114,825 research outputs found
A Linear-Optical Proof that the Permanent is #P-Hard
One of the crown jewels of complexity theory is Valiant's 1979 theorem that
computing the permanent of an n*n matrix is #P-hard. Here we show that, by
using the model of linear-optical quantum computing---and in particular, a
universality theorem due to Knill, Laflamme, and Milburn---one can give a
different and arguably more intuitive proof of this theorem.Comment: 12 pages, 2 figures, to appear in Proceedings of the Royal Society A.
doi: 10.1098/rspa.2011.023
Simple Wriggling is Hard unless You Are a Fat Hippo
We prove that it is NP-hard to decide whether two points in a polygonal
domain with holes can be connected by a wire. This implies that finding any
approximation to the shortest path for a long snake amidst polygonal obstacles
is NP-hard. On the positive side, we show that snake's problem is
"length-tractable": if the snake is "fat", i.e., its length/width ratio is
small, the shortest path can be computed in polynomial time.Comment: A shorter version is to be presented at FUN 201
Polynomial algorithms that prove an NP-hard hypothesis implies an NP-hard conclusion
A number of results in Hamiltonian graph theory are of the form implies , where is a property of graphs that is NP-hard and is a cycle structure property of graphs that is also NP-hard. Such a theorem is the well-known Chv\'{a}tal-Erd\"{o}s Theorem, which states that every graph with is Hamiltonian. Here is the vertex connectivity of and is the cardinality of a largest set of independent vertices of . In another paper Chv\'{a}tal points out that the proof of this result is in fact a polynomial time construction that either produces a Hamilton cycle or a set of more than independent vertices. In this note we point out that other theorems in Hamiltonian graph theory have a similar character. In particular, we present a constructive proof of the well-known theorem of Jung for graphs on or more vertices.. \u
Computing the interleaving distance is NP-hard
We show that computing the interleaving distance between two multi-graded
persistence modules is NP-hard. More precisely, we show that deciding whether
two modules are -interleaved is NP-complete, already for bigraded, interval
decomposable modules. Our proof is based on previous work showing that a
constrained matrix invertibility problem can be reduced to the interleaving
distance computation of a special type of persistence modules. We show that
this matrix invertibility problem is NP-complete. We also give a slight
improvement of the above reduction, showing that also the approximation of the
interleaving distance is NP-hard for any approximation factor smaller than .
Additionally, we obtain corresponding hardness results for the case that the
modules are indecomposable, and in the setting of one-sided stability.
Furthermore, we show that checking for injections (resp. surjections) between
persistence modules is NP-hard. In conjunction with earlier results from
computational algebra this gives a complete characterization of the
computational complexity of one-sided stability. Lastly, we show that it is in
general NP-hard to approximate distances induced by noise systems within a
factor of 2.Comment: 25 pages. Several expository improvements and minor corrections. Also
added a section on noise system
Settling the Sample Complexity of Single-parameter Revenue Maximization
This paper settles the sample complexity of single-parameter revenue
maximization by showing matching upper and lower bounds, up to a
poly-logarithmic factor, for all families of value distributions that have been
considered in the literature. The upper bounds are unified under a novel
framework, which builds on the strong revenue monotonicity by Devanur, Huang,
and Psomas (STOC 2016), and an information theoretic argument. This is
fundamentally different from the previous approaches that rely on either
constructing an -net of the mechanism space, explicitly or implicitly
via statistical learning theory, or learning an approximately accurate version
of the virtual values. To our knowledge, it is the first time information
theoretical arguments are used to show sample complexity upper bounds, instead
of lower bounds. Our lower bounds are also unified under a meta construction of
hard instances.Comment: 49 pages, Accepted by STOC1
Minimum-weight triangulation is NP-hard
A triangulation of a planar point set S is a maximal plane straight-line
graph with vertex set S. In the minimum-weight triangulation (MWT) problem, we
are looking for a triangulation of a given point set that minimizes the sum of
the edge lengths. We prove that the decision version of this problem is
NP-hard. We use a reduction from PLANAR-1-IN-3-SAT. The correct working of the
gadgets is established with computer assistance, using dynamic programming on
polygonal faces, as well as the beta-skeleton heuristic to certify that certain
edges belong to the minimum-weight triangulation.Comment: 45 pages (including a technical appendix of 13 pages), 28 figures.
This revision contains a few improvements in the expositio
A Simple Model to Generate Hard Satisfiable Instances
In this paper, we try to further demonstrate that the models of random CSP
instances proposed by [Xu and Li, 2000; 2003] are of theoretical and practical
interest. Indeed, these models, called RB and RD, present several nice
features. First, it is quite easy to generate random instances of any arity
since no particular structure has to be integrated, or property enforced, in
such instances. Then, the existence of an asymptotic phase transition can be
guaranteed while applying a limited restriction on domain size and on
constraint tightness. In that case, a threshold point can be precisely located
and all instances have the guarantee to be hard at the threshold, i.e., to have
an exponential tree-resolution complexity. Next, a formal analysis shows that
it is possible to generate forced satisfiable instances whose hardness is
similar to unforced satisfiable ones. This analysis is supported by some
representative results taken from an intensive experimentation that we have
carried out, using complete and incomplete search methods.Comment: Proc. of 19th IJCAI, pp.337-342, Edinburgh, Scotland, 2005. For more
information, please click
http://www.nlsde.buaa.edu.cn/~kexu/papers/ijcai05-abstract.ht
Inserting one edge into a simple drawing is hard
A simple drawing D(G) of a graph G is one where each pair of edges share at most one point: either a common endpoint or a proper crossing. An edge e in the complement of G can be inserted into D(G) if there exists a simple drawing of G + e extending D(G). As a result of Levi’s Enlargement Lemma, if a drawing is rectilinear (pseudolinear), that is, the edges can be extended into an arrangement of lines (pseudolines), then any edge in the complement of G can be inserted. In contrast, we show that it is NP-complete to decide whether one edge can be inserted into a simple drawing. This remains true even if we assume that the drawing is pseudocircular, that is, the edges can be extended to an arrangement of pseudocircles. On the positive side, we show that, given an arrangement of pseudocircles A and a pseudosegment s, it can be decided in polynomial time whether there exists a pseudocircle Fs extending s for which A ¿ {Fs} is again an arrangement of pseudocircles.Peer ReviewedPostprint (published version
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