7,478 research outputs found
Lower Bounds for the Graph Homomorphism Problem
The graph homomorphism problem (HOM) asks whether the vertices of a given
-vertex graph can be mapped to the vertices of a given -vertex graph
such that each edge of is mapped to an edge of . The problem
generalizes the graph coloring problem and at the same time can be viewed as a
special case of the -CSP problem. In this paper, we prove several lower
bound for HOM under the Exponential Time Hypothesis (ETH) assumption. The main
result is a lower bound .
This rules out the existence of a single-exponential algorithm and shows that
the trivial upper bound is almost asymptotically
tight.
We also investigate what properties of graphs and make it difficult
to solve HOM. An easy observation is that an upper
bound can be improved to where
is the minimum size of a vertex cover of . The second
lower bound shows that the upper bound is
asymptotically tight. As to the properties of the "right-hand side" graph ,
it is known that HOM can be solved in time and
where is the maximum degree of
and is the treewidth of . This gives
single-exponential algorithms for graphs of bounded maximum degree or bounded
treewidth. Since the chromatic number does not exceed
and , it is natural to ask whether similar
upper bounds with respect to can be obtained. We provide a negative
answer to this question by establishing a lower bound for any
function . We also observe that similar lower bounds can be obtained for
locally injective homomorphisms.Comment: 19 page
On Index Coding and Graph Homomorphism
In this work, we study the problem of index coding from graph homomorphism
perspective. We show that the minimum broadcast rate of an index coding problem
for different variations of the problem such as non-linear, scalar, and vector
index code, can be upper bounded by the minimum broadcast rate of another index
coding problem when there exists a homomorphism from the complement of the side
information graph of the first problem to that of the second problem. As a
result, we show that several upper bounds on scalar and vector index code
problem are special cases of one of our main theorems.
For the linear scalar index coding problem, it has been shown in [1] that the
binary linear index of a graph is equal to a graph theoretical parameter called
minrank of the graph. For undirected graphs, in [2] it is shown that
if and only if there exists a homomorphism from
to a predefined graph . Combining these two results, it
follows that for undirected graphs, all the digraphs with linear index of at
most k coincide with the graphs for which there exists a homomorphism from
to . In this paper, we give a direct proof to this result
that works for digraphs as well.
We show how to use this classification result to generate lower bounds on
scalar and vector index. In particular, we provide a lower bound for the scalar
index of a digraph in terms of the chromatic number of its complement.
Using our framework, we show that by changing the field size, linear index of
a digraph can be at most increased by a factor that is independent from the
number of the nodes.Comment: 5 pages, to appear in "IEEE Information Theory Workshop", 201
Full Complexity Classification of the List Homomorphism Problem for Bounded-Treewidth Graphs
A homomorphism from a graph G to a graph H is an edge-preserving mapping from V(G) to V(H). Let H be a fixed graph with possible loops. In the list homomorphism problem, denoted by LHom(H), we are given a graph G, whose every vertex v is assigned with a list L(v) of vertices of H. We ask whether there exists a homomorphism h from G to H, which respects lists L, i.e., for every v ? V(G) it holds that h(v) ? L(v).
The complexity dichotomy for LHom(H) was proven by Feder, Hell, and Huang [JGT 2003]. The authors showed that the problem is polynomial-time solvable if H belongs to the class called bi-arc graphs, and for all other graphs H it is NP-complete.
We are interested in the complexity of the LHom(H) problem, parameterized by the treewidth of the input graph. This problem was investigated by Egri, Marx, and Rz??ewski [STACS 2018], who obtained tight complexity bounds for the special case of reflexive graphs H, i.e., if every vertex has a loop.
In this paper we extend and generalize their results for all relevant graphs H, i.e., those, for which the LHom(H) problem is NP-hard. For every such H we find a constant k = k(H), such that the LHom(H) problem on instances G with n vertices and treewidth t
- can be solved in time k^t ? n^?(1), provided that G is given along with a tree decomposition of width t,
- cannot be solved in time (k-?)^t ? n^?(1), for any ? > 0, unless the SETH fails. For some graphs H the value of k(H) is much smaller than the trivial upper bound, i.e., |V(H)|.
Obtaining matching upper and lower bounds shows that the set of algorithmic tools that we have discovered cannot be extended in order to obtain faster algorithms for LHom(H) in bounded-treewidth graphs. Furthermore, neither the algorithm, nor the proof of the lower bound, is very specific to treewidth. We believe that they can be used for other variants of the LHom(H) problem, e.g. with different parameterizations
Sparsification Lower Bounds for List H-Coloring
We investigate the List H-Coloring problem, the generalization of graph coloring that asks whether an input graph G admits a homomorphism to the undirected graph H (possibly with loops), such that each vertex v ? V(G) is mapped to a vertex on its list L(v) ? V(H). An important result by Feder, Hell, and Huang [JGT 2003] states that List H-Coloring is polynomial-time solvable if H is a so-called bi-arc graph, and NP-complete otherwise. We investigate the NP-complete cases of the problem from the perspective of polynomial-time sparsification: can an n-vertex instance be efficiently reduced to an equivalent instance of bitsize ?(n^(2-?)) for some ? > 0? We prove that if H is not a bi-arc graph, then List H-Coloring does not admit such a sparsification algorithm unless NP ? coNP/poly. Our proofs combine techniques from kernelization lower bounds with a study of the structure of graphs H which are not bi-arc graphs
Quantum Query Complexity of Subgraph Isomorphism and Homomorphism
Let be a fixed graph on vertices. Let iff the input
graph on vertices contains as a (not necessarily induced) subgraph.
Let denote the cardinality of a maximum independent set of . In
this paper we show:
where
denotes the quantum query complexity of .
As a consequence we obtain a lower bounds for in terms of several
other parameters of such as the average degree, minimum vertex cover,
chromatic number, and the critical probability.
We also use the above bound to show that for any
, improving on the previously best known bound of . Until
very recently, it was believed that the quantum query complexity is at least
square root of the randomized one. Our bound for
matches the square root of the current best known bound for the randomized
query complexity of , which is due to Gr\"oger.
Interestingly, the randomized bound of for
still remains open.
We also study the Subgraph Homomorphism Problem, denoted by , and
show that .
Finally we extend our results to the -uniform hypergraphs. In particular,
we show an bound for quantum query complexity of the Subgraph
Isomorphism, improving on the previously known bound. For the
Subgraph Homomorphism, we obtain an bound for the same.Comment: 16 pages, 2 figure
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