91,656 research outputs found
P vs NP
In this paper, we discuss the problem using the Henkin's
Theory and the Herbrand Theory in the first-order logic, and prove that
is a proper subset of
On inefficient special cases of NP-complete problems
AbstractEvery intractable set A has a polynomial complexity core, a set H such that for any P-subset S of A or of Δ, Sβ©H is finite. A complexity core H of A is proper if HβA. It is shown here that if Pβ NP, then every currently known (i.e., either invertibly paddable or k-creative) NP-complete set A and its complement Δ have proper polynomial complexity cores that are nonsparse and are accepted by deterministic machines in time 2cn for some constant c. Turning to the intractable class DEXT=βͺc>0DTIME(2cn), it is shown that every set that is β©½pm-complete for DEXT has an infinite proper polynomial complexity core that is nonsparse and recursive
On W[1]-Hardness as Evidence for Intractability
The central conjecture of parameterized complexity states that FPT !=W[1], and is generally regarded as the parameterized counterpart to P !=NP. We revisit the issue of the plausibility of FPT !=W[1], focusing on two aspects: the difficulty of proving the conjecture (assuming it holds), and how the relation between the two classes might differ from the one between P and NP. Regarding the first aspect, we give new evidence that separating FPT from W[1] would be considerably harder than doing the same for P and NP. Our main result regarding the relation between FPT and W[1] states that the closure of W[1] under relativization with FPT-oracles is precisely the class W[P], implying that either FPT is not low for W[1], or the W-Hierarchy collapses. This theorem also has consequences for the A-Hierarchy (a parameterized version of the Polynomial Hierarchy), namely that unless W[P] is a subset of some level A[t], there are structural differences between the A-Hierarchy and the Polynomial Hierarchy. We also prove that under the unlikely assumption that W[P] collapses to W[1] in a specific way, the collapse of any two consecutive levels of the A-Hierarchy implies the collapse of the entire hierarchy to a finite level; this extends a result of Chen, Flum, and Grohe (2005). Finally, we give weak (oracle-based) evidence that the inclusion W[t]subseteqA[t] is strict for t>1, and that the W-Hierarchy is proper. The latter result answers a question of Downey and Fellows (1993)
Convexity theorems for semisimple symmetric spaces
We prove a remarkable generalization of a convexity theorem for semisimple
symmetric spaces G/H established earlier in 1986 by the second named author.
The latter result generalized Kostant's non-linear convexity theorem for the
Iwasawa decomposition of a real semisimple Lie group.
The present generalization involves Iwasawa decompositions related to minimal
parabolic subgroups of G of arbitrary type instead of the particular type
relative to H considered in 1986.Comment: LaTeX, 52 pages, v2 contains minor corrections and modification
Approximate Hypergraph Coloring under Low-discrepancy and Related Promises
A hypergraph is said to be -colorable if its vertices can be colored
with colors so that no hyperedge is monochromatic. -colorability is a
fundamental property (called Property B) of hypergraphs and is extensively
studied in combinatorics. Algorithmically, however, given a -colorable
-uniform hypergraph, it is NP-hard to find a -coloring miscoloring fewer
than a fraction of hyperedges (which is achieved by a random
-coloring), and the best algorithms to color the hypergraph properly require
colors, approaching the trivial bound of as
increases.
In this work, we study the complexity of approximate hypergraph coloring, for
both the maximization (finding a -coloring with fewest miscolored edges) and
minimization (finding a proper coloring using fewest number of colors)
versions, when the input hypergraph is promised to have the following stronger
properties than -colorability:
(A) Low-discrepancy: If the hypergraph has discrepancy ,
we give an algorithm to color the it with colors.
However, for the maximization version, we prove NP-hardness of finding a
-coloring miscoloring a smaller than (resp. )
fraction of the hyperedges when (resp. ). Assuming
the UGC, we improve the latter hardness factor to for almost
discrepancy- hypergraphs.
(B) Rainbow colorability: If the hypergraph has a -coloring such
that each hyperedge is polychromatic with all these colors, we give a
-coloring algorithm that miscolors at most of the
hyperedges when , and complement this with a matching UG
hardness result showing that when , it is hard to even beat the
bound achieved by a random coloring.Comment: Approx 201
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