We study the problem of deleting a minimum cost set of vertices from a given vertex-weighted graph in such a way that the resulting graph has no induced path on three vertices. This problem is often called cluster ver...
详细信息
In this paper, we propose a successive convex approximation framework for sparse optimization where the nondifferentiable regularization in the objective function is nonconvex and it can be written as the difference o...
详细信息
In this paper, we propose a successive convex approximation framework for sparse optimization where the nondifferentiable regularization in the objective function is nonconvex and it can be written as the difference of two convex functions. The proposed framework is based on a nontrivial combination of the majorization-minimization method and successive convex approximation for nonconvex optimization where the regularization function is convex. The proposed framework is flexible and it leads to algorithms that exploit the problem structure and have a low complexity. We demonstrate these advantages by an example application where the nonconvex regularization is the capped l1 -norm function. Customizing the proposed framework, we obtain a best-response type algorithm for which all elements of the unknown parameter are updated in parallel according to closed-form expressions. Finally, the proposed algorithms are numerically tested.
We present a new 4-approximation algorithm for the Combinatorial Motion Planning problem which runs in O(n2α(n2, n)) time, where α is the functional inverse of the Ackermann function, and a fully distributed version...
详细信息
We develop a new framework for generalizing approximation algorithms from the structural graph algorithm literature so that they apply to graphs somewhat close to that class (a scenario we expect is common when workin...
详细信息
Given an undirected graph G = (V, E) and a weight function w: E → R, the Minimum Dominating Tree problem asks to find a minimum weight sub-tree of G, T = (U, F), such that every v ∈ V \ U is adjacent to at least one...
详细信息
Given an n-vertex m-edge graph G with non negative edge-weights, a shortest cycle of G is one minimizing the sum of the weights on its edges. The girth of G is the weight of such a shortest cycle. We obtain several ne...
详细信息
Given an n-vertex m-edge graph G with non negative edge-weights, a shortest cycle of G is one minimizing the sum of the weights on its edges. The girth of G is the weight of such a shortest cycle. We obtain several new approximation algorithms for computing the girth of weighted graphs: • For any graph G with polynomially bounded integer weights, we present a deterministic algorithm that computes, in Õ(n5/3 + m)-time, a cycle of weight at most twice the girth of G. This matches both the approximation factor and – almost – the running time of the best known subquadratic-time approximation algorithm for the girth of unweighted graphs (Roditty and Vassilevska Williams, SODA’12). Our approach combines some new insights on the previous approximation algorithms for this problem (Lingas and Lundell, IPL’09;Roditty and Tov, TALG’13) with Hitting Set based methods that are used for approximate distance oracles and date back from (Thorup and Zwick, JACM’05). • Then, we turn our algorithm into a deterministic (2 + Ε)-approximation for graphs with arbitrary non negative edge-weights, at the price of a slightly worse running-time in Õ(n5/3polylog(1/Ε) + m). For that we introduce a novel polynomial-factor approximation of the girth, that makes more amenable the passing from the graphs with bounded integer edge-weights to the general case and is of independent interest. • Finally, if we insist in removing the dependency in the number m of edges, we can transform our algorithms into an Õ(n5/3)-time randomized 4-approximation for the graphs with non negative edge-weights – assuming the adjacency lists are sorted. Combined with the aforementioned Hitting Set based methods, this algorithm can be derandomized, thereby yielding an Õ(n5/3)-time deterministic 4-approximation for the graphs with polynomially bounded integer weights, and an Õ(n5/3polylog(1/Ε))-time deterministic (4 + Ε)approximation for the graphs with non negative edge-weights. To the best of our knowledge, these are the f
Given a weighted graph G = (V, E) with weight functions c: E → R+ and π: V → R+, and a subset U ⊆ V, the normalized cut value for U is defined as the sum of the weights of edges exiting U divided by the weight of v...
详细信息
Given a weighted graph G = (V, E) with weight functions c: E → R+ and π: V → R+, and a subset U ⊆ V, the normalized cut value for U is defined as the sum of the weights of edges exiting U divided by the weight of vertices in U. The mean isoperimetry problem, ISO1(G, k), for a weighted graph G is a generalization of the classical uniform sparsest cut problem in which, given a parameter k, the objective is to find k disjoint nonempty subsets of V minimizing the average normalized cut value of the parts. The robust version of the problem seeks an optimizer where the number of vertices that fall out of the subpartition is bounded by some given integer 0 ≤ ρ ≤ |V |. Our main result states that ISO1(G, k), as well as its robust version, CRISO1(G, k, ρ), subjected to the condition that each part of the subpartition induces a connected subgraph, are solvable in time O(k2ρ2 π(V (T)3) on any weighted tree T, in which π(V (T)) is the sum of the vertex-weights. This result implies that ISO1(G, k) is strongly polynomial-time solvable on weighted trees when the vertex-weights are polynomially bounded and may be compared to the fact that the problem is NP-Hard for weighted trees in general. As far as applications are concerned, the connectivity requirement may be interpreted as an approach to model the practical consistency of the parts, which together with having control on the size of the outlier set and applying a smooth "mean" cost function (as opposed to, say, the "max" version), characterizes our solution to CRISO1(G, k, ρ) on trees as one of the most flexible and accurate procedures within the framework of isoperimetry-based clustering. Also, using this, we show that both mentioned problems, ISO1(G, k) and CRISO1(G, k, ρ) as well as the ordinary robust mean isoperimetry problem RISO1(G, k, ρ), admit polynomial-time O(log1.5 |V | log log |V |)-approximation algorithms for weighted graphs with polynomially bounded weights, using the Räcke-Shah tree cut sparsifier. To the best
A top-list is a possibly incomplete ranking of elements: only a subset of the elements are ranked, with all unranked elements tied for last. Top-list aggregation, a generalization of the well-known rank aggregation pr...
详细信息
A top-list is a possibly incomplete ranking of elements: only a subset of the elements are ranked, with all unranked elements tied for last. Top-list aggregation, a generalization of the well-known rank aggregation problem, takes as input a collection of top-lists and aggregates them into a single complete ranking, aiming to minimize the number of upsets (pairs ranked in opposite order in the input and in the output). In this paper, we give simple approximation algorithms for top-list aggregation. • We generalize the footrule algorithm for rank aggregation (which minimizes Spearman’s footrule distance), yielding a simple 2-approximation algorithm for top-list aggregation. • Ailon’s RepeatChoice algorithm for bucket-orders aggregation yields a 2-approximation algorithm for top-list aggregation. Using inspiration from approval voting, we define the score of an element as the frequency with which it is ranked, i.e. appears in an input top-list. We reinterpret RepeatChoice for top-list aggregation as a randomized algorithm using variables whose expectations correspond to score and to the average rank of an element given that it is ranked. • Using average ranks, we generalize and analyze Borda’s algorithm for rank aggregation. We observe that the natural generalization is not a constant approximation. • We design a simple 2-phase variant of the Generalized Borda’s algorithm, roughly sorting by scores and breaking ties by average ranks, yielding another simple constant-approximation algorithm for top-list aggregation. • We then design another 2-phase variant in which in order to break ties we use, as a black box, the Mathieu-Schudy PTAS for rank aggregation, yielding a PTAS for top-list aggregation. This solves an open problem posed by Ailon. • Finally, in the special case in which all input lists have length at most k, we design another simple 2-phase algorithm based on sorting by scores, and prove that it is an EPTAS – the complexity is O(n log n) when k = o(log n). Cop
We describe approximation algorithms in Linial's classic LOCAL model of distributed computing to find maximum-weight matchings in a hypergraph of rank r. Our main result is a deterministic algorithm to generate a ...
详细信息
The notions of bounded expansion and nowhere denseness not only offer robust and general definitions of uniform sparseness of graphs, they also describe the tractability boundary for several important algorithmic ques...
详细信息
暂无评论