# Difference between revisions of "Discrete Time-Cost Tradeoff Problem"

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= Problem Statement = | = Problem Statement = | ||

− | Let <math> | + | Let <math>G</math> be an acyclic digraph. |

− | + | For each edge <math>e</math> we have execution times <math> 0 \le \tau_1(e) \le \tau_2(e) </math> | |

+ | and an acceleration cost <math> c(e) </math> (that we pay if we make the edge fast). | ||

+ | Moreover, there is a deadline <math>D\ge 0</math>. | ||

+ | Let <math>\mathcal{P}</math> be the set of all maximal paths in <math>G</math> | ||

− | = | + | The task is to find an edge set <math> F\subseteq E(G) </math> (the fast edges) such that |

+ | <math> \sum_{e\in E(P)\cap F} \tau_1(e) + \sum_{e\in E(P)\setminus F} \tau_2(e) \le D</math> | ||

+ | for all <math> P\in\mathcal{P}</math>. | ||

+ | Among all such solutions we would like to minimize <math>c(F):=\sum_{e\in F} c(e)</math>. | ||

+ | |||

+ | = What is known? = | ||

+ | |||

+ | The best known approximation algorithm has approximation ratio <math> \max_{P\in\mathcal{P}} |E(P)| </math>. | ||

+ | The problem cannot be approximated better than VERTEX COVER, and there is no constant-factor | ||

+ | approximation algorithm if the Unique Games Conjecture holds. | ||

+ | |||

+ | The standard LP (via set covering) has integrality ratio <math>\Omega(n)</math>. | ||

+ | Knapsack cover inequalities do not help. | ||

+ | |||

+ | = Challenge = | ||

+ | |||

+ | Find an <math>o(n)</math>-approximation algorithm. | ||

+ | Find an LP (or SDP) formulation with an <math>o(n)</math> integrality ratio. | ||

+ | |||

+ | = Please add comments here = |

## Latest revision as of 08:58, 22 September 2015

(suggested by Jens Vygen)

# Problem Statement

Let be an acyclic digraph. For each edge we have execution times and an acceleration cost (that we pay if we make the edge fast). Moreover, there is a deadline . Let be the set of all maximal paths in

The task is to find an edge set (the fast edges) such that for all . Among all such solutions we would like to minimize .

# What is known?

The best known approximation algorithm has approximation ratio . The problem cannot be approximated better than VERTEX COVER, and there is no constant-factor approximation algorithm if the Unique Games Conjecture holds.

The standard LP (via set covering) has integrality ratio . Knapsack cover inequalities do not help.

# Challenge

Find an -approximation algorithm. Find an LP (or SDP) formulation with an integrality ratio.