50 research outputs found
Counting and enumerating optimum cut sets for hypergraph -partitioning problems for fixed
We consider the problem of enumerating optimal solutions for two hypergraph
-partitioning problems -- namely, Hypergraph--Cut and
Minmax-Hypergraph--Partition. The input in hypergraph -partitioning
problems is a hypergraph with positive hyperedge costs along with a
fixed positive integer . The goal is to find a partition of into
non-empty parts -- known as a -partition -- so as
to minimize an objective of interest.
1. If the objective of interest is the maximum cut value of the parts, then
the problem is known as Minmax-Hypergraph--Partition. A subset of hyperedges
is a minmax--cut-set if it is the subset of hyperedges crossing an optimum
-partition for Minmax-Hypergraph--Partition.
2. If the objective of interest is the total cost of hyperedges crossing the
-partition, then the problem is known as Hypergraph--Cut. A subset of
hyperedges is a min--cut-set if it is the subset of hyperedges crossing an
optimum -partition for Hypergraph--Cut.
We give the first polynomial bound on the number of minmax--cut-sets and a
polynomial-time algorithm to enumerate all of them in hypergraphs for every
fixed . Our technique is strong enough to also enable an -time
deterministic algorithm to enumerate all min--cut-sets in hypergraphs, thus
improving on the previously known -time deterministic algorithm,
where is the number of vertices and is the size of the hypergraph. The
correctness analysis of our enumeration approach relies on a structural result
that is a strong and unifying generalization of known structural results for
Hypergraph--Cut and Minmax-Hypergraph--Partition. We believe that our
structural result is likely to be of independent interest in the theory of
hypergraphs (and graphs).Comment: Accepted to ICALP'22. arXiv admin note: text overlap with
arXiv:2110.1481
Hypergraphs with Edge-Dependent Vertex Weights: p-Laplacians and Spectral Clustering
We study p-Laplacians and spectral clustering for a recently proposed
hypergraph model that incorporates edge-dependent vertex weights (EDVW). These
weights can reflect different importance of vertices within a hyperedge, thus
conferring the hypergraph model higher expressivity and flexibility. By
constructing submodular EDVW-based splitting functions, we convert hypergraphs
with EDVW into submodular hypergraphs for which the spectral theory is better
developed. In this way, existing concepts and theorems such as p-Laplacians and
Cheeger inequalities proposed under the submodular hypergraph setting can be
directly extended to hypergraphs with EDVW. For submodular hypergraphs with
EDVW-based splitting functions, we propose an efficient algorithm to compute
the eigenvector associated with the second smallest eigenvalue of the
hypergraph 1-Laplacian. We then utilize this eigenvector to cluster the
vertices, achieving higher clustering accuracy than traditional spectral
clustering based on the 2-Laplacian. More broadly, the proposed algorithm works
for all submodular hypergraphs that are graph reducible. Numerical experiments
using real-world data demonstrate the effectiveness of combining spectral
clustering based on the 1-Laplacian and EDVW
Hardness of submodular cost allocation : lattice matching and a simplex coloring conjecture
We consider the Minimum Submodular Cost Allocation (MSCA) problem. In this problem, we are given k submodular cost functions f1, ... , fk: 2V -> R+ and the goal is to partition V into k sets A1, ..., Ak so as to minimize the total cost sumi = 1,k fi(Ai). We show that MSCA is inapproximable within any multiplicative factor even in very restricted settings; prior to our work, only Set Cover hardness was known. In light of this negative result, we turn our attention to special cases of the problem. We consider the setting in which each function fi satisfies fi = gi + h, where each gi is monotone submodular and h is (possibly non-monotone) submodular. We give an O(k log |V|) approximation for this problem. We provide some evidence that a factor of k may be necessary, even in the special case of HyperLabel. In particular, we formulate a simplex-coloring conjecture that implies a Unique-Games-hardness of (k - 1 - epsilon) for k-uniform HyperLabel and label set [k]. We provide a proof of the simplex-coloring conjecture for k=3
Hypergraph Diffusions and Resolvents for Norm-Based Hypergraph Laplacians
The development of simple and fast hypergraph spectral methods has been
hindered by the lack of numerical algorithms for simulating heat diffusions and
computing fundamental objects, such as Personalized PageRank vectors, over
hypergraphs. In this paper, we overcome this challenge by designing two novel
algorithmic primitives. The first is a simple, easy-to-compute discrete-time
heat diffusion that enjoys the same favorable properties as the discrete-time
heat diffusion over graphs. This diffusion can be directly applied to speed up
existing hypergraph partitioning algorithms.
Our second contribution is the novel application of mirror descent to compute
resolvents of non-differentiable squared norms, which we believe to be of
independent interest beyond hypergraph problems. Based on this new primitive,
we derive the first nearly-linear-time algorithm that simulates the
discrete-time heat diffusion to approximately compute resolvents of the
hypergraph Laplacian operator, which include Personalized PageRank vectors and
solutions to the hypergraph analogue of Laplacian systems. Our algorithm runs
in time that is linear in the size of the hypergraph and inversely proportional
to the hypergraph spectral gap , matching the complexity of
analogous diffusion-based algorithms for the graph version of the problem
Hardness of Submodular Cost Allocation: Lattice Matching and a Simplex Coloring Conjecture
We consider the Minimum Submodular Cost Allocation (MSCA) problem.
In this problem, we are given k submodular cost functions f_1, ... ,
f_k: 2^V -> R_+ and the goal is to partition V into k sets A_1, ...,
A_k so as to minimize the total cost sum_{i = 1}^k f_i(A_i). We show
that MSCA is inapproximable within any multiplicative factor even in
very restricted settings; prior to our work, only Set Cover hardness
was known. In light of this negative result, we turn our attention
to special cases of the problem. We consider the setting in which
each function f_i satisfies f_i = g_i + h, where each g_i is monotone
submodular and h is (possibly non-monotone) submodular. We give an
O(k log |V|) approximation for this problem. We provide some evidence
that a factor of k may be necessary, even in the special case of
HyperLabel. In particular, we formulate a simplex-coloring
conjecture that implies a Unique-Games-hardness of (k - 1 - epsilon)
for k-uniform HyperLabel and label set [k]. We provide a proof of the
simplex-coloring conjecture for k=3
Approximating submodular -partition via principal partition sequence
In submodular -partition, the input is a non-negative submodular function
defined over a finite ground set (given by an evaluation oracle) along
with a positive integer and the goal is to find a partition of the ground
set into non-empty parts in order to minimize
. Narayanan, Roy, and Patkar (Journal of Algorithms, 1996)
designed an algorithm for submodular -partition based on the principal
partition sequence and showed that the approximation factor of their algorithm
is for the special case of graph cut functions (subsequently rediscovered
by Ravi and Sinha (Journal of Operational Research, 2008)). In this work, we
study the approximation factor of their algorithm for three subfamilies of
submodular functions -- monotone, symmetric, and posimodular, and show the
following results:
1. The approximation factor of their algorithm for monotone submodular
-partition is . This result improves on the -factor achievable via
other algorithms. Moreover, our upper bound of matches the recently shown
lower bound under polynomial number of function evaluation queries (Santiago,
IWOCA 2021). Our upper bound of is also the first improvement beyond
for a certain graph partitioning problem that is a special case of monotone
submodular -partition.
2. The approximation factor of their algorithm for symmetric submodular
-partition is . This result generalizes their approximation factor
analysis beyond graph cut functions.
3. The approximation factor of their algorithm for posimodular submodular
-partition is .
We also construct an example to show that the approximation factor of their
algorithm for arbitrary submodular functions is .Comment: Accepted to APPROX'2