1,912 research outputs found
Knapsack Problems in Groups
We generalize the classical knapsack and subset sum problems to arbitrary
groups and study the computational complexity of these new problems. We show
that these problems, as well as the bounded submonoid membership problem, are
P-time decidable in hyperbolic groups and give various examples of finitely
presented groups where the subset sum problem is NP-complete.Comment: 28 pages, 12 figure
Knapsack problems in products of groups
The classic knapsack and related problems have natural generalizations to
arbitrary (non-commutative) groups, collectively called knapsack-type problems
in groups. We study the effect of free and direct products on their time
complexity. We show that free products in certain sense preserve time
complexity of knapsack-type problems, while direct products may amplify it. Our
methods allow to obtain complexity results for rational subset membership
problem in amalgamated free products over finite subgroups.Comment: 15 pages, 5 figures. Updated to include more general results, mostly
in Section
Collapsing Superstring Conjecture
In the Shortest Common Superstring (SCS) problem, one is given a collection of strings, and needs to find a shortest string containing each of them as a substring. SCS admits 2 11/23-approximation in polynomial time (Mucha, SODA\u2713). While this algorithm and its analysis are technically involved, the 30 years old Greedy Conjecture claims that the trivial and efficient Greedy Algorithm gives a 2-approximation for SCS.
We develop a graph-theoretic framework for studying approximation algorithms for SCS. The framework is reminiscent of the classical 2-approximation for Traveling Salesman: take two copies of an optimal solution, apply a trivial edge-collapsing procedure, and get an approximate solution. In this framework, we observe two surprising properties of SCS solutions, and we conjecture that they hold for all input instances. The first conjecture, that we call Collapsing Superstring conjecture, claims that there is an elementary way to transform any solution repeated twice into the same graph G. This conjecture would give an elementary 2-approximate algorithm for SCS. The second conjecture claims that not only the resulting graph G is the same for all solutions, but that G can be computed by an elementary greedy procedure called Greedy Hierarchical Algorithm.
While the second conjecture clearly implies the first one, perhaps surprisingly we prove their equivalence. We support these equivalent conjectures by giving a proof for the special case where all input strings have length at most 3 (which until recently had been the only case where the Greedy Conjecture was proven). We also tested our conjectures on millions of instances of SCS.
We prove that the standard Greedy Conjecture implies Greedy Hierarchical Conjecture, while the latter is sufficient for an efficient greedy 2-approximate approximation of SCS. Except for its (conjectured) good approximation ratio, the Greedy Hierarchical Algorithm provably finds a 3.5-approximation, and finds exact solutions for the special cases where we know polynomial time (not greedy) exact algorithms: (1) when the input strings form a spectrum of a string (2) when all input strings have length at most 2
On subset sum problem in branch groups
We consider a group-theoretic analogue of the classic subset sum problem. In
this brief note, we show that the subset sum problem is NP-complete in the
first Grigorchuk group. More generally, we show NP-hardness of that problem in
weakly regular branch groups, which implies NP-completeness if the group is, in
addition, contracting.Comment: v3: final version for journal of Groups, Complexity, Cryptology.
arXiv admin note: text overlap with arXiv:1703.0740
Nanoskyrmion engineering with -electron materials: Sn monolayer on SiC(0001) surface
Materials with -magnetism demonstrate strongly nonlocal Coulomb
interactions, which opens a way to probe correlations in the regimes not
achievable in transition metal compounds. By the example of Sn monolayer on
SiC(0001) surface, we show that such systems exhibit unusual but intriguing
magnetic properties at the nanoscale. Physically, this is attributed to the
presence of a significant ferromagnetic coupling, the so-called direct
exchange, which fully compensates ubiquitous antiferromagnetic interactions of
the superexchange origin. Having a nonlocal nature, the direct exchange was
previously ignored because it cannot be captured within the conventional
density functional methods and significantly challenges ground state models
earlier proposed for Sn/SiC(0001). Furthermore, heavy adatoms induce strong
spin-orbit coupling, which leads to a highly anisotropic form of the spin
Hamiltonian, in which the Dzyaloshinskii-Moriya interaction is dominant. The
latter is suggested to be responsible for the formation of a nanoskyrmion state
at realistic magnetic fields and temperatures.Comment: 4 pages, supplemental materia
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