11,843 research outputs found

    The Complexity of Computing Minimal Unidirectional Covering Sets

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    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

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    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 (Ī»,Ī¼,Ļ€)(\lambda, \mu, \pi) such that the Kronecker coefficient kĪ¼,Ļ€Ī»=0k^\lambda_{\mu, \pi} = 0 but the Kronecker coefficient klĪ¼,lĻ€lĪ»>0k^{l \lambda}_{l \mu, l \pi} > 0 for some integer l>1l>1. 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 0<Ļµā‰¤10<\epsilon\leq1, there exists 0<a<10<a<1 such that, for all mm, there exist Ī©(2ma)\Omega(2^{m^a}) partition triples (Ī»,Ī¼,Ī¼)(\lambda,\mu,\mu) in the Kronecker cone such that: (a) the Kronecker coefficient kĪ¼,Ī¼Ī»k^\lambda_{\mu,\mu} is zero, (b) the height of Ī¼\mu is mm, (c) the height of Ī»\lambda is ā‰¤mĻµ\le m^\epsilon, and (d) āˆ£Ī»āˆ£=āˆ£Ī¼āˆ£ā‰¤m3|\lambda|=|\mu| \le m^3. 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

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    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

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    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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