21,794 research outputs found
The matching relaxation for a class of generalized set partitioning problems
This paper introduces a discrete relaxation for the class of combinatorial
optimization problems which can be described by a set partitioning formulation
under packing constraints. We present two combinatorial relaxations based on
computing maximum weighted matchings in suitable graphs. Besides providing dual
bounds, the relaxations are also used on a variable reduction technique and a
matheuristic. We show how that general method can be tailored to sample
applications, and also perform a successful computational evaluation with
benchmark instances of a problem in maritime logistics.Comment: 33 pages. A preliminary (4-page) version of this paper was presented
at CTW 2016 (Cologne-Twente Workshop on Graphs and Combinatorial
Optimization), with proceedings on Electronic Notes in Discrete Mathematic
Pareto Optimal Allocation under Uncertain Preferences
The assignment problem is one of the most well-studied settings in social
choice, matching, and discrete allocation. We consider the problem with the
additional feature that agents' preferences involve uncertainty. The setting
with uncertainty leads to a number of interesting questions including the
following ones. How to compute an assignment with the highest probability of
being Pareto optimal? What is the complexity of computing the probability that
a given assignment is Pareto optimal? Does there exist an assignment that is
Pareto optimal with probability one? We consider these problems under two
natural uncertainty models: (1) the lottery model in which each agent has an
independent probability distribution over linear orders and (2) the joint
probability model that involves a joint probability distribution over
preference profiles. For both of the models, we present a number of algorithmic
and complexity results.Comment: Preliminary Draft; new results & new author
Online Assignment Algorithms for Dynamic Bipartite Graphs
This paper analyzes the problem of assigning weights to edges incrementally
in a dynamic complete bipartite graph consisting of producer and consumer
nodes. The objective is to minimize the overall cost while satisfying certain
constraints. The cost and constraints are functions of attributes of the edges,
nodes and online service requests. Novelty of this work is that it models
real-time distributed resource allocation using an approach to solve this
theoretical problem. This paper studies variants of this assignment problem
where the edges, producers and consumers can disappear and reappear or their
attributes can change over time. Primal-Dual algorithms are used for solving
these problems and their competitive ratios are evaluated
Derandomization of Online Assignment Algorithms for Dynamic Graphs
This paper analyzes different online algorithms for the problem of assigning
weights to edges in a fully-connected bipartite graph that minimizes the
overall cost while satisfying constraints. Edges in this graph may disappear
and reappear over time. Performance of these algorithms is measured using
simulations. This paper also attempts to derandomize the randomized online
algorithm for this problem
Matchings with lower quotas: Algorithms and complexity
We study a natural generalization of the maximum weight many-to-one matching problem. We are given an undirected bipartite graph G=(A∪˙P,E)G=(A∪˙P,E) with weights on the edges in E, and with lower and upper quotas on the vertices in P. We seek a maximum weight many-to-one matching satisfying two sets of constraints: vertices in A are incident to at most one matching edge, while vertices in P are either unmatched or they are incident to a number of matching edges between their lower and upper quota. This problem, which we call maximum weight many-to-one matching with lower and upper quotas (WMLQ), has applications to the assignment of students to projects within university courses, where there are constraints on the minimum and maximum numbers of students that must be assigned to each project. In this paper, we provide a comprehensive analysis of the complexity of WMLQ from the viewpoints of classical polynomial time algorithms, fixed-parameter tractability, as well as approximability. We draw the line between NPNP-hard and polynomially tractable instances in terms of degree and quota constraints and provide efficient algorithms to solve the tractable ones. We further show that the problem can be solved in polynomial time for instances with bounded treewidth; however, the corresponding runtime is exponential in the treewidth with the maximum upper quota umaxumax as basis, and we prove that this dependence is necessary unless FPT=W[1]FPT=W[1]. The approximability of WMLQ is also discussed: we present an approximation algorithm for the general case with performance guarantee umax+1umax+1, which is asymptotically best possible unless P=NPP=NP. Finally, we elaborate on how most of our positive results carry over to matchings in arbitrary graphs with lower quotas
Activity recognition from videos with parallel hypergraph matching on GPUs
In this paper, we propose a method for activity recognition from videos based
on sparse local features and hypergraph matching. We benefit from special
properties of the temporal domain in the data to derive a sequential and fast
graph matching algorithm for GPUs.
Traditionally, graphs and hypergraphs are frequently used to recognize
complex and often non-rigid patterns in computer vision, either through graph
matching or point-set matching with graphs. Most formulations resort to the
minimization of a difficult discrete energy function mixing geometric or
structural terms with data attached terms involving appearance features.
Traditional methods solve this minimization problem approximately, for instance
with spectral techniques.
In this work, instead of solving the problem approximatively, the exact
solution for the optimal assignment is calculated in parallel on GPUs. The
graphical structure is simplified and regularized, which allows to derive an
efficient recursive minimization algorithm. The algorithm distributes
subproblems over the calculation units of a GPU, which solves them in parallel,
allowing the system to run faster than real-time on medium-end GPUs
Computing Similarity between a Pair of Trajectories
With recent advances in sensing and tracking technology, trajectory data is
becoming increasingly pervasive and analysis of trajectory data is becoming
exceedingly important. A fundamental problem in analyzing trajectory data is
that of identifying common patterns between pairs or among groups of
trajectories. In this paper, we consider the problem of identifying similar
portions between a pair of trajectories, each observed as a sequence of points
sampled from it.
We present new measures of trajectory similarity --- both local and global
--- between a pair of trajectories to distinguish between similar and
dissimilar portions. Our model is robust under noise and outliers, it does not
make any assumptions on the sampling rates on either trajectory, and it works
even if they are partially observed. Additionally, the model also yields a
scalar similarity score which can be used to rank multiple pairs of
trajectories according to similarity, e.g. in clustering applications. We also
present efficient algorithms for computing the similarity under our measures;
the worst-case running time is quadratic in the number of sample points.
Finally, we present an extensive experimental study evaluating the
effectiveness of our approach on real datasets, comparing with it with earlier
approaches, and illustrating many issues that arise in trajectory data. Our
experiments show that our approach is highly accurate in distinguishing similar
and dissimilar portions as compared to earlier methods even with sparse
sampling
Fair assignment of indivisible objects under ordinal preferences
We consider the discrete assignment problem in which agents express ordinal
preferences over objects and these objects are allocated to the agents in a
fair manner. We use the stochastic dominance relation between fractional or
randomized allocations to systematically define varying notions of
proportionality and envy-freeness for discrete assignments. The computational
complexity of checking whether a fair assignment exists is studied for these
fairness notions. We also characterize the conditions under which a fair
assignment is guaranteed to exist. For a number of fairness concepts,
polynomial-time algorithms are presented to check whether a fair assignment
exists. Our algorithmic results also extend to the case of unequal entitlements
of agents. Our NP-hardness result, which holds for several variants of
envy-freeness, answers an open question posed by Bouveret, Endriss, and Lang
(ECAI 2010). We also propose fairness concepts that always suggest a non-empty
set of assignments with meaningful fairness properties. Among these concepts,
optimal proportionality and optimal weak proportionality appear to be desirable
fairness concepts.Comment: extended version of a paper presented at AAMAS 201
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