Fabrizio Grandoni - Academia.edu (original) (raw)
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Papers by Fabrizio Grandoni
Mathematics of Operations Research, Apr 15, 2011
Eprint Arxiv 1004 3051, Apr 18, 2010
In this paper we present a PTAS for the highway problem, hence closing the complexity status of t... more In this paper we present a PTAS for the highway problem, hence closing the complexity status of the problem. Our result is based on a novel randomized dissection approach, which has some points in common with Arora's quadtree dissection for Euclidean network design [Arora-'98]. The basic idea is enclosing the highway in a bounding path, such that both the size of the bounding path and the position of the highway in it are random variables. Then we consider a recursive O(1)-ary dissection of the bounding path, in subpaths of uniform optimal weight. Since the optimal weights are unknown, we construct the dissection in a bottom-up fashion via dynamic programming, while computing the approximate solution at the same time. Our algorithm can be easily derandomized. We demonstrate the versatility of our technique by presenting PTASs for two variants of the highway problem: the tollbooth problem with a constant number of leaves and the maximum-feasibility subsystem problem on interval matrices. In both cases the previous best approximation factors are polylogarithmic [Gamzu,Segev-'10,Elbassioni,Raman,Ray,Sitters-'09].
Information Processing Letters, Jul 1, 2003
Automata Languages and Programming 37th International Colloquium Icalp 2010 Bordeaux France July 6 10 2010 Proceedings Part I, Jul 6, 2010
Symposium on Discrete Algorithms, 2011
Corr, Feb 10, 2010
A natural way to deal with multiple, partially conflicting objectives is turning all the objectiv... more A natural way to deal with multiple, partially conflicting objectives is turning all the objectives but one into budget constraints. Some classical polynomial-time optimization problems, such as spanning tree and forest, shortest path, (perfect) matching, independent set (basis) in a matroid or in the intersection of two matroids, become NP-hard even with one budget constraint. Still, for most of these problems deterministic and randomized polynomial-time approximation schemes are known. In the case of two or more budgets, typically only multi-criteria approximation schemes are available, which return slightly infeasible solutions. Not much is known however for the case of strict budget constraints: filling this gap is the main goal of this paper. We show that shortest path, perfect matching, and spanning tree (and hence matroid basis and matroid intersection basis) are inapproximable already with two budget constraints. For the remaining problems, whose set of solutions forms an independence system, we present deterministic and randomized polynomial-time approximation schemes for a constant number k of budget constraints. Our results are based on a variety of techniques: 1. We present a simple and powerful mechanism to transform multi-criteria approximation schemes into pure approximation schemes. 2. We show that points in low dimensional faces of any matroid polytope are almost integral, an interesting result on its own. This gives a deterministic approximation scheme for k-budgeted matroid independent set. 3. We present a deterministic approximation scheme for 2-budgeted matching. The backbone of this result is a purely topological property of curves in R^2.
Lecture Notes in Computer Science, 2003
Siam Journal on Computing, 2008
Proceedings of the 17th International Conference on Algorithms and Computation, 2006
Proceedings of the Twenty Second Annual Acm Siam Symposium, 2011
Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, 2013
Proceedings of the 20th Annual European Conference on Algorithms, May 16, 2012
In the classical (min-cost) Steiner tree problem, we are given an edge-weighted undirected graph ... more In the classical (min-cost) Steiner tree problem, we are given an edge-weighted undirected graph and a set of terminal nodes. The goal is to compute a min-cost tree S which spans all terminals. In this paper we consider the min-power version of the problem, which is better suited for wireless applications. Here, the goal is to minimize the total power consumption of nodes, where the power of a node v is the maximum cost of any edge of S incident to v. Intuitively, nodes are antennas (part of which are terminals that we need to connect) and edge costs define the power to connect their endpoints via bidirectional links (so as to support protocols with ack messages). Differently from its min-cost counterpart, min-power Steiner tree is NP-hard even in the spanning tree case, i.e. when all nodes are terminals. Since the power of any tree is within once and twice its cost, computing a rho \leq ln(4)+eps [Byrka et al.'10] approximate min-cost Steiner tree provides a 2rho<2.78 approximation for the problem. For min-power spanning tree the same approach provides a 2 approximation, which was improved to 5/3+eps with a non-trivial approach in [Althaus et al.'06]. Here we present an improved approximation algorithm for min-power Steiner tree. Our result is based on two main ingredients. We prove the first decomposition theorem for min-power Steiner tree, in the spirit of analogous structural results for min-cost Steiner tree and min-power spanning tree. Based on this theorem, we define a proper LP relaxation, that we exploit within the iterative randomized rounding framework in [Byrka et al.'10]. A careful analysis provides a 3ln 4-9/4+eps<1.91 approximation factor. The same approach gives an improved 1.5+eps approximation for min-power spanning tree as well, matching the approximation factor in [Nutov and Yaroshevitch'09] for the special case of min-power spanning tree with edge weights in {0,1}.
Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, 2011
In this paper we define the multi-commodity connected facility location (MCFL) problem, a natural... more In this paper we define the multi-commodity connected facility location (MCFL) problem, a natural generalization of the well-known multi-commodity rent-or-buy (MROB) and connected facility loca-tion (CFL) network design problems. Like in MROB, we wish to send one unit of flow from a set of sources to a set of sinks. This flow is supported by renting or buying edges. For each bought edge e, we pay M ≥ 1 times the cost of e, and we are then free to route an unbounded amount of flow along e. For each rented edge e, we pay its cost times the amount of flow routed along e. However, differently from MROB and similarly to CFL, the rented paths cannot reach and leave the connected components of bought edges at any node. This must happen only at specific nodes, the open facilities, for which we have to pay an opening cost. MCFL models, e.g., a transportation system consisting of several disconnected networks (the bought edges). Users can switch from one network to the other on foot or by bik...
Lecture Notes in Computer Science, 2012
ABSTRACT In this paper we focus on problems characterized by an input n-node tree and a collectio... more ABSTRACT In this paper we focus on problems characterized by an input n-node tree and a collection of subpaths. Motivated by the fact that some of these problems admit a very good approximation (or even a poly-time exact algorithm) when the input tree is a path, we develop a decomposition theorem of trees into paths. Our decomposition allows us to partition the input problem into a collection of O(loglogn) subproblems, where in each subproblem either the input tree is a path or there exists a hitting set F of edges such that each path has a non-empty, small intersection with F. When both kinds of subproblems admit constant approximations, our method implies an O(loglogn) approximation for the original problem. We illustrate the above technique by considering two natural problems of the mentioned kind, namely Uniform Tree Tollbooth and Unique Tree Coverage. In Uniform Tree Tollbooth each subpath has a budget, where budgets are within a constant factor from each other, and we have to choose non-negative edge prices so that we maximize the total price of subpaths whose budget is not exceeded. In Unique Tree Coverage each subpath has a weight, and the goal is to select a subset X of edges so that we maximize the total weight of subpaths containing exactly one edge of X. We obtain O(loglogn) approximation algorithms for both problems. The previous best approximations are O(logn/loglogn) by Gamzu and Segev [ICALP'10] and O(logn) by Demaine et al. [SICOMP'08] for the first and second problem, respectively, however both previous results were obtained for much more general problems with arbitrary budgets (weights).
Proceedings of the 42nd ACM symposium on Theory of computing - STOC '10, 2010
Mathematics of Operations Research, Apr 15, 2011
Eprint Arxiv 1004 3051, Apr 18, 2010
In this paper we present a PTAS for the highway problem, hence closing the complexity status of t... more In this paper we present a PTAS for the highway problem, hence closing the complexity status of the problem. Our result is based on a novel randomized dissection approach, which has some points in common with Arora's quadtree dissection for Euclidean network design [Arora-'98]. The basic idea is enclosing the highway in a bounding path, such that both the size of the bounding path and the position of the highway in it are random variables. Then we consider a recursive O(1)-ary dissection of the bounding path, in subpaths of uniform optimal weight. Since the optimal weights are unknown, we construct the dissection in a bottom-up fashion via dynamic programming, while computing the approximate solution at the same time. Our algorithm can be easily derandomized. We demonstrate the versatility of our technique by presenting PTASs for two variants of the highway problem: the tollbooth problem with a constant number of leaves and the maximum-feasibility subsystem problem on interval matrices. In both cases the previous best approximation factors are polylogarithmic [Gamzu,Segev-'10,Elbassioni,Raman,Ray,Sitters-'09].
Information Processing Letters, Jul 1, 2003
Automata Languages and Programming 37th International Colloquium Icalp 2010 Bordeaux France July 6 10 2010 Proceedings Part I, Jul 6, 2010
Symposium on Discrete Algorithms, 2011
Corr, Feb 10, 2010
A natural way to deal with multiple, partially conflicting objectives is turning all the objectiv... more A natural way to deal with multiple, partially conflicting objectives is turning all the objectives but one into budget constraints. Some classical polynomial-time optimization problems, such as spanning tree and forest, shortest path, (perfect) matching, independent set (basis) in a matroid or in the intersection of two matroids, become NP-hard even with one budget constraint. Still, for most of these problems deterministic and randomized polynomial-time approximation schemes are known. In the case of two or more budgets, typically only multi-criteria approximation schemes are available, which return slightly infeasible solutions. Not much is known however for the case of strict budget constraints: filling this gap is the main goal of this paper. We show that shortest path, perfect matching, and spanning tree (and hence matroid basis and matroid intersection basis) are inapproximable already with two budget constraints. For the remaining problems, whose set of solutions forms an independence system, we present deterministic and randomized polynomial-time approximation schemes for a constant number k of budget constraints. Our results are based on a variety of techniques: 1. We present a simple and powerful mechanism to transform multi-criteria approximation schemes into pure approximation schemes. 2. We show that points in low dimensional faces of any matroid polytope are almost integral, an interesting result on its own. This gives a deterministic approximation scheme for k-budgeted matroid independent set. 3. We present a deterministic approximation scheme for 2-budgeted matching. The backbone of this result is a purely topological property of curves in R^2.
Lecture Notes in Computer Science, 2003
Siam Journal on Computing, 2008
Proceedings of the 17th International Conference on Algorithms and Computation, 2006
Proceedings of the Twenty Second Annual Acm Siam Symposium, 2011
Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms, 2013
Proceedings of the 20th Annual European Conference on Algorithms, May 16, 2012
In the classical (min-cost) Steiner tree problem, we are given an edge-weighted undirected graph ... more In the classical (min-cost) Steiner tree problem, we are given an edge-weighted undirected graph and a set of terminal nodes. The goal is to compute a min-cost tree S which spans all terminals. In this paper we consider the min-power version of the problem, which is better suited for wireless applications. Here, the goal is to minimize the total power consumption of nodes, where the power of a node v is the maximum cost of any edge of S incident to v. Intuitively, nodes are antennas (part of which are terminals that we need to connect) and edge costs define the power to connect their endpoints via bidirectional links (so as to support protocols with ack messages). Differently from its min-cost counterpart, min-power Steiner tree is NP-hard even in the spanning tree case, i.e. when all nodes are terminals. Since the power of any tree is within once and twice its cost, computing a rho \leq ln(4)+eps [Byrka et al.'10] approximate min-cost Steiner tree provides a 2rho<2.78 approximation for the problem. For min-power spanning tree the same approach provides a 2 approximation, which was improved to 5/3+eps with a non-trivial approach in [Althaus et al.'06]. Here we present an improved approximation algorithm for min-power Steiner tree. Our result is based on two main ingredients. We prove the first decomposition theorem for min-power Steiner tree, in the spirit of analogous structural results for min-cost Steiner tree and min-power spanning tree. Based on this theorem, we define a proper LP relaxation, that we exploit within the iterative randomized rounding framework in [Byrka et al.'10]. A careful analysis provides a 3ln 4-9/4+eps<1.91 approximation factor. The same approach gives an improved 1.5+eps approximation for min-power spanning tree as well, matching the approximation factor in [Nutov and Yaroshevitch'09] for the special case of min-power spanning tree with edge weights in {0,1}.
Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, 2011
In this paper we define the multi-commodity connected facility location (MCFL) problem, a natural... more In this paper we define the multi-commodity connected facility location (MCFL) problem, a natural generalization of the well-known multi-commodity rent-or-buy (MROB) and connected facility loca-tion (CFL) network design problems. Like in MROB, we wish to send one unit of flow from a set of sources to a set of sinks. This flow is supported by renting or buying edges. For each bought edge e, we pay M ≥ 1 times the cost of e, and we are then free to route an unbounded amount of flow along e. For each rented edge e, we pay its cost times the amount of flow routed along e. However, differently from MROB and similarly to CFL, the rented paths cannot reach and leave the connected components of bought edges at any node. This must happen only at specific nodes, the open facilities, for which we have to pay an opening cost. MCFL models, e.g., a transportation system consisting of several disconnected networks (the bought edges). Users can switch from one network to the other on foot or by bik...
Lecture Notes in Computer Science, 2012
ABSTRACT In this paper we focus on problems characterized by an input n-node tree and a collectio... more ABSTRACT In this paper we focus on problems characterized by an input n-node tree and a collection of subpaths. Motivated by the fact that some of these problems admit a very good approximation (or even a poly-time exact algorithm) when the input tree is a path, we develop a decomposition theorem of trees into paths. Our decomposition allows us to partition the input problem into a collection of O(loglogn) subproblems, where in each subproblem either the input tree is a path or there exists a hitting set F of edges such that each path has a non-empty, small intersection with F. When both kinds of subproblems admit constant approximations, our method implies an O(loglogn) approximation for the original problem. We illustrate the above technique by considering two natural problems of the mentioned kind, namely Uniform Tree Tollbooth and Unique Tree Coverage. In Uniform Tree Tollbooth each subpath has a budget, where budgets are within a constant factor from each other, and we have to choose non-negative edge prices so that we maximize the total price of subpaths whose budget is not exceeded. In Unique Tree Coverage each subpath has a weight, and the goal is to select a subset X of edges so that we maximize the total weight of subpaths containing exactly one edge of X. We obtain O(loglogn) approximation algorithms for both problems. The previous best approximations are O(logn/loglogn) by Gamzu and Segev [ICALP'10] and O(logn) by Demaine et al. [SICOMP'08] for the first and second problem, respectively, however both previous results were obtained for much more general problems with arbitrary budgets (weights).
Proceedings of the 42nd ACM symposium on Theory of computing - STOC '10, 2010