10 research outputs found
Realizable paths and the NL vs L problem
A celebrated theorem of Savitch [Savitch'70] states that NSPACE(S) is contained in DSPACE(S²). In particular, Savitch gave a deterministic algorithm to solve ST-Connectivity (an NL-complete problem) using O({log}²{n}) space, implying NL (non-deterministic logspace) is contained in DSPACE({log}²{n}). While Savitch's theorem itself has not been improved in the last four decades, several graph connectivity problems are shown to lie between L and NL, providing new insights into the space-bounded complexity classes. All the connectivity problems considered in the literature so far are essentially special cases of ST-Connectivity.
In this dissertation, we initiate the study of auxiliary PDAs as graph connectivity problems and define sixteen different "graph realizability problems" and study their relationships. The complexity of these connectivity problems lie between L (logspace) and P (polynomial time). ST-Realizability, the most general graph realizability problem is P-complete. 1DSTREAL(poly), the most specific graph realizability problem is L-complete. As special cases of our graph realizability problems we define two natural problems, Balanced ST-Connectivity and Positive Balanced ST-Connectivity, that lie between L and NL.
We study the space complexity of SGSLOGCFL, a graph realizability problem lying between L and LOGCFL. We define generalizations of graph squaring and transitive closure, present efficient parallel algorithms for SGSLOGCFL and use the techniques of Trifonov to show that SGSLOGCFL is contained in DSPACE(lognloglogn). This implies that Balanced ST-Connectivity is contained in DSPACE(lognloglogn). We conclude with several interesting new research directions.PhDCommittee Chair: Richard Lipton; Committee Member: Anna Gal; Committee Member: Maria-Florina Balcan; Committee Member: Merrick Furst; Committee Member: William Coo
Three lectures on random proper colorings of
A proper -coloring of a graph is an assignment of one of colors to
each vertex of the graph so that adjacent vertices are colored differently.
Sample uniformly among all proper -colorings of a large discrete cube in the
integer lattice . Does the random coloring obtained exhibit any
large-scale structure? Does it have fast decay of correlations? We discuss
these questions and the way their answers depend on the dimension and the
number of colors . The questions are motivated by statistical physics
(anti-ferromagnetic materials, square ice), combinatorics (proper colorings,
independent sets) and the study of random Lipschitz functions on a lattice. The
discussion introduces a diverse set of tools, useful for this purpose and for
other problems, including spatial mixing, entropy and coupling methods, Gibbs
measures and their classification and refined contour analysis.Comment: 53 pages, 10 figures; Based on lectures given at the workshop on
Random Walks, Random Graphs and Random Media, September 2019, Munich and at
the school Lectures on Probability and Stochastic Processes XIV, December
2019, Delh
Probabilistic methods and coloring problems in graphs
Aquest projecte estĂ dedicat a estudiar el k-èssim nombre cromĂ tic generalitzat que sorgeix de les descomposicions Low Tree--Depth en grafs usant mètodes probabilĂstics.. Una extensiĂł natural del nombre cromĂ tic d'un graf Ă©s l'estudi de particions de grafs en les
que cada i parts indueixen un subgraf amb un cert parĂ metre acotat en funciĂł de i, per exemple cada i parts tenen com a molt i-1 arestes. En particular el nombre cromĂ tic generalitzat Ă©s le mĂnim nombre de parts per tal que cada i parts tĂ© 'treedepth' com a molt i.
Resultats recents proven que grans classes de grafs tenen parĂ metres d'aquest tipus acotats. L'objectiu del projecte Ă©s
(i) fer servie mètodes probabilĂstics per donar cotas ajustades d'aquests parĂ metres i (ii) estudiar el seu valor
per grafs aleatoris
The Complexity of Approximately Counting Retractions
Let be a graph that contains an induced subgraph . A retraction from
to is a homomorphism from to that is the identity function on
. Retractions are very well-studied: Given , the complexity of deciding
whether there is a retraction from an input graph to is completely
classified, in the sense that it is known for which this problem is
tractable (assuming ). Similarly, the complexity of
(exactly) counting retractions from to is classified (assuming
). However, almost nothing is known about
approximately counting retractions. Our first contribution is to give a
complete trichotomy for approximately counting retractions to graphs of girth
at least . Our second contribution is to locate the retraction counting
problem for each in the complexity landscape of related approximate
counting problems. Interestingly, our results are in contrast to the situation
in the exact counting context. We show that the problem of approximately
counting retractions is separated both from the problem of approximately
counting homomorphisms and from the problem of approximately counting list
homomorphisms --- whereas for exact counting all three of these problems are
interreducible. We also show that the number of retractions is at least as hard
to approximate as both the number of surjective homomorphisms and the number of
compactions. In contrast, exactly counting compactions is the hardest of all of
these exact counting problems