155 research outputs found
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
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
Learning and Testing Variable Partitions
Let be a multivariate function from a product set to an
Abelian group . A -partition of with cost is a partition of
the set of variables into non-empty subsets such that is -close to
for some with
respect to a given error metric. We study algorithms for agnostically learning
partitions and testing -partitionability over various groups and error
metrics given query access to . In particular we show that
Given a function that has a -partition of cost , a partition
of cost can be learned in time
for any .
In contrast, for and learning a partition of cost is NP-hard.
When is real-valued and the error metric is the 2-norm, a
2-partition of cost can be learned in time
.
When is -valued and the error metric is Hamming
weight, -partitionability is testable with one-sided error and
non-adaptive queries. We also show that even
two-sided testers require queries when .
This work was motivated by reinforcement learning control tasks in which the
set of control variables can be partitioned. The partitioning reduces the task
into multiple lower-dimensional ones that are relatively easier to learn. Our
second algorithm empirically increases the scores attained over previous
heuristic partitioning methods applied in this context.Comment: Innovations in Theoretical Computer Science (ITCS) 202
Maximizing Symmetric Submodular Functions
Symmetric submodular functions are an important family of submodular
functions capturing many interesting cases including cut functions of graphs
and hypergraphs. Maximization of such functions subject to various constraints
receives little attention by current research, unlike similar minimization
problems which have been widely studied. In this work, we identify a few
submodular maximization problems for which one can get a better approximation
for symmetric objectives than the state of the art approximation for general
submodular functions.
We first consider the problem of maximizing a non-negative symmetric
submodular function subject to a
down-monotone solvable polytope . For
this problem we describe an algorithm producing a fractional solution of value
at least , where is the optimal integral solution.
Our second result considers the problem for a
non-negative symmetric submodular function . For this problem, we give an approximation ratio that depends on
the value and is always at least . Our method can
also be applied to non-negative non-symmetric submodular functions, in which
case it produces approximation, improving over the best known
result for this problem. For unconstrained maximization of a non-negative
symmetric submodular function we describe a deterministic linear-time
-approximation algorithm. Finally, we give a -approximation algorithm for Submodular Welfare with players having
identical non-negative submodular utility functions, and show that this is the
best possible approximation ratio for the problem.Comment: 31 pages, an extended abstract appeared in ESA 201
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