11,843 research outputs found
The Complexity of Computing Minimal Unidirectional Covering Sets
Given a binary dominance relation on a set of alternatives, a common thread
in the social sciences is to identify subsets of alternatives that satisfy
certain notions of stability. Examples can be found in areas as diverse as
voting theory, game theory, and argumentation theory. Brandt and Fischer [BF08]
proved that it is NP-hard to decide whether an alternative is contained in some
inclusion-minimal upward or downward covering set. For both problems, we raise
this lower bound to the Theta_{2}^{p} level of the polynomial hierarchy and
provide a Sigma_{2}^{p} upper bound. Relatedly, we show that a variety of other
natural problems regarding minimal or minimum-size covering sets are hard or
complete for either of NP, coNP, and Theta_{2}^{p}. An important consequence of
our results is that neither minimal upward nor minimal downward covering sets
(even when guaranteed to exist) can be computed in polynomial time unless P=NP.
This sharply contrasts with Brandt and Fischer's result that minimal
bidirectional covering sets (i.e., sets that are both minimal upward and
minimal downward covering sets) are polynomial-time computable.Comment: 27 pages, 7 figure
On vanishing of Kronecker coefficients
We show that the problem of deciding positivity of Kronecker coefficients is
NP-hard. Previously, this problem was conjectured to be in P, just as for the
Littlewood-Richardson coefficients. Our result establishes in a formal way that
Kronecker coefficients are more difficult than Littlewood-Richardson
coefficients, unless P=NP.
We also show that there exists a #P-formula for a particular subclass of
Kronecker coefficients whose positivity is NP-hard to decide. This is an
evidence that, despite the hardness of the positivity problem, there may well
exist a positive combinatorial formula for the Kronecker coefficients. Finding
such a formula is a major open problem in representation theory and algebraic
combinatorics.
Finally, we consider the existence of the partition triples such that the Kronecker coefficient but the
Kronecker coefficient for some integer
. Such "holes" are of great interest as they witness the failure of the
saturation property for the Kronecker coefficients, which is still poorly
understood. Using insight from computational complexity theory, we turn our
hardness proof into a positive result: We show that not only do there exist
many such triples, but they can also be found efficiently. Specifically, we
show that, for any , there exists such that, for all
, there exist partition triples in the
Kronecker cone such that: (a) the Kronecker coefficient
is zero, (b) the height of is , (c) the height of is , and (d) . The proof of the last result
illustrates the effectiveness of the explicit proof strategy of GCT.Comment: 43 pages, 1 figur
Algorithms and Throughput Analysis for MDS-Coded Switches
Network switches and routers need to serve packet writes and reads at rates
that challenge the most advanced memory technologies. As a result, scaling the
switching rates is commonly done by parallelizing the packet I/Os using
multiple memory units. For improved read rates, packets can be coded with an
[n,k] MDS code, thus giving more flexibility at read time to achieve higher
utilization of the memory units. In the paper, we study the usage of [n,k] MDS
codes in a switching environment. In particular, we study the algorithmic
problem of maximizing the instantaneous read rate given a set of packet
requests and the current layout of the coded packets in memory. The most
interesting results from practical standpoint show how the complexity of
reaching optimal read rate depends strongly on the writing policy of the coded
packets.Comment: 6 pages, an extended version of a paper accepted to the 2015 IEEE
International Symposium on Information Theory (ISIT
Routing Games with Progressive Filling
Max-min fairness (MMF) is a widely known approach to a fair allocation of
bandwidth to each of the users in a network. This allocation can be computed by
uniformly raising the bandwidths of all users without violating capacity
constraints. We consider an extension of these allocations by raising the
bandwidth with arbitrary and not necessarily uniform time-depending velocities
(allocation rates). These allocations are used in a game-theoretic context for
routing choices, which we formalize in progressive filling games (PFGs).
We present a variety of results for equilibria in PFGs. We show that these
games possess pure Nash and strong equilibria. While computation in general is
NP-hard, there are polynomial-time algorithms for prominent classes of
Max-Min-Fair Games (MMFG), including the case when all users have the same
source-destination pair. We characterize prices of anarchy and stability for
pure Nash and strong equilibria in PFGs and MMFGs when players have different
or the same source-destination pairs. In addition, we show that when a designer
can adjust allocation rates, it is possible to design games with optimal strong
equilibria. Some initial results on polynomial-time algorithms in this
direction are also derived
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