Submodular Maximization through the Lens of Linear Programming

The simplex algorithm for linear programming is based on the fact that any local optimum with respect to the polyhedral neighborhood is also a global optimum. We show that a similar result carries over to submodular maximization. In particular, every local optimum of a constrained monotone submodular maximization problem yields a $1/2$-approximation, and we also present an appropriate extension to the non-monotone setting. However, reaching a local optimum quickly is a non-trivial task. Moreover, we describe a fast and very general local search procedure that applies to a wide range of constraint families, and unifies as well as extends previous methods. In our framework, we match known approximation guarantees while disentangling and simplifying previous approaches. Moreover, despite its generality, we are able to show that our local search procedure is slightly faster than previous specialized methods. Furthermore, we resolve an open question on the relation between linear optimization and submodular maximization; namely, whether a linear optimization oracle may be enough to obtain strong approximation algorithms for submodular maximization. We show that this is not the case by providing an example of a constraint family on a ground set of size $n$ for which, if only given a linear optimization oracle, any algorithm for submodular maximization with a polynomial number of calls to the linear optimization oracle will have an approximation ratio of only $O ( \frac{1}{\sqrt{n}} \cdot \frac{\log n}{\log\log n} )$

Efficient Computation of Representative Sets with Applications in Parameterized and Exact Algorithms

We give two algorithms computing representative families of linear and uniform matroids and demonstrate how to use representative families for designing single-exponential parameterized and exact exponential time algorithms. The applications of our approach include - LONGEST DIRECTED CYCLE - MINIMUM EQUIVALENT GRAPH (MEG) - Algorithms on graphs of bounded treewidth -k-PATH, k-TREE, and more generally, k-SUBGRAPH ISOMORPHISM, where the k-vertex pattern graph is of constant treewidth.Comment: 61 page

Matrix approach to rough sets through vector matroids over a field

Rough sets were proposed to deal with the vagueness and incompleteness of knowledge in information systems. There are may optimization issues in this field such as attribute reduction. Matroids generalized from matrices are widely used in optimization. Therefore, it is necessary to connect matroids with rough sets. In this paper, we take field into consideration and introduce matrix to study rough sets through vector matroids. First, a matrix representation of an equivalence relation is proposed, and then a matroidal structure of rough sets over a field is presented by the matrix. Second, the properties of the matroidal structure including circuits, bases and so on are studied through two special matrix solution spaces, especially null space. Third, over a binary field, we construct an equivalence relation from matrix null space, and establish an algebra isomorphism from the collection of equivalence relations to the collection of sets, which any member is a family of the minimal non-empty sets that are supports of members of null space of a binary dependence matrix. In a word, matrix provides a new viewpoint to study rough sets

Inferring the strength of social ties: a community-driven approach

Online social networks are growing and becoming denser. The social connections of a given person may have very high variability: from close friends and relatives to acquaintances to people who hardly know. Inferring the strength of social ties is an important ingredient for modeling the interaction of users in a network and understanding their behavior. Furthermore, the problem has applications in computational social science, viral marketing, and people recommendation. In this paper we study the problem of inferring the strength of social ties in a given network. Our work is motivated by a recent approach [27], which leverages the strong triadic closure (STC) principle, a hypothesis rooted in social psychology [13]. To guide our inference process, in addition to the network structure, we also consider as input a collection of tight communities. Those are sets of vertices that we expect to be connected via strong ties. Such communities appear in different situations, e.g., when being part of a community implies a strong connection to one of the existing members. We consider two related problem formalizations that reflect the assumptions of our setting: small number of STC violations and strong-tie connectivity in the input communities. We show that both problem formulations are NP-hard. We also show that one problem formulation is hard to approximate, while for the second we develop an algorithm with approximation guarantee. We validate the proposed method on real-world datasets by comparing with baselines that optimize STC violations and community connectivity separately

The complexity of computing the minimum rank of a sign pattern matrix

We show that computing the minimum rank of a sign pattern matrix is NP hard. Our proof is based on a simple but useful connection between minimum ranks of sign pattern matrices and the stretchability problem for pseudolines arrangements. In fact, our hardness result shows that it is already hard to determine if the minimum rank of a sign pattern matrix is $\leq 3$. We complement this by giving a polynomial time algorithm for determining if a given sign pattern matrix has minimum rank $\leq 2$. Our result answers one of the open problems from Linial et al. [Combinatorica, 27(4):439--463, 2007].Comment: 16 page

Optimally approximating exponential families

This article studies exponential families $\mathcal{E}$ on finite sets such that the information divergence $D(P\|\mathcal{E})$ of an arbitrary probability distribution from $\mathcal{E}$ is bounded by some constant $D>0$. A particular class of low-dimensional exponential families that have low values of $D$ can be obtained from partitions of the state space. The main results concern optimality properties of these partition exponential families. Exponential families where $D=\log(2)$ are studied in detail. This case is special, because if $D<\log(2)$, then $\mathcal{E}$ contains all probability measures with full support.Comment: 17 page

Polyhedral aspects of score equivalence in Bayesian network structure learning

This paper deals with faces and facets of the family-variable polytope and the characteristic-imset polytope, which are special polytopes used in integer linear programming approaches to statistically learn Bayesian network structure. A common form of linear objectives to be maximized in this area leads to the concept of score equivalence (SE), both for linear objectives and for faces of the family-variable polytope. We characterize the linear space of SE objectives and establish a one-to-one correspondence between SE faces of the family-variable polytope, the faces of the characteristic-imset polytope, and standardized supermodular functions. The characterization of SE facets in terms of extremality of the corresponding supermodular function gives an elegant method to verify whether an inequality is SE-facet-defining for the family-variable polytope. We also show that when maximizing an SE objective one can eliminate linear constraints of the family-variable polytope that correspond to non-SE facets. However, we show that solely considering SE facets is not enough as a counter-example shows; one has to consider the linear inequality constraints that correspond to facets of the characteristic-imset polytope despite the fact that they may not define facets in the family-variable mode.Comment: 37 page

Ample completions of OMs and CUOMs

This paper considers completions of COMs (complexes oriented matroids) to ample partial cubes of the same VC-dimension. We show that these exist for OMs (oriented matroids) and CUOMs (complexes of uniform oriented matroids). This implies that OMs and CUOMs satisfy the sample compression conjecture -- one of the central open questions of learning theory. We conjecture that every COM can be completed to an ample partial cube without increasing the VC-dimension.Comment: 19 pages, 3 figure

Non-negative submodular stochastic probing via stochastic contention resolution schemes

The abstract model of stochastic probing was presented by Gupta and Nagarajan (IPCO'13), and provides a unified view of a number of problems. Adamczyk, Sviridenko, Ward (STACS'14) gave better approximation for matroid environments and linear objectives. At the same time this method was easily extendable to settings, where the objective function was monotone submodular. However, the case of non-negative submodular function could not be handled by previous techniques. In this paper we address this problem, and our results are twofold. First, we adapt the notion of contention resolution schemes of Chekuri, Vondr\'ak, Zenklusen (SICOMP'14) to show that we can optimize non-negative submodular functions in this setting with a constant factor loss with respect to the deterministic setting. Second, we show a new contention resolution scheme for transversal matroids, which yields better approximations in the stochastic probing setting than the previously known tools. The rounding procedure underlying the scheme can be of independent interest --- Bansal, Gupta, Li, Mestre, Nagarajan, Rudra (Algorithmica'12) gave two seemingly different algorithms for stochastic matching and stochastic $k$-set packing problems with two different analyses, but we show that our single technique can be used to analyze both their algorithms

Submodular Function Maximization over Distributive and Integer Lattices

The problem of maximizing non-negative submodular functions has been studied extensively in the last few years. However, most papers consider submodular set functions. Recently, several advances have been made for the more general case of submodular functions on the integer lattice. In this paper, we present a deterministic $\frac{1}{3}$-approximation for maximizing a submodular function on a bounded integer lattice $\{0, \ldots, C\}^n$ using a Double Greedy framework. Moreover, we show that the analysis is tight and that other ideas used for approximating set functions cannot easily be extended. In contrast to set functions, submodularity on the integer lattice does not imply the so-called diminishing returns property. Assuming this property, it was shown that many results for set functions can also be obtained for the integer lattice. In this paper, we consider a further generalization. Instead of the integer lattice, we consider a distributive lattice as the function domain and assume the diminishing returns (DR) property. On the one hand, we show that some approximation algorithms match the set functions setting. In particular, we can obtain a $\frac{1}{2}$-approximation for unconstrained maximization, a $(1-\frac{1}{e})$-approximation for monotone functions under a cardinality constraint and a $\frac{1}{2}$-approximation for a poset matroid constraint. On the other hand, for a knapsack constraint, the problem becomes significantly harder: even for monotone DR-submodular functions, we show that there is no $2^{(\log (n^{1/2} - 1))^\delta - 1}$-approximation for every $\delta > 0$ under the assumption that $3-SAT$ cannot be solved in time $2^{n^{3/4 + \epsilon}}$