88,599 research outputs found
On Vanishing of {K}ronecker Coefficients
It is shown that: (1) The problem of deciding positivity of Kronecker coefficients is NP-hard. (2) There exists a positive ()-formula for a subclass of Kronecker coefficients whose positivity is NP-hard to decide. (3) 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 last result takes a step towards proving the existence of occurrence-based representation-theoretic obstructions in the context of the GCT approach to the permanent vs. determinant problem. Its proof also illustrates the effectiveness of the explicit proof strategy of GCT
P vs NP: P is Equal to NP: Desired Proof
Computations and computational complexity are fundamental for mathematics and all computer science, including web load time, cryptography (cryptocurrency mining), cybersecurity, artificial intelligence, game theory, multimedia processing, computational physics, biology (for instance, in protein structure prediction), chemistry, and the P vs. NP problem that has been singled out as one of the most challenging open problems in computer science and has great importance as this would essentially solve all the algorithmic problems that we have today if the problem is solved, but the existing complexity is deprecated and does not solve complex computations of tasks that appear in the new digital age as efficiently as it needs. Therefore, we need to realize a new complexity to solve these tasks more rapidly and easily. This paper presents proof of the equality of P and NP complexity classes when the NP problem is not harder to compute than to verify in polynomial time if we forget recursion that takes exponential running time and goes to regress only (every problem in NP can be solved in exponential time, and so it is recursive, this is a key concept that exists, but recursion does not solve the NP problems efficiently). The paperâs goal is to prove the existence of an algorithm solving the NP task in polynomial running time. We get the desired reduction of the exponential problem to the polynomial problem that takes O(log n) complexity
Uniformly generated submodules of permutation modules Over fields of characteristic 0
AbstractThis paper is motivated by a link between algebraic proof complexity and the representation theory of the finite symmetric groups. Our perspective leads to a new avenue of investigation in the representation theory of Sn. Most of our technical results concern the structure of âuniformlyâ generated submodules of permutation modules. For example, we consider sequences {Wn}nâN of submodules of the permutation modules M(nâk,1k) and prove that if the sequence Wn is given in a uniform (in n) way â which we make precise â the dimension p(n) of Wn (as a vector space) is a single polynomial with rational coefficients, for all but finitely many âsingularâ values of n. Furthermore, we show that dim(Wn)<p(n) for each singular value of nâ„4k. The results have a non-traditional flavor arising from the study of the irreducible structure of the submodules Wn beyond isomorphism types. We sketch the link between our structure theorems and proof complexity questions, which are motivated by the famous NP vs. co-NP problem in complexity theory. In particular, we focus on the complexity of showing membership in polynomial ideals, in various proof systems, for example, based on Hilbert's Nullstellensatz
Algorithmic aspects of disjunctive domination in graphs
For a graph , a set is called a \emph{disjunctive
dominating set} of if for every vertex , is either
adjacent to a vertex of or has at least two vertices in at distance
from it. The cardinality of a minimum disjunctive dominating set of is
called the \emph{disjunctive domination number} of graph , and is denoted by
. The \textsc{Minimum Disjunctive Domination Problem} (MDDP)
is to find a disjunctive dominating set of cardinality .
Given a positive integer and a graph , the \textsc{Disjunctive
Domination Decision Problem} (DDDP) is to decide whether has a disjunctive
dominating set of cardinality at most . In this article, we first propose a
linear time algorithm for MDDP in proper interval graphs. Next we tighten the
NP-completeness of DDDP by showing that it remains NP-complete even in chordal
graphs. We also propose a -approximation
algorithm for MDDP in general graphs and prove that MDDP can not be
approximated within for any unless NP
DTIME. Finally, we show that MDDP is
APX-complete for bipartite graphs with maximum degree
On the Complexity of Spill Everywhere under SSA Form
Compilation for embedded processors can be either aggressive (time consuming
cross-compilation) or just in time (embedded and usually dynamic). The
heuristics used in dynamic compilation are highly constrained by limited
resources, time and memory in particular. Recent results on the SSA form open
promising directions for the design of new register allocation heuristics for
embedded systems and especially for embedded compilation. In particular,
heuristics based on tree scan with two separated phases -- one for spilling,
then one for coloring/coalescing -- seem good candidates for designing
memory-friendly, fast, and competitive register allocators. Still, also because
of the side effect on power consumption, the minimization of loads and stores
overhead (spilling problem) is an important issue. This paper provides an
exhaustive study of the complexity of the ``spill everywhere'' problem in the
context of the SSA form. Unfortunately, conversely to our initial hopes, many
of the questions we raised lead to NP-completeness results. We identify some
polynomial cases but that are impractical in JIT context. Nevertheless, they
can give hints to simplify formulations for the design of aggressive
allocators.Comment: 10 page
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