67 research outputs found
Deterministically Isolating a Perfect Matching in Bipartite Planar Graphs
We present a deterministic way of assigning small (log bit) weights to the
edges of a bipartite planar graph so that the minimum weight perfect matching
becomes unique. The isolation lemma as described in (Mulmuley et al. 1987)
achieves the same for general graphs using a randomized weighting scheme,
whereas we can do it deterministically when restricted to bipartite planar
graphs. As a consequence, we reduce both decision and construction versions of
the matching problem to testing whether a matrix is singular, under the promise
that its determinant is 0 or 1, thus obtaining a highly parallel SPL algorithm
for bipartite planar graphs. This improves the earlier known bounds of
non-uniform SPL by (Allender et al. 1999) and by (Miller and Naor 1995,
Mahajan and Varadarajan 2000). It also rekindles the hope of obtaining a
deterministic parallel algorithm for constructing a perfect matching in
non-bipartite planar graphs, which has been open for a long time. Our
techniques are elementary and simple
Solving graph problems with single-photons and linear optics
An important challenge for current and near-term quantum devices is finding
useful tasks that can be preformed on them. We first show how to efficiently
encode a bounded matrix into a linear optical circuit with
modes. We then apply this encoding to the case where is a matrix
containing information about a graph . We show that a photonic quantum
processor consisting of single-photon sources, a linear optical circuit
encoding , and single-photon detectors can solve a range of graph problems
including finding the number of perfect matchings of bipartite graphs,
computing permanental polynomials, determining whether two graphs are
isomorphic, and the -densest subgraph problem. We also propose
pre-processing methods to boost the probabilities of observing the relevant
detection events and thus improve performance. Finally, we present various
numerical simulations which validate our findings.Comment: 6 pages + 9 pages appendix. Comments Welcome
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