2,329 research outputs found
Inapproximability of Combinatorial Optimization Problems
We survey results on the hardness of approximating combinatorial optimization
problems
Inapproximability of Truthful Mechanisms via Generalizations of the VC Dimension
Algorithmic mechanism design (AMD) studies the delicate interplay between
computational efficiency, truthfulness, and optimality. We focus on AMD's
paradigmatic problem: combinatorial auctions. We present a new generalization
of the VC dimension to multivalued collections of functions, which encompasses
the classical VC dimension, Natarajan dimension, and Steele dimension. We
present a corresponding generalization of the Sauer-Shelah Lemma and harness
this VC machinery to establish inapproximability results for deterministic
truthful mechanisms. Our results essentially unify all inapproximability
results for deterministic truthful mechanisms for combinatorial auctions to
date and establish new separation gaps between truthful and non-truthful
algorithms
Minimum-weight Cycle Covers and Their Approximability
A cycle cover of a graph is a set of cycles such that every vertex is part of
exactly one cycle. An L-cycle cover is a cycle cover in which the length of
every cycle is in the set L.
We investigate how well L-cycle covers of minimum weight can be approximated.
For undirected graphs, we devise a polynomial-time approximation algorithm that
achieves a constant approximation ratio for all sets L. On the other hand, we
prove that the problem cannot be approximated within a factor of 2-eps for
certain sets L.
For directed graphs, we present a polynomial-time approximation algorithm
that achieves an approximation ratio of O(n), where is the number of
vertices. This is asymptotically optimal: We show that the problem cannot be
approximated within a factor of o(n).
To contrast the results for cycle covers of minimum weight, we show that the
problem of computing L-cycle covers of maximum weight can, at least in
principle, be approximated arbitrarily well.Comment: To appear in the Proceedings of the 33rd Workshop on Graph-Theoretic
Concepts in Computer Science (WG 2007). Minor change
The matching polytope does not admit fully-polynomial size relaxation schemes
The groundbreaking work of Rothvo{\ss} [arxiv:1311.2369] established that
every linear program expressing the matching polytope has an exponential number
of inequalities (formally, the matching polytope has exponential extension
complexity). We generalize this result by deriving strong bounds on the
polyhedral inapproximability of the matching polytope: for fixed , every polyhedral -approximation
requires an exponential number of inequalities, where is the number of
vertices. This is sharp given the well-known -approximation of size
provided by the odd-sets of size up to
. Thus matching is the first problem in , whose natural
linear encoding does not admit a fully polynomial-size relaxation scheme (the
polyhedral equivalent of an FPTAS), which provides a sharp separation from the
polynomial-size relaxation scheme obtained e.g., via constant-sized odd-sets
mentioned above.
Our approach reuses ideas from Rothvo{\ss} [arxiv:1311.2369], however the
main lower bounding technique is different. While the original proof is based
on the hyperplane separation bound (also called the rectangle corruption
bound), we employ the information-theoretic notion of common information as
introduced in Braun and Pokutta [http://eccc.hpi-web.de/report/2013/056/],
which allows to analyze perturbations of slack matrices. It turns out that the
high extension complexity for the matching polytope stem from the same source
of hardness as for the correlation polytope: a direct sum structure.Comment: 21 pages, 3 figure
Scheduling over Scenarios on Two Machines
We consider scheduling problems over scenarios where the goal is to find a
single assignment of the jobs to the machines which performs well over all
possible scenarios. Each scenario is a subset of jobs that must be executed in
that scenario and all scenarios are given explicitly. The two objectives that
we consider are minimizing the maximum makespan over all scenarios and
minimizing the sum of the makespans of all scenarios. For both versions, we
give several approximation algorithms and lower bounds on their
approximability. With this research into optimization problems over scenarios,
we have opened a new and rich field of interesting problems.Comment: To appear in COCOON 2014. The final publication is available at
link.springer.co
Complexity of Discrete Energy Minimization Problems
Discrete energy minimization is widely-used in computer vision and machine
learning for problems such as MAP inference in graphical models. The problem,
in general, is notoriously intractable, and finding the global optimal solution
is known to be NP-hard. However, is it possible to approximate this problem
with a reasonable ratio bound on the solution quality in polynomial time? We
show in this paper that the answer is no. Specifically, we show that general
energy minimization, even in the 2-label pairwise case, and planar energy
minimization with three or more labels are exp-APX-complete. This finding rules
out the existence of any approximation algorithm with a sub-exponential
approximation ratio in the input size for these two problems, including
constant factor approximations. Moreover, we collect and review the
computational complexity of several subclass problems and arrange them on a
complexity scale consisting of three major complexity classes -- PO, APX, and
exp-APX, corresponding to problems that are solvable, approximable, and
inapproximable in polynomial time. Problems in the first two complexity classes
can serve as alternative tractable formulations to the inapproximable ones.
This paper can help vision researchers to select an appropriate model for an
application or guide them in designing new algorithms.Comment: ECCV'16 accepte
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