2,816 research outputs found
A Local Search Algorithm for the Min-Sum Submodular Cover Problem
We consider the problem of solving the Min-Sum Submodular Cover problem using
local search. The Min-Sum Submodular Cover problem generalizes the NP-complete
Min-Sum Set Cover problem, replacing the input set cover instance with a
monotone submodular set function. A simple greedy algorithm achieves an
approximation factor of 4, which is tight unless P=NP [Streeter and Golovin,
NeurIPS, 2008]. We complement the greedy algorithm with analysis of a local
search algorithm. Building on work of Munagala et al. [ICDT, 2005], we show
that, using simple initialization, a straightforward local search algorithm
achieves a -approximate solution in time
, provided that the monotone submodular set function is
also second-order supermodular. Second-order supermodularity has been shown to
hold for a number of submodular functions of practical interest, including
functions associated with set cover, matching, and facility location. We
present experiments on two special cases of Min-Sum Submodular Cover and find
that the local search algorithm can outperform the greedy algorithm on small
data sets
An Efficient Streaming Algorithm for the Submodular Cover Problem
We initiate the study of the classical Submodular Cover (SC) problem in the
data streaming model which we refer to as the Streaming Submodular Cover (SSC).
We show that any single pass streaming algorithm using sublinear memory in the
size of the stream will fail to provide any non-trivial approximation
guarantees for SSC. Hence, we consider a relaxed version of SSC, where we only
seek to find a partial cover.
We design the first Efficient bicriteria Submodular Cover Streaming
(ESC-Streaming) algorithm for this problem, and provide theoretical guarantees
for its performance supported by numerical evidence. Our algorithm finds
solutions that are competitive with the near-optimal offline greedy algorithm
despite requiring only a single pass over the data stream. In our numerical
experiments, we evaluate the performance of ESC-Streaming on active set
selection and large-scale graph cover problems.Comment: To appear in NIPS'1
Approximation Algorithms for Stochastic Boolean Function Evaluation and Stochastic Submodular Set Cover
Stochastic Boolean Function Evaluation is the problem of determining the
value of a given Boolean function f on an unknown input x, when each bit of x_i
of x can only be determined by paying an associated cost c_i. The assumption is
that x is drawn from a given product distribution, and the goal is to minimize
the expected cost. This problem has been studied in Operations Research, where
it is known as "sequential testing" of Boolean functions. It has also been
studied in learning theory in the context of learning with attribute costs. We
consider the general problem of developing approximation algorithms for
Stochastic Boolean Function Evaluation. We give a 3-approximation algorithm for
evaluating Boolean linear threshold formulas. We also present an approximation
algorithm for evaluating CDNF formulas (and decision trees) achieving a factor
of O(log kd), where k is the number of terms in the DNF formula, and d is the
number of clauses in the CNF formula. In addition, we present approximation
algorithms for simultaneous evaluation of linear threshold functions, and for
ranking of linear functions.
Our function evaluation algorithms are based on reductions to the Stochastic
Submodular Set Cover (SSSC) problem. This problem was introduced by Golovin and
Krause. They presented an approximation algorithm for the problem, called
Adaptive Greedy. Our main technical contribution is a new approximation
algorithm for the SSSC problem, which we call Adaptive Dual Greedy. It is an
extension of the Dual Greedy algorithm for Submodular Set Cover due to Fujito,
which is a generalization of Hochbaum's algorithm for the classical Set Cover
Problem. We also give a new bound on the approximation achieved by the Adaptive
Greedy algorithm of Golovin and Krause
Algorithms for covering multiple submodular constraints and applications
We consider the problem of covering multiple submodular constraints. Given a finite ground set N, a weight function , r monotone submodular functions over N and requirements the goal is to find a minimum weight subset such that for . We refer to this problem as Multi-Submod-Cover and it was recently considered by Har-Peled and Jones (Few cuts meet many point sets. CoRR. arxiv:abs1808.03260Har-Peled and Jones 2018) who were motivated by an application in geometry. Even with Multi-Submod-Cover generalizes the well-known Submodular Set Cover problem (Submod-SC), and it can also be easily reduced to Submod-SC. A simple greedy algorithm gives an approximation where and this ratio cannot be improved in the general case. In this paper, motivated by several concrete applications, we consider two ways to improve upon the approximation given by the greedy algorithm. First, we give a bicriteria approximation algorithm for Multi-Submod-Cover that covers each constraint to within a factor of while incurring an approximation of in the cost. Second, we consider the special case when each is a obtained from a truncated coverage function and obtain an algorithm that generalizes previous work on partial set cover (Partial-SC), covering integer programs (CIPs) and multiple vertex cover constraints Bera et al. (Theoret Comput Sci 555:2–8 Bera et al. 2014). Both these algorithms are based on mathematical programming relaxations that avoid the limitations of the greedy algorithm. We demonstrate the implications of our algorithms and related ideas to several applications ranging from geometric covering problems to clustering with outliers. Our work highlights the utility of the high-level model and the lens of submodularity in addressing this class of covering problems.publishedVersio
Hardness and Approximation of Submodular Minimum Linear Ordering Problems
The minimum linear ordering problem (MLOP) generalizes well-known
combinatorial optimization problems such as minimum linear arrangement and
minimum sum set cover. MLOP seeks to minimize an aggregated cost due
to an ordering of the items (say ), i.e., , where is the set of items
mapped by to indices . Despite an extensive literature on MLOP
variants and approximations for these, it was unclear whether the graphic
matroid MLOP was NP-hard. We settle this question through non-trivial
reductions from mininimum latency vertex cover and minimum sum vertex cover
problems. We further propose a new combinatorial algorithm for approximating
monotone submodular MLOP, using the theory of principal partitions. This is in
contrast to the rounding algorithm by Iwata, Tetali, and Tripathi [ITT2012],
using Lov\'asz extension of submodular functions. We show a
-approximation for monotone submodular MLOP where
satisfies . Our theory provides new approximation bounds for special cases of the
problem, in particular a -approximation for the
matroid MLOP, where is the rank function of a matroid. We further show
that minimum latency vertex cover (MLVC) is -approximable, by
which we also lower bound the integrality gap of its natural LP relaxation,
which might be of independent interest
Submodularity Property for Facility Locations of Dynamic Flow Networks
This paper considers facility location problems within dynamic flow networks, shifting the focus from minimizing evacuation time to handling situations with a constrained evacuation timeframe. Our study sets two main goals: 1) Determining a fixed-size set of locations that can maximize the number of evacuees, and 2) Identifying the smallest set of locations capable of accommodating all evacuees within the time constraint. We introduce flow_t(S) to represent the number of evacuees for given locations S within a fixed time limit t. We prove that flow_t functions is a monotone submodular function, which allows us to apply an approximation algorithm specifically designed for maximizing such functions with size restrictions. For the second objective, we implement an approximation algorithm tailored to solving the submodular cover problem. We conduct experiments on the real datasets of Chiang Mai, and demonstrate that the approximation algorithms give solutions which are close to optimal for all of the experiments we have conducted
Submodular Optimization with Submodular Cover and Submodular Knapsack Constraints
We investigate two new optimization problems -- minimizing a submodular
function subject to a submodular lower bound constraint (submodular cover) and
maximizing a submodular function subject to a submodular upper bound constraint
(submodular knapsack). We are motivated by a number of real-world applications
in machine learning including sensor placement and data subset selection, which
require maximizing a certain submodular function (like coverage or diversity)
while simultaneously minimizing another (like cooperative cost). These problems
are often posed as minimizing the difference between submodular functions [14,
35] which is in the worst case inapproximable. We show, however, that by
phrasing these problems as constrained optimization, which is more natural for
many applications, we achieve a number of bounded approximation guarantees. We
also show that both these problems are closely related and an approximation
algorithm solving one can be used to obtain an approximation guarantee for the
other. We provide hardness results for both problems thus showing that our
approximation factors are tight up to log-factors. Finally, we empirically
demonstrate the performance and good scalability properties of our algorithms.Comment: 23 pages. A short version of this appeared in Advances of NIPS-201
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