17 research outputs found
A simple combinatorial algorithm for submodular function minimization
This paper presents a new simple algorithm for minimizing submodular functions. For integer valued submodular functions, the algorithm runs in O(n6EO log nM) [O (n superscript 6 E O log nM)] time, where n is the cardinality of the ground set, M is the maximum absolute value of the function value, and EO is the time for function evaluation. The algorithm can be improved to run in O ((n4EO+n5)log nM) [O ((n superscript 4 EO + n superscript 5) log nM)] time. The strongly polynomial version of this faster algorithm runs in O((n5EO + n6) log n) [O ((n superscript 5 EO + n superscript 6) log n)] time for real valued general submodular functions. These are comparable to the best known running time bounds for submodular function minimization. The algorithm can also be implemented in strongly polynomial time using only additions, subtractions, comparisons, and the oracle calls for function evaluation. This is the first fully combinatorial submodular function minimization algorithm that does not rely on the scaling method.United States. Office of Naval Research ( ONR grant N00014-08-1-0029
Provable Submodular Minimization using Wolfe's Algorithm
Owing to several applications in large scale learning and vision problems,
fast submodular function minimization (SFM) has become a critical problem.
Theoretically, unconstrained SFM can be performed in polynomial time [IFF 2001,
IO 2009]. However, these algorithms are typically not practical. In 1976, Wolfe
proposed an algorithm to find the minimum Euclidean norm point in a polytope,
and in 1980, Fujishige showed how Wolfe's algorithm can be used for SFM. For
general submodular functions, this Fujishige-Wolfe minimum norm algorithm seems
to have the best empirical performance.
Despite its good practical performance, very little is known about Wolfe's
minimum norm algorithm theoretically. To our knowledge, the only result is an
exponential time analysis due to Wolfe himself. In this paper we give a maiden
convergence analysis of Wolfe's algorithm. We prove that in iterations,
Wolfe's algorithm returns an -approximate solution to the min-norm
point on {\em any} polytope. We also prove a robust version of Fujishige's
theorem which shows that an -approximate solution to the min-norm
point on the base polytope implies {\em exact} submodular minimization. As a
corollary, we get the first pseudo-polynomial time guarantee for the
Fujishige-Wolfe minimum norm algorithm for unconstrained submodular function
minimization
Curvature and Optimal Algorithms for Learning and Minimizing Submodular Functions
We investigate three related and important problems connected to machine
learning: approximating a submodular function everywhere, learning a submodular
function (in a PAC-like setting [53]), and constrained minimization of
submodular functions. We show that the complexity of all three problems depends
on the 'curvature' of the submodular function, and provide lower and upper
bounds that refine and improve previous results [3, 16, 18, 52]. Our proof
techniques are fairly generic. We either use a black-box transformation of the
function (for approximation and learning), or a transformation of algorithms to
use an appropriate surrogate function (for minimization). Curiously, curvature
has been known to influence approximations for submodular maximization [7, 55],
but its effect on minimization, approximation and learning has hitherto been
open. We complete this picture, and also support our theoretical claims by
empirical results.Comment: 21 pages. A shorter version appeared in Advances of NIPS-201
Constrained Monotone Function Maximization and the Supermodular Degree
The problem of maximizing a constrained monotone set function has many
practical applications and generalizes many combinatorial problems.
Unfortunately, it is generally not possible to maximize a monotone set function
up to an acceptable approximation ratio, even subject to simple constraints.
One highly studied approach to cope with this hardness is to restrict the set
function. An outstanding disadvantage of imposing such a restriction on the set
function is that no result is implied for set functions deviating from the
restriction, even slightly. A more flexible approach, studied by Feige and
Izsak, is to design an approximation algorithm whose approximation ratio
depends on the complexity of the instance, as measured by some complexity
measure. Specifically, they introduced a complexity measure called supermodular
degree, measuring deviation from submodularity, and designed an algorithm for
the welfare maximization problem with an approximation ratio that depends on
this measure.
In this work, we give the first (to the best of our knowledge) algorithm for
maximizing an arbitrary monotone set function, subject to a k-extendible
system. This class of constraints captures, for example, the intersection of
k-matroids (note that a single matroid constraint is sufficient to capture the
welfare maximization problem). Our approximation ratio deteriorates gracefully
with the complexity of the set function and k. Our work can be seen as
generalizing both the classic result of Fisher, Nemhauser and Wolsey, for
maximizing a submodular set function subject to a k-extendible system, and the
result of Feige and Izsak for the welfare maximization problem. Moreover, when
our algorithm is applied to each one of these simpler cases, it obtains the
same approximation ratio as of the respective original work.Comment: 23 page
Discrete Newtonâs Algorithm for Parametric Submodular Function Minimization
We consider the line search problem in a submodular polyhedron P (f) â â n : Given an arbitrary a â â n and x 0 â P (f), compute max{δ: x 0 + δa â P (f)}. The use of the discrete Newtonâs algorithm for this line search problem is very natural, but no strongly polynomial bound on its number of iterations was known (Iwata 2008). We solve this open problem by providing a quadratic bound of n 2 + O(n log 2 n) on its number of iterations. Our result considerably improves upon the only other known strongly polynomial time algorithm, which is based on Megiddoâs parametric search framework and which requires Ă(n 8 ) submodular function minimizations (Nagano 2007). As a by-product of our study, we prove (tight) bounds on the length of chains of ring families and geometrically increasing sequences of sets, which might be of independent interest. Keywords:
Discrete Newtonâs algorithm; Submodular functions; Line search; Ring families; Geometrically increasing sequence of sets; Fractional combinatorial optimizatio
Submodular Minimization Under Congruency Constraints
Submodular function minimization (SFM) is a fundamental and efficiently
solvable problem class in combinatorial optimization with a multitude of
applications in various fields. Surprisingly, there is only very little known
about constraint types under which SFM remains efficiently solvable. The
arguably most relevant non-trivial constraint class for which polynomial SFM
algorithms are known are parity constraints, i.e., optimizing only over sets of
odd (or even) cardinality. Parity constraints capture classical combinatorial
optimization problems like the odd-cut problem, and they are a key tool in a
recent technique to efficiently solve integer programs with a constraint matrix
whose subdeterminants are bounded by two in absolute value.
We show that efficient SFM is possible even for a significantly larger class
than parity constraints, by introducing a new approach that combines techniques
from Combinatorial Optimization, Combinatorics, and Number Theory. In
particular, we can show that efficient SFM is possible over all sets (of any
given lattice) of cardinality r mod m, as long as m is a constant prime power.
This covers generalizations of the odd-cut problem with open complexity status,
and with relevance in the context of integer programming with higher
subdeterminants. To obtain our results, we establish a connection between the
correctness of a natural algorithm, and the inexistence of set systems with
specific combinatorial properties. We introduce a general technique to disprove
the existence of such set systems, which allows for obtaining extensions of our
results beyond the above-mentioned setting. These extensions settle two open
questions raised by Geelen and Kapadia [Combinatorica, 2017] in the context of
computing the girth and cogirth of certain types of binary matroids