Alina Ene

Fast exact and heuristic methods for role minimization problems

We describe several new bottom-up approaches to problems in role engineering for Role-Based Access Control (RBAC). The salient problems are all NP-complete, even to approximate, yet we find that in instances that arise in practice these problems can be solved in minutes. We first consider role minimization, the process of finding a smallest collection of roles that can be used to implement a pre-existing user-to-permission relation. We introduce fast graph reductions that allow recovery of the solution from the solution to a problem on a smaller input graph. For our test cases, these reductions either solve the problem, or reduce the problem enough that we find the optimum solution with a (worst-case) exponential method. We introduce lower bounds that are sharp for seven of nine test cases and are within 3.4% on the other two. We introduce and test a new polynomial-time approximation that on average yields 2% more roles than the optimum. We next consider the related problem of minimizing the number of connections between roles and users or permissions, and we develop effective heuristic methods for this problem as well. Finally, we propose methods for several related problems.

Unsplittable Flow in Paths and Trees and Column-Restricted Packing Integer Programs

We consider the unsplittable flow problem (UFP) and the closely related column-restricted packing integer programs (CPIPs). In UFP we are given an edge-capacitated graph G = (V ,E ) and k request pairs R 1 , ..., R k , where each R i consists of a source-destination pair (s i ,t i ), a demand d i and a weight w i . The goal is to find a maximum weight subset of requests that can be routed unsplittably in G . Most previous work on UFP has focused on the no-bottleneck case in which the maximum demand of the requests is at most the smallest edge capacity. Inspired by the recent work of Bansal et al . [3] on UFP on a path without the above assumption, we consider UFP on paths as well as trees. We give a simple O (logn ) approximation for UFP on trees when all weights are identical; this yields an O (log2 n ) approximation for the weighted case. These are the first non-trivial approximations for UFP on trees. We develop an LP relaxation for UFP on paths that has an integrality gap of O (log2 n ); previously there was no relaxation with o (n ) gap. We also consider UFP in general graphs and CPIPs without the no-bottleneck assumption and obtain new and useful results.

Submodular cost allocation problem and applications

We study the Minimum Submodular-Cost Allocation problem (MSCA). In this problem we are given a finite ground set V and k non-negative submodular set functions f1, ... , fk on V. The objective is to partition V into k (possibly empty) sets A1, ... , Ak such that the sum Σi=1k fi(Ai) is minimized. Several well-studied problems such as the non-metric facility location problem, multiway-cut in graphs and hypergraphs, and uniform metric labeling and its generalizations can be shown to be special cases of MSCA. In this paper we consider a convexprogramming relaxation obtained via the Lovasz-extension for submodular functions. This allows us to understand several previous relaxations and rounding procedures in a unified fashion and also develop new formulations and approximation algorithms for related problems. In particular, we give a (1.5 - 1/k)-approximation for the hypergraph multiway partition problem. We also give a min{2(1-1/k), HΔ}-approximation for the hypergraph multiway cut problem when Δ is the maximum hyperedge size. Both problems generalize the multiway cut problem in graphs and the hypergraph cut problem is approximation equivalent to the nodeweighted multiway cut problem in graphs.

Constrained Submodular Maximization: Beyond 1/e

In this work, we present a new algorithm for maximizing a non-monotone submodular function subject to a general constraint. Our algorithm finds an approximate fractional solution for maximizing the multilinear extension of the function over a down-closed polytope. The approximation guarantee is 0.372 and it is the first improvement over the 1/e approximation achieved by the unified Continuous Greedy algorithm [Feldman et al., FOCS 2011].

Geometric packing under non-uniform constraints

We study the problem of discrete geometric packing. Here, given weighted regions (say in the plane) and points (with capacities), one has to pick a maximum weight subset of the regions such that no point is covered more than its capacity. We provide a general framework and an algorithm for approximating the optimal solution for packing in hypergraphs arising out of such geometric settings. Using this framework we get a flotilla of results on this problem (and also on its dual, where one wants to pick a maximum weight subset of the points when the regions have capacities). For example, for the case of fat triangles of similar size, we show an (1)-approximation and prove that no PTAS is possible. See [ehr-gpnuc-11] for the full version of the paper.

Prize-Collecting Survivable Network Design in Node-Weighted Graphs

We consider node-weighted network design problems, in particular the survivable network design problem (SNDP) and its prize-collecting version (PC-SNDP). The input consists of a node-weighted undirected graph G = (V,E) and integral connectivity requirements r(st) for each pair of nodes st. The goal is to find a minimum node-weighted subgraph H of G such that, for each pair st, H contains r(st) edge-disjoint paths between s and t. PC-SNDP is a generalization in which the input also includes a penalty π(st) for each pair, and the goal is to find a subgraph H to minimize the sum of the weight of H and the sum of the penalties for all pairs whose connectivity requirements are not fully satisfied by H. Let k = max st r(st) be the maximum requirement. There has been no non-trivial approximation for node-weighted PC-SNDP for k > 1, the main reason being the lack of an LP relaxation based approach for node-weighted SNDP. In this paper we describe multiroute-flow based relaxations for the two problems and obtain approximation algorithms for PC-SNDP through them. The approximation ratios we obtain for PC-SNDP are similar to those that were previously known for SNDP via combinatorial algorithms. Specifically, we obtain an O(k 2 logn)-approximation in general graphs and an O(k 2)-approximation in graphs that exclude a fixed minor. The approximation ratios can be improved by a factor of k but the running times of the algorithms depend polynomially on n k .

Node-weighted network design in planar and minor-closed families of graphs

We consider node-weighted network design in planar and minor-closed families of graphs. In particular we focus on the edge-connectivity survivable network design problem (EC-SNDP). The input consists of a node-weighted undirected graph G=(V,E) and integral connectivity requirements r(uv) for each pair of nodes uv. The goal is to find a minimum node-weighted subgraph H of G such that, for each pair uv, H contains r(uv) edge-disjoint paths between u and v. Our main result is an O(k)-approximation algorithm for EC-SNDP where k= max uvr(uv) is the maximum requirement. This improves the O(k logn)-approximation known for node-weighted EC-SNDP in general graphs [15]. Our algorithm and analysis applies to the more general problem of covering a proper function with maximum requirement k. Our result is inspired by, and generalizes, the work of Demaine, Hajiaghayi and Klein [5] who gave constant factor approximation algorithms for node-weighted Steiner tree and Steiner forest problems (and more generally covering 0-1 proper functions) in planar and minor-closed families of graphs.

On Approximating Maximum Independent Set of Rectangles

We study the Maximum Independent Set of Rectangles (MISR) problem: given a set of n axis-parallel rectangles, find a largest-cardinality subset of the rectangles, such that no two of them overlap. MISR is a basic geometric optimization problem with many applications, that has been studied extensively. Until recently, the best approximation algorithm for it achieved an O(log log n)-approximation factor. In a recent breakthrough, Adamaszek and Wiese provided a quasi-polynomial time approximation scheme: a (1-ε)-approximation algorithm with running time nO(poly(log n)/ε). Despite this result, obtaining a PTAS or even a polynomial-time constant-factor approximation remains a challenging open problem. In this paper we make progress towards this goal by providing an algorithm for MISR that achieves a (1 - ε)-approximation in time nO(poly(log logn/ε)). We introduce several new technical ideas, that we hope will lead to further progress on this and related problems.

Submodular unsplittable flow on trees

AbstractWe study the Unsplittable Flow problem (