14,492 research outputs found
Homomorphism Tensors and Linear Equations
Lov\'asz (1967) showed that two graphs and are isomorphic if and only
if they are homomorphism indistinguishable over the class of all graphs, i.e.
for every graph , the number of homomorphisms from to equals the
number of homomorphisms from to . Recently, homomorphism
indistinguishability over restricted classes of graphs such as bounded
treewidth, bounded treedepth and planar graphs, has emerged as a surprisingly
powerful framework for capturing diverse equivalence relations on graphs
arising from logical equivalence and algebraic equation systems.
In this paper, we provide a unified algebraic framework for such results by
examining the linear-algebraic and representation-theoretic structure of
tensors counting homomorphisms from labelled graphs. The existence of certain
linear transformations between such homomorphism tensor subspaces can be
interpreted both as homomorphism indistinguishability over a graph class and as
feasibility of an equational system. Following this framework, we obtain
characterisations of homomorphism indistinguishability over two natural graph
classes, namely trees of bounded degree and graphs of bounded pathwidth,
answering a question of Dell et al. (2018).Comment: 33 pages, accepted for ICALP 202
The Complexity of Homomorphism Indistinguishability
For every graph class {F}, let HomInd({F}) be the problem of deciding whether two given graphs are homomorphism-indistinguishable over {F}, i.e., for every graph F in {F}, the number hom(F, G) of homomorphisms from F to G equals the corresponding number hom(F, H) for H. For several natural graph classes (such as paths, trees, bounded treewidth graphs), homomorphism-indistinguishability over the class has an efficient structural characterization, resulting in polynomial time solvability [H. Dell et al., 2018].
In particular, it is known that two non-isomorphic graphs are homomorphism-indistinguishable over the class {T}_k of graphs of treewidth k if and only if they are not distinguished by k-dimensional Weisfeiler-Leman algorithm, a central heuristic for isomorphism testing: this characterization implies a polynomial time algorithm for HomInd({T}_k), for every fixed k in N. In this paper, we show that there is a polynomial-time-decidable class {F} of undirected graphs of bounded treewidth such that HomInd({F}) is undecidable.
Our second hardness result concerns the class {K} of complete graphs. We show that HomInd({K}) is co-NP-hard, and in fact, we show completeness for the class C_=P (under P-time Turing reductions). On the algorithmic side, we show that HomInd({P}) can be solved in polynomial time for the class {P} of directed paths. We end with a brief study of two variants of the HomInd({F}) problem: (a) the problem of lexographic-comparison of homomorphism numbers of two graphs, and (b) the problem of computing certain distance-measures (defined via homomorphism numbers) between two graphs
Antimatroids and Balanced Pairs
We generalize the 1/3-2/3 conjecture from partially ordered sets to
antimatroids: we conjecture that any antimatroid has a pair of elements x,y
such that x has probability between 1/3 and 2/3 of appearing earlier than y in
a uniformly random basic word of the antimatroid. We prove the conjecture for
antimatroids of convex dimension two (the antimatroid-theoretic analogue of
partial orders of width two), for antimatroids of height two, for antimatroids
with an independent element, and for the perfect elimination antimatroids and
node search antimatroids of several classes of graphs. A computer search shows
that the conjecture is true for all antimatroids with at most six elements.Comment: 16 pages, 5 figure
Detecting Activations over Graphs using Spanning Tree Wavelet Bases
We consider the detection of activations over graphs under Gaussian noise,
where signals are piece-wise constant over the graph. Despite the wide
applicability of such a detection algorithm, there has been little success in
the development of computationally feasible methods with proveable theoretical
guarantees for general graph topologies. We cast this as a hypothesis testing
problem, and first provide a universal necessary condition for asymptotic
distinguishability of the null and alternative hypotheses. We then introduce
the spanning tree wavelet basis over graphs, a localized basis that reflects
the topology of the graph, and prove that for any spanning tree, this approach
can distinguish null from alternative in a low signal-to-noise regime. Lastly,
we improve on this result and show that using the uniform spanning tree in the
basis construction yields a randomized test with stronger theoretical
guarantees that in many cases matches our necessary conditions. Specifically,
we obtain near-optimal performance in edge transitive graphs, -nearest
neighbor graphs, and -graphs
Relative Expressive Power of Navigational Querying on Graphs
Motivated by both established and new applications, we study navigational
query languages for graphs (binary relations). The simplest language has only
the two operators union and composition, together with the identity relation.
We make more powerful languages by adding any of the following operators:
intersection; set difference; projection; coprojection; converse; and the
diversity relation. All these operators map binary relations to binary
relations. We compare the expressive power of all resulting languages. We do
this not only for general path queries (queries where the result may be any
binary relation) but also for boolean or yes/no queries (expressed by the
nonemptiness of an expression). For both cases, we present the complete Hasse
diagram of relative expressiveness. In particular the Hasse diagram for boolean
queries contains some nontrivial separations and a few surprising collapses.Comment: An extended abstract announcing the results of this paper was
presented at the 14th International Conference on Database Theory, Uppsala,
Sweden, March 201
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