1,978 research outputs found
Truthful Mechanisms for Matching and Clustering in an Ordinal World
We study truthful mechanisms for matching and related problems in a partial
information setting, where the agents' true utilities are hidden, and the
algorithm only has access to ordinal preference information. Our model is
motivated by the fact that in many settings, agents cannot express the
numerical values of their utility for different outcomes, but are still able to
rank the outcomes in their order of preference. Specifically, we study problems
where the ground truth exists in the form of a weighted graph of agent
utilities, but the algorithm can only elicit the agents' private information in
the form of a preference ordering for each agent induced by the underlying
weights. Against this backdrop, we design truthful algorithms to approximate
the true optimum solution with respect to the hidden weights. Our techniques
yield universally truthful algorithms for a number of graph problems: a
1.76-approximation algorithm for Max-Weight Matching, 2-approximation algorithm
for Max k-matching, a 6-approximation algorithm for Densest k-subgraph, and a
2-approximation algorithm for Max Traveling Salesman as long as the hidden
weights constitute a metric. We also provide improved approximation algorithms
for such problems when the agents are not able to lie about their preferences.
Our results are the first non-trivial truthful approximation algorithms for
these problems, and indicate that in many situations, we can design robust
algorithms even when the agents may lie and only provide ordinal information
instead of precise utilities.Comment: To appear in the Proceedings of WINE 201
General Bounds for Incremental Maximization
We propose a theoretical framework to capture incremental solutions to
cardinality constrained maximization problems. The defining characteristic of
our framework is that the cardinality/support of the solution is bounded by a
value that grows over time, and we allow the solution to be
extended one element at a time. We investigate the best-possible competitive
ratio of such an incremental solution, i.e., the worst ratio over all
between the incremental solution after steps and an optimum solution of
cardinality . We define a large class of problems that contains many
important cardinality constrained maximization problems like maximum matching,
knapsack, and packing/covering problems. We provide a general
-competitive incremental algorithm for this class of problems, and show
that no algorithm can have competitive ratio below in general.
In the second part of the paper, we focus on the inherently incremental
greedy algorithm that increases the objective value as much as possible in each
step. This algorithm is known to be -competitive for submodular objective
functions, but it has unbounded competitive ratio for the class of incremental
problems mentioned above. We define a relaxed submodularity condition for the
objective function, capturing problems like maximum (weighted) (-)matching
and a variant of the maximum flow problem. We show that the greedy algorithm
has competitive ratio (exactly) for the class of problems that satisfy
this relaxed submodularity condition.
Note that our upper bounds on the competitive ratios translate to
approximation ratios for the underlying cardinality constrained problems.Comment: fixed typo
Diverse Weighted Bipartite b-Matching
Bipartite matching, where agents on one side of a market are matched to
agents or items on the other, is a classical problem in computer science and
economics, with widespread application in healthcare, education, advertising,
and general resource allocation. A practitioner's goal is typically to maximize
a matching market's economic efficiency, possibly subject to some fairness
requirements that promote equal access to resources. A natural balancing act
exists between fairness and efficiency in matching markets, and has been the
subject of much research.
In this paper, we study a complementary goal---balancing diversity and
efficiency---in a generalization of bipartite matching where agents on one side
of the market can be matched to sets of agents on the other. Adapting a
classical definition of the diversity of a set, we propose a quadratic
programming-based approach to solving a supermodular minimization problem that
balances diversity and total weight of the solution. We also provide a scalable
greedy algorithm with theoretical performance bounds. We then define the price
of diversity, a measure of the efficiency loss due to enforcing diversity, and
give a worst-case theoretical bound. Finally, we demonstrate the efficacy of
our methods on three real-world datasets, and show that the price of diversity
is not bad in practice
Effect of correlations on network controllability
A dynamical system is controllable if by imposing appropriate external
signals on a subset of its nodes, it can be driven from any initial state to
any desired state in finite time. Here we study the impact of various network
characteristics on the minimal number of driver nodes required to control a
network. We find that clustering and modularity have no discernible impact, but
the symmetries of the underlying matching problem can produce linear, quadratic
or no dependence on degree correlation coefficients, depending on the nature of
the underlying correlations. The results are supported by numerical simulations
and help narrow the observed gap between the predicted and the observed number
of driver nodes in real networks
Coresets Meet EDCS: Algorithms for Matching and Vertex Cover on Massive Graphs
As massive graphs become more prevalent, there is a rapidly growing need for
scalable algorithms that solve classical graph problems, such as maximum
matching and minimum vertex cover, on large datasets. For massive inputs,
several different computational models have been introduced, including the
streaming model, the distributed communication model, and the massively
parallel computation (MPC) model that is a common abstraction of
MapReduce-style computation. In each model, algorithms are analyzed in terms of
resources such as space used or rounds of communication needed, in addition to
the more traditional approximation ratio.
In this paper, we give a single unified approach that yields better
approximation algorithms for matching and vertex cover in all these models. The
highlights include:
* The first one pass, significantly-better-than-2-approximation for matching
in random arrival streams that uses subquadratic space, namely a
-approximation streaming algorithm that uses space
for constant .
* The first 2-round, better-than-2-approximation for matching in the MPC
model that uses subquadratic space per machine, namely a
-approximation algorithm with memory per
machine for constant .
By building on our unified approach, we further develop parallel algorithms
in the MPC model that give a -approximation to matching and an
-approximation to vertex cover in only MPC rounds and
memory per machine. These results settle multiple open
questions posed in the recent paper of Czumaj~et.al. [STOC 2018]
Higher-order Projected Power Iterations for Scalable Multi-Matching
The matching of multiple objects (e.g. shapes or images) is a fundamental
problem in vision and graphics. In order to robustly handle ambiguities, noise
and repetitive patterns in challenging real-world settings, it is essential to
take geometric consistency between points into account. Computationally, the
multi-matching problem is difficult. It can be phrased as simultaneously
solving multiple (NP-hard) quadratic assignment problems (QAPs) that are
coupled via cycle-consistency constraints. The main limitations of existing
multi-matching methods are that they either ignore geometric consistency and
thus have limited robustness, or they are restricted to small-scale problems
due to their (relatively) high computational cost. We address these
shortcomings by introducing a Higher-order Projected Power Iteration method,
which is (i) efficient and scales to tens of thousands of points, (ii)
straightforward to implement, (iii) able to incorporate geometric consistency,
(iv) guarantees cycle-consistent multi-matchings, and (iv) comes with
theoretical convergence guarantees. Experimentally we show that our approach is
superior to existing methods
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