361 research outputs found
Space Saving by Dynamic Algebraization
Dynamic programming is widely used for exact computations based on tree
decompositions of graphs. However, the space complexity is usually exponential
in the treewidth. We study the problem of designing efficient dynamic
programming algorithm based on tree decompositions in polynomial space. We show
how to construct a tree decomposition and extend the algebraic techniques of
Lokshtanov and Nederlof such that the dynamic programming algorithm runs in
time , where is the maximum number of vertices in the union of
bags on the root to leaf paths on a given tree decomposition, which is a
parameter closely related to the tree-depth of a graph. We apply our algorithm
to the problem of counting perfect matchings on grids and show that it
outperforms other polynomial-space solutions. We also apply the algorithm to
other set covering and partitioning problems.Comment: 14 pages, 1 figur
Beyond Hypertree Width: Decomposition Methods Without Decompositions
The general intractability of the constraint satisfaction problem has
motivated the study of restrictions on this problem that permit polynomial-time
solvability. One major line of work has focused on structural restrictions,
which arise from restricting the interaction among constraint scopes. In this
paper, we engage in a mathematical investigation of generalized hypertree
width, a structural measure that has up to recently eluded study. We obtain a
number of computational results, including a simple proof of the tractability
of CSP instances having bounded generalized hypertree width
Tensor network states in time-bin quantum optics
The current shift in the quantum optics community towards large-size
experiments -- with many modes and photons -- necessitates new classical
simulation techniques that go beyond the usual phase space formulation of
quantum mechanics. To address this pressing demand we formulate linear quantum
optics in the language of tensor network states. As a toy model, we extensively
analyze the quantum and classical correlations of time-bin interference in a
single fiber loop. We then generalize our results to more complex time-bin
quantum setups and identify different classes of architectures for
high-complexity and low-overhead boson sampling experiments
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