7,605 research outputs found
Optimal Vertex Fault Tolerant Spanners (for fixed stretch)
A -spanner of a graph is a sparse subgraph whose shortest path
distances match those of up to a multiplicative error . In this paper we
study spanners that are resistant to faults. A subgraph is an
vertex fault tolerant (VFT) -spanner if is a -spanner
of for any small set of vertices that might "fail." One
of the main questions in the area is: what is the minimum size of an fault
tolerant -spanner that holds for all node graphs (as a function of ,
and )? This question was first studied in the context of geometric
graphs [Levcopoulos et al. STOC '98, Czumaj and Zhao SoCG '03] and has more
recently been considered in general undirected graphs [Chechik et al. STOC '09,
Dinitz and Krauthgamer PODC '11].
In this paper, we settle the question of the optimal size of a VFT spanner,
in the setting where the stretch factor is fixed. Specifically, we prove
that every (undirected, possibly weighted) -node graph has a
-spanner resilient to vertex faults with edges, and this is fully optimal (unless the famous Erdos Girth
Conjecture is false). Our lower bound even generalizes to imply that no data
structure capable of approximating similarly can
beat the space usage of our spanner in the worst case. We also consider the
edge fault tolerant (EFT) model, defined analogously with edge failures rather
than vertex failures. We show that the same spanner upper bound applies in this
setting. Our data structure lower bound extends to the case (and hence we
close the EFT problem for -approximations), but it falls to for . We leave it as an open problem to
close this gap.Comment: To appear in SODA 201
Subgraph complementation and minimum rank
Any finite simple graph can be represented by a collection
of subsets of such that if and only if and
appear together in an odd number of sets in . Let denote
the minimum cardinality of such a collection. This invariant is equivalent to
the minimum dimension of a faithful orthogonal representation of over
and is closely connected to the minimum rank of . We show
that when
is odd, or when is a forest. Otherwise,
. Furthermore, we show that the following
are equivalent for any graph with at least one edge: i.
; ii. the adjacency matrix of
is the unique matrix of rank which fits
over ; iii. there is a minimum collection as
described in which every vertex appears an even number of times; and iv. for
every component of , . We also show that, for these graphs, is
twice the minimum number of tricliques whose symmetric difference of edge sets
is . Additionally, we provide a set of upper bounds on in terms of
the order, size, and vertex cover number of . Finally, we show that the
class of graphs with is hereditary and finitely defined. For odd
, the sets of minimal forbidden induced subgraphs are the same as those for
the property , and we exhibit this set
for
How unproportional must a graph be?
Let be the maximum over all -vertex graphs of by how much
the number of induced copies of in differs from its expectation in the
binomial random graph with the same number of vertices as and with edge
probability . This may be viewed as a measure of how close is to being
-quasirandom. For a positive integer and , let be the
distance from to the nearest integer. Our main result is that,
for fixed and for large, the minimum of over -vertex
graphs has order of magnitude
provided that
On Minrank and Forbidden Subgraphs
The minrank over a field of a graph on the vertex set
is the minimum possible rank of a matrix such that for every , and
for every distinct non-adjacent vertices and in . For an
integer , a graph , and a field , let
denote the maximum possible minrank over of an -vertex graph
whose complement contains no copy of . In this paper we study this quantity
for various graphs and fields . For finite fields, we prove by
a probabilistic argument a general lower bound on , which
yields a nearly tight bound of for the triangle
. For the real field, we prove by an explicit construction that for
every non-bipartite graph , for some
. As a by-product of this construction, we disprove a
conjecture of Codenotti, Pudl\'ak, and Resta. The results are motivated by
questions in information theory, circuit complexity, and geometry.Comment: 15 page
Finding Large H-Colorable Subgraphs in Hereditary Graph Classes
We study the \textsc{Max Partial -Coloring} problem: given a graph ,
find the largest induced subgraph of that admits a homomorphism into ,
where is a fixed pattern graph without loops. Note that when is a
complete graph on vertices, the problem reduces to finding the largest
induced -colorable subgraph, which for is equivalent (by
complementation) to \textsc{Odd Cycle Transversal}.
We prove that for every fixed pattern graph without loops, \textsc{Max
Partial -Coloring} can be solved:
in -free graphs in polynomial time, whenever is a
threshold graph;
in -free graphs in polynomial time;
in -free graphs in time ;
in -free graphs in time
.
Here, is the number of vertices of the input graph and is
the maximum size of a clique in~. Furthermore, combining the mentioned
algorithms for -free and for -free
graphs with a simple branching procedure, we obtain subexponential-time
algorithms for \textsc{Max Partial -Coloring} in these classes of graphs.
Finally, we show that even a restricted variant of \textsc{Max Partial
-Coloring} is -hard in the considered subclasses of -free
graphs, if we allow loops on
On the decomposition threshold of a given graph
We study the -decomposition threshold for a given graph .
Here an -decomposition of a graph is a collection of edge-disjoint
copies of in which together cover every edge of . (Such an
-decomposition can only exist if is -divisible, i.e. if and each vertex degree of can be expressed as a linear combination of
the vertex degrees of .)
The -decomposition threshold is the smallest value ensuring
that an -divisible graph on vertices with
has an -decomposition. Our main results imply
the following for a given graph , where is the fractional
version of and :
(i) ;
(ii) if , then
;
(iii) we determine if is bipartite.
In particular, (i) implies that . Our proof
involves further developments of the recent `iterative' absorbing approach.Comment: Final version, to appear in the Journal of Combinatorial Theory,
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