238 research outputs found
Provably Good Solutions to the Knapsack Problem via Neural Networks of Bounded Size
The development of a satisfying and rigorous mathematical understanding of
the performance of neural networks is a major challenge in artificial
intelligence. Against this background, we study the expressive power of neural
networks through the example of the classical NP-hard Knapsack Problem. Our
main contribution is a class of recurrent neural networks (RNNs) with rectified
linear units that are iteratively applied to each item of a Knapsack instance
and thereby compute optimal or provably good solution values. We show that an
RNN of depth four and width depending quadratically on the profit of an optimum
Knapsack solution is sufficient to find optimum Knapsack solutions. We also
prove the following tradeoff between the size of an RNN and the quality of the
computed Knapsack solution: for Knapsack instances consisting of items, an
RNN of depth five and width computes a solution of value at least
times the optimum solution value. Our results
build upon a classical dynamic programming formulation of the Knapsack Problem
as well as a careful rounding of profit values that are also at the core of the
well-known fully polynomial-time approximation scheme for the Knapsack Problem.
A carefully conducted computational study qualitatively supports our
theoretical size bounds. Finally, we point out that our results can be
generalized to many other combinatorial optimization problems that admit
dynamic programming solution methods, such as various Shortest Path Problems,
the Longest Common Subsequence Problem, and the Traveling Salesperson Problem.Comment: A short version of this paper appears in the proceedings of AAAI 202
Approximation Schemes for Subset Sum Ratio Problems
We consider the Subset Sum Ratio Problem (), in which given a set of integers the goal is to find two subsets such that the ratio of their sums is as close to~1 as possible, and introduce a family of variations that capture additional meaningful requirements. Our main contribution is a generic framework that yields fully polynomial time approximation schemes (FPTAS) for problems in this family that meet certain conditions. We use our framework to design explicit FPTASs for two such problems, namely Two-Set Subset-Sum Ratio and Factor- Subset-Sum Ratio, with running time , which coincides with the best known running time for the original problem [15]
On Approximating Multi-Criteria TSP
We present approximation algorithms for almost all variants of the
multi-criteria traveling salesman problem (TSP).
First, we devise randomized approximation algorithms for multi-criteria
maximum traveling salesman problems (Max-TSP). For multi-criteria Max-STSP,
where the edge weights have to be symmetric, we devise an algorithm with an
approximation ratio of 2/3 - eps. For multi-criteria Max-ATSP, where the edge
weights may be asymmetric, we present an algorithm with a ratio of 1/2 - eps.
Our algorithms work for any fixed number k of objectives. Furthermore, we
present a deterministic algorithm for bi-criteria Max-STSP that achieves an
approximation ratio of 7/27.
Finally, we present a randomized approximation algorithm for the asymmetric
multi-criteria minimum TSP with triangle inequality Min-ATSP. This algorithm
achieves a ratio of log n + eps.Comment: Preliminary version at STACS 2009. This paper is a revised full
version, where some proofs are simplifie
A Relaxed FPTAS for Chance-Constrained Knapsack
The stochastic knapsack problem is a stochastic version of the well known deterministic knapsack problem, in which some of the input values are random variables. There are several variants of the stochastic problem. In this paper we concentrate on the chance-constrained variant, where item values are deterministic and item sizes are stochastic. The goal is to find a maximum value allocation subject to the constraint that the overflow probability is at most a given value. Previous work showed a PTAS for the problem for various distributions (Poisson, Exponential, Bernoulli and Normal). Some strictly respect the constraint and some relax the constraint by a factor of (1+epsilon). All algorithms use Omega(n^{1/epsilon}) time. A very recent work showed a "almost FPTAS" algorithm for Bernoulli distributions with O(poly(n) * quasipoly(1/epsilon)) time.
In this paper we present a FPTAS for normal distributions with a solution that satisfies the chance constraint in a relaxed sense. The normal distribution is particularly important, because by the Berry-Esseen theorem, an algorithm solving the normal distribution also solves, under mild conditions, arbitrary independent distributions. To the best of our knowledge, this is the first (relaxed or non-relaxed) FPTAS for the problem. In fact, our algorithm runs in poly(n/epsilon) time. We achieve the FPTAS by a delicate combination of previous techniques plus a new alternative solution to the non-heavy elements that is based on a non-convex program with a simple structure and an O(n^2 log {n/epsilon}) running time. We believe this part is also interesting on its own right
Knapsack Problems with Side Constraints
The thesis considers a specific class of resource allocation problems in Combinatorial Optimization: the Knapsack Problems. These are paradigmatic NP-hard problems where a set of items with given profits and weights is available. The aim is to select a subset of the items in order to maximize the total profit without exceeding a known knapsack capacity. In the classical 0-1 Knapsack Problem (KP), each item can be picked at most once.
The focus of the thesis is on four generalizations of KP involving side constraints beyond the capacity bound. More precisely, we provide solution approaches and insights for the following problems: The Knapsack Problem with Setups; the Collapsing Knapsack Problem; the Penalized Knapsack Problem; the Incremental Knapsack Problem.
These problems reveal challenging research topics with many real-life applications. The scientific contributions we provide are both from a theoretical and a practical perspective. On the one hand, we give insights into structural elements and properties of the problems and derive a series of approximation results for some of them. On the other hand, we offer valuable solution approaches for direct applications of practical interest or when the problems considered arise as sub-problems in broader contexts
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