We consider a broad class of stochastic dynamic programming problems that are amenable to relaxation via decomposition. These problems comprise multiple subproblems that are independent of each other except for a collection of coupling constraints on the action space. We fit an additively separable value function approximation using two techniques, namely, Lagrangian relaxation and the linear p...

We consider the feasibility problem OPP in higher-dimensional orthogonal packing: given a set of d-dimensional (d ≥ 2) rectangular items, decide whether all of them can be orthogonally packed in the given rectangular container without rotation. The 1D ‘bar’ LP relaxation of OPP reduces the latter to a 1D cutting-stock problem where the packing of each stock bar represents a possible 1D stitch t...

We study the Maximum Budgeted Allocation problem, which is the problem of assigning indivisible items to players with budget constraints. In its most general form, an instance of the MBA problem might include many different prices for the same item among different players, and different budget constraints for every player. So far, the best approximation algorithms we know for the MBA problem ac...

In the Directed Steiner Tree problem, we are given a directed graph G = (V,E) with edge costs, a root vertex r ∈ V , and a terminal set X ⊆ V . The goal is to find the cheapest subset of edges that contains an r-t path for every terminal t ∈ X. The only known polylogarithmic approximations for Directed Steiner Tree run in quasi-polynomial time and the best polynomial time approximations only ac...

When solving the linear programming (LP) relaxation of a mixed-integer program (MIP) with column generation, columns might be generated that are not needed to express any integer optimal solution. Such called strongly redundant and dual bound obtained by LP is potentially stronger if these generated. We introduce sufficient condition for strong redundancy can checked compact LP. Using solution ...

We study some linear programming relaxations for the Unsplittable Flow problem on trees (UFPtree). Inspired by results obtained by Chekuri, Ene, and Korula for Unsplittable Flow on paths (UFP-path), we present a relaxation with polynomially many constraints that has an integrality gap bound of O(logn ·min{logm, logn}) where n denotes the number of tasks and m denotes the number of edges in the ...

We study the ATSP (Asymmetric Traveling Salesman Problem), and our focus is on negative results in the framework of the Sherali-Adams (SA) Lift and Project method. Our main result pertains to the standard LP (linear programming) relaxation of ATSP, due to Dantzig, Fulkerson, and Johnson. For any fixed integer t ≥ 0 and small ǫ, 0 < ǫ ≪ 1, there exists a digraph G on ν = ν(t, ǫ) = O(t/ǫ) vertice...

LP relaxation-based message passing algorithms provide an effective tool for MAP inference over Probabilistic Graphical Models. However, different LP relaxations often have different objective functions and variables of differing dimensions, which presents a barrier to effective comparison and analysis. In addition, the computational complexity of LP relaxation-based methods grows quickly with ...

Graph-based dependency parsers suffer from the sheer number of higher order edges they need to (a) score and (b) consider during optimization. Here we show that when working with LP relaxations, large fractions of these edges can be pruned before they are fully scored—without any loss of optimality guarantees and, hence, accuracy. This is achieved by iteratively parsing with a subset of highero...

Our motivation is to improve on the best approximation guarantee known for problem of finding a minimum-cost 2-node connected spanning subgraph given undirected graph with nonnegative edge costs. We present an LP (Linear Programming) relaxation based partition constraints. The special case where input contains tree zero cost called 2NC-TAP. greedy algorithm 2NC-TAP, and we analyze it via dual-f...