In the Priority Steiner Tree (PST) problem, we are given an undirected graph \(G=(V,E)\) with a source \(s \in V\) and terminals \(T \subseteq V \setminus \{s\}\), where each terminal \(v T\) requires nonnegative priority P(v). The goal is to compute minimum weight tree containing edges of varying rates such that path from s v consists rate greater than or equal PST problem k priorities admits ...

We revisit the problem of online linear optimization in case where set feasible actions is accessible through an approximated oracle with a factor α multiplicative approximation guarantee. This setting particular interesting because it captures natural extensions well-studied offline problems that are NP-hard yet admit efficient algorithms. The goal here to minimize α-regret, which extension th...

For any column A:,i the best response vector is 1, so A:,i1 T − A 0 = 2 n − 1 = 2(1 − 1/n) OPTF 1 OPTF 1 = n Boolean l0-rank-1 Theorem 3. (Sublinear) Given A ∈ 0,1 m×n with column adjacency arrays and with row and column sums, we can compute w.h.p. in time O min A 0 +m + n, ψB −1 m + n log(mn) vectors u, v such that A − uv 0 ≤ 1 + O ψB OPTB . Theorem 4. (Exact) Given A ∈ 0,1 m×n with OPTB / A 0...

Chandra Chekuri (Dept. of Computer Science, University of Illinois at Urbana-Champaign, Urbana, IL), Joseph Cheriyan (Dept. of Combinatorics and Optimization, University of Waterloo, Waterloo, ON), Ryan O’Donnell (School of Computer Science, Carnegie Mellon University, Pittsburgh, PA), Mohammad R. Salavatipour (Dept. of Computing Science, University of Alberta, Edmonton, AL), David Williamson (...

S (in alphabetic order by speaker surname) Speaker: Hyung-Chan An (EPFL, Lausanne) Title: LP-Based Algorithms for Capacitated Facility Location Abstract: Linear programming has played a key role in the study of algorithms for combinatorial optimization problems. In the field of approximation algorithms, this is well illustrated by the uncapacitated facility location problem. A variety of algori...

In the previous lecture we saw examples of greedy algorithms that made locally optimal decisions at each step to arrive at a solution that wasn’t too far from the optimal solution in the end. Specifically for the case of Set Cover we saw that this strategy leads to the best possible approximation algorithm we could hope for (unless NP ⊂ DTIME(n ), which is very unlikely). In general, we also no...

We study the l0-Low Rank Approximation Problem, where the goal is, given anm×nmatrix A, to output a rank-k matrix A for which ‖A′ −A‖0 is minimized. Here, for a matrix B, ‖B‖0 denotes the number of its non-zero entries. This NP-hard variant of low rank approximation is natural for problems with no underlying metric, and its goal is to minimize the number of disagreeing data positions. We provid...