want to maximize the total weight of vertices in the tour/tree subject to the same budget constraint
Scalable Robust Kidney Exchange
In barter exchanges, participants directly trade their endowed goods in a
constrained economic setting without money. Transactions in barter exchanges
are often facilitated via a central clearinghouse that must match participants
even in the face of uncertainty---over participants, existence and quality of
potential trades, and so on. Leveraging robust combinatorial optimization
techniques, we address uncertainty in kidney exchange, a real-world barter
market where patients swap (in)compatible paired donors. We provide two
scalable robust methods to handle two distinct types of uncertainty in kidney
exchange---over the quality and the existence of a potential match. The latter
case directly addresses a weakness in all stochastic-optimization-based methods
to the kidney exchange clearing problem, which all necessarily require explicit
estimates of the probability of a transaction existing---a still-unsolved
problem in this nascent market. We also propose a novel, scalable kidney
exchange formulation that eliminates the need for an exponential-time
constraint generation process in competing formulations, maintains provable
optimality, and serves as a subsolver for our robust approach. For each type of
uncertainty we demonstrate the benefits of robustness on real data from a
large, fielded kidney exchange in the United States. We conclude by drawing
parallels between robustness and notions of fairness in the kidney exchange
setting.Comment: Presented at AAAI1
A Swarm of Salesmen: Algorithmic Approaches to Multiagent Modeling
This honors thesis describes the algorithmic abstraction of a problem modeling a swarm of Mars rovers, where many agents must together achieve a goal. The algorithmic formulation of this problem is based on the traveling salesman problem (TSP), and so in this thesis I offer a review of the mathematical technique of linear programming in the context of its application to the TSP, an overview of some variations of the TSP and algorithms for approximating and solving them, and formulations without solutions of two novel TSP variations which are useful for modeling the original problem
A Swarm of Salesmen: Algorithmic Approaches to Multiagent Modeling
This honors thesis describes the algorithmic abstraction of a problem modeling a swarm of Mars rovers, where many agents must together achieve a goal. The algorithmic formulation of this problem is based on the traveling salesman problem (TSP), and so in this thesis I offer a review of the mathematical technique of linear programming in the context of its application to the TSP, an overview of some variations of the TSP and algorithms for approximating and solving them, and formulations without solutions of two novel TSP variations which are useful for modeling the original problem
Algorithms and Adaptivity Gaps for Stochastic k-TSP
Given a metric and a , the classic
\textsf{k-TSP} problem is to find a tour originating at the
of minimum length that visits at least nodes in . In this work,
motivated by applications where the input to an optimization problem is
uncertain, we study two stochastic versions of \textsf{k-TSP}.
In Stoch-Reward -TSP, originally defined by Ene-Nagarajan-Saket [ENS17],
each vertex in the given metric contains a stochastic reward .
The goal is to adaptively find a tour of minimum expected length that collects
at least reward ; here "adaptively" means our next decision may depend on
previous outcomes. Ene et al. give an -approximation adaptive
algorithm for this problem, and left open if there is an -approximation
algorithm. We totally resolve their open question and even give an
-approximation \emph{non-adaptive} algorithm for this problem.
We also introduce and obtain similar results for the Stoch-Cost -TSP
problem. In this problem each vertex has a stochastic cost , and the
goal is to visit and select at least vertices to minimize the expected
\emph{sum} of tour length and cost of selected vertices. This problem
generalizes the Price of Information framework [Singla18] from deterministic
probing costs to metric probing costs.
Our techniques are based on two crucial ideas: "repetitions" and "critical
scaling". We show using Freedman's and Jogdeo-Samuels' inequalities that for
our problems, if we truncate the random variables at an ideal threshold and
repeat, then their expected values form a good surrogate. Unfortunately, this
ideal threshold is adaptive as it depends on how far we are from achieving our
target , so we truncate at various different scales and identify a
"critical" scale.Comment: ITCS 202
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