396 research outputs found
Hypergraph Connectivity Augmentation in Strongly Polynomial Time
We consider hypergraph network design problems where the goal is to construct
a hypergraph that satisfies certain connectivity requirements. For graph
network design problems where the goal is to construct a graph that satisfies
certain connectivity requirements, the number of edges in every feasible
solution is at most quadratic in the number of vertices. In contrast, for
hypergraph network design problems, we might have feasible solutions in which
the number of hyperedges is exponential in the number of vertices. This
presents an additional technical challenge in hypergraph network design
problems compared to graph network design problems: in order to solve the
problem in polynomial time, we first need to show that there exists a feasible
solution in which the number of hyperedges is polynomial in the input size.
The central theme of this work is to show that certain hypergraph network
design problems admit solutions in which the number of hyperedges is polynomial
in the number of vertices and moreover, can be solved in strongly polynomial
time. Our work improves on the previous fastest pseudo-polynomial run-time for
these problems. In addition, we develop strongly polynomial time algorithms
that return near-uniform hypergraphs as solutions (i.e., every pair of
hyperedges differ in size by at most one). As applications of our results, we
derive the first strongly polynomial time algorithms for (i) degree-specified
hypergraph connectivity augmentation using hyperedges, (ii) degree-specified
hypergraph node-to-area connectivity augmentation using hyperedges, and (iii)
degree-constrained mixed-hypergraph connectivity augmentation using hyperedges.Comment: arXiv admin note: substantial text overlap with arXiv:2307.0855
Non-Uniform Robust Network Design in Planar Graphs
Robust optimization is concerned with constructing solutions that remain
feasible also when a limited number of resources is removed from the solution.
Most studies of robust combinatorial optimization to date made the assumption
that every resource is equally vulnerable, and that the set of scenarios is
implicitly given by a single budget constraint. This paper studies a robustness
model of a different kind. We focus on \textbf{bulk-robustness}, a model
recently introduced~\cite{bulk} for addressing the need to model non-uniform
failure patterns in systems.
We significantly extend the techniques used in~\cite{bulk} to design
approximation algorithm for bulk-robust network design problems in planar
graphs. Our techniques use an augmentation framework, combined with linear
programming (LP) rounding that depends on a planar embedding of the input
graph. A connection to cut covering problems and the dominating set problem in
circle graphs is established. Our methods use few of the specifics of
bulk-robust optimization, hence it is conceivable that they can be adapted to
solve other robust network design problems.Comment: 17 pages, 2 figure
Approximating Source Location and Star Survivable Network Problems
In Source Location (SL) problems the goal is to select a mini-mum cost source
set such that the connectivity (or flow) from
to any node is at least the demand of . In many SL problems
if , namely, the demand of nodes selected to is
completely satisfied. In a node-connectivity variant suggested recently by
Fukunaga, every node gets a "bonus" if it is selected to
. Fukunaga showed that for undirected graphs one can achieve ratio for his variant, where is the maximum demand. We
improve this by achieving ratio \min\{p^*\lnk,k\}\cdot O(\ln (k/q^*)) for a
more general version with node capacities, where is
the maximum bonus and is the minimum capacity. In
particular, for the most natural case considered by Fukunaga, we
improve the ratio from to . We also get ratio
for the edge-connectivity version, for which no ratio that depends on only
was known before. To derive these results, we consider a particular case of the
Survivable Network (SN) problem when all edges of positive cost form a star. We
give ratio for this variant, improving over the best
ratio known for the general case of Chuzhoy and Khanna
Generalized Kneser coloring theorems with combinatorial proofs
The Kneser conjecture (1955) was proved by Lov\'asz (1978) using the
Borsuk-Ulam theorem; all subsequent proofs, extensions and generalizations also
relied on Algebraic Topology results, namely the Borsuk-Ulam theorem and its
extensions. Only in 2000, Matou\v{s}ek provided the first combinatorial proof
of the Kneser conjecture.
Here we provide a hypergraph coloring theorem, with a combinatorial proof,
which has as special cases the Kneser conjecture as well as its extensions and
generalization by (hyper)graph coloring theorems of Dol'nikov,
Alon-Frankl-Lov\'asz, Sarkaria, and Kriz. We also give a combinatorial proof of
Schrijver's theorem.Comment: 19 pages, 4 figure
Approximating subset -connectivity problems
A subset of terminals is -connected to a root in a
directed/undirected graph if has internally-disjoint -paths for
every ; is -connected in if is -connected to every
. We consider the {\sf Subset -Connectivity Augmentation} problem:
given a graph with edge/node-costs, node subset , and
a subgraph of such that is -connected in , find a
minimum-cost augmenting edge-set such that is
-connected in . The problem admits trivial ratio .
We consider the case and prove that for directed/undirected graphs and
edge/node-costs, a -approximation for {\sf Rooted Subset -Connectivity
Augmentation} implies the following ratios for {\sf Subset -Connectivity
Augmentation}: (i) ; (ii) , where
b=1 for undirected graphs and b=2 for directed graphs, and is the th
harmonic number. The best known values of on undirected graphs are
for edge-costs and for
node-costs; for directed graphs for both versions. Our results imply
that unless , {\sf Subset -Connectivity Augmentation} admits
the same ratios as the best known ones for the rooted version. This improves
the ratios in \cite{N-focs,L}
Decoding Across the Quantum LDPC Code Landscape
We show that belief propagation combined with ordered statistics
post-processing is a general decoder for quantum low density parity check codes
constructed from the hypergraph product. To this end, we run numerical
simulations of the decoder applied to three families of hypergraph product
code: topological codes, fixed-rate random codes and a new class of codes that
we call semi-topological codes. Our new code families share properties of both
topological and random hypergraph product codes, with a construction that
allows for a finely-controlled trade-off between code threshold and stabilizer
locality. Our results indicate thresholds across all three families of
hypergraph product code, and provide evidence of exponential suppression in the
low error regime. For the Toric code, we observe a threshold in the range
. This result improves upon previous quantum decoders based on
belief propagation, and approaches the performance of the minimum weight
perfect matching algorithm. We expect semi-topological codes to have the same
threshold as Toric codes, as they are identical in the bulk, and we present
numerical evidence supporting this observation.Comment: The code for the BP+OSD decoder used in this work can be found on
Github: https://github.com/quantumgizmos/bp_os
Covering symmetric skew-supermodular functions with hyperedges
In this paper we give results related to a theorem of Szigeti that concerns
the covering of symmetric skew-supermodular set functions with hyperedges of
minimum total size. In particular, we show the following generalization using a
variation of Schrijver’s supermodular colouring theorem: if p1 and p2 are skewsupermodular
functions whose maximum value is the same, then it is possible to
find in polynomial time a hypergraph of minimum total size that covers both of
them. Note that without the assumption on the maximum values this problem
is NP-hard. The result has applications concerning the local edge-connectivity
augmentation problem of hypergraphs and the global edge-connectivity augmentation
problem of mixed hypergraphs. We also present some results on the case
when the hypergraph must be obtained by merging given hyperedges
- …