Davide Bilò

Reoptimization of Steiner Trees

In this paper we study the problem of finding a minimum Steiner Tree given a minimum Steiner Tree for similar problem instance. We consider scenarios of altering an instance by locally changing the terminal set or the weight of an edge. For all modification scenarios we provide approximation algorithms that improve best currently known corresponding approximation ratios.

Reoptimization of Weighted Graph and Covering Problems

Given an instance of an optimization problem and a good solution of that instance, the reoptimization is a concept of analyzing how does the solution change when the instance is locally modified. We investigate reoptimization of the following problems: Maximum Weighted Independent Set, Maximum Weighted Clique, Minimum Weighted Dominating Set, Minimum Weighted Set Cover and Minimum Weighted Vertex Cover. The local modifications we consider are addition or removal of a constant number of edges to the graph, or elements to the covering sets in case of Set Cover problem. We present the following results: 1 We provide a PTAS for reoptimization of the unweighted versions of the aforementioned problems when the input solution is optimal. 1 We provide two general techniques for analyzing approximation ratio of the weighted reoptimization problems. 1 We apply our techniques to reoptimization of the considered optimization problems and obtain tight approximation ratios in all the cases.

Reconstructing visibility graphs with simple robots

We consider the problem of finding a minimalistic configuration of sensors that enable a simple robot inside an initially unknown polygon P on n vertices to reconstruct the visibility graph of P. The robot can sense features of its environment through its sensors, and it is allowed to move from vertex to vertex. We aim at understanding which sensorial capabilities are sufficient for the reconstruction of the visibility graph of P. We are able to show that the combinatorial visibilities at every vertex do not contain enough information even when combined with the knowledge of the exact interior angle at each vertex. Using sensors that can put distant vertices into a spatial relation on the other hand can in some cases enable our robot to reconstruct the visibility graph of P. We show that this is true for a sensor that can distinguish whether the angle between the lines toward two visible vertices is convex or reflex, as long as the robot is capable of identifying the vertex it last visited. We also show that measuring angles exactly is enough, if the robot has a compass.

Bounded-Distance Network Creation Games

A <i>network creation game</i> simulates a decentralized and noncooperative construction of a communication network. Informally, there are <i>n</i> players sitting on the network nodes, which attempt to establish a reciprocal communication by activating, thereby incurring a certain cost, any of their incident links. The goal of each player is to have all the other nodes as close as possible in the resulting network, while buying as few links as possible. According to this intuition, any model of the game must then appropriately address a balance between these two conflicting objectives. Motivated by the fact that a player might have a strong requirement about her centrality in the network, we introduce a new setting in which a player who maintains her (maximum or average) distance to the other nodes within a given bound incurs a cost equal to the number of activated edges; otherwise her cost is unbounded. We study the problem of understanding the structure of pure Nash equilibria of the resulting games, which we call M<scp>ax</scp>BD and S<scp>um</scp>BD, respectively. For both games, we show that when distance bounds associated with players are nonuniform, then equilibria can be arbitrarily bad. On the other hand, for M<scp>ax</scp>BD, we show that when nodes have a uniform bound <i>D</i> ≥ 3 on the maximum distance, then the <i>price of anarchy</i> (PoA) is lower and upper bounded by 2 and <i>O</i>(<i>n</i>^{1/⌊log₃ <i>D</i> ⌋+1}), respectively (i.e., PoA is constant as soon as <i>D</i> is Ω(<i>n</i>^ε), for any ε > 0), while for the interesting case <i>D</i>=2, we are able to prove that the PoA is Ω(&sqrt;<i>n</i>) and <i>O</i>(&sqrt;<i>n</i> log <i>n</i>). For the uniform S<scp>um</scp>BD, we obtain similar (asymptotically) results and moreover show that PoA becomes constant as soon as the bound on the average distance is 2^{<i>ω</i>(&sqrt;log <i>n</i>)}.

Improved approximability and non-approximability results for graph diameter decreasing problems

In this paper, we study two variants of the problem of adding edges to a graph so as to reduce the resulting diameter. More precisely, given a graph G=(V,E), and two positive integers D and B, the Minimum-Cardinality Bounded-Diameter Edge Addition (MCBD) problem is to find a minimum-cardinality set F of edges to be added to G in such a way that the diameter of G+F is less than or equal to D, while the Bounded-Cardinality Minimum-Diameter Edge Addition (BCMD) problem is to find a set F of B edges to be added to G in such a way that the diameter of G+F is minimized. Both problems are well known to be NP-hard, as well as approximable within O(lognlogD) and 4 (up to an additive term of 2), respectively. In this paper, we improve these long-standing approximation ratios to O(logn) and to 2 (up to an additive term of 2), respectively. As a consequence, we close, in an asymptotic sense, the gap on the approximability of MCBD, which was known to be not approximable within clogn, for some constant c>0, unless P=NP. Remarkably, as we further show in the paper, our approximation ratio remains asymptotically tight even if we allow for a solution whose diameter is optimal up to a multiplicative factor approaching 53. On the other hand, on the positive side, we show that at most twice of the minimal number of additional edges suffices to get at most twice of the required diameter. Some of our results extend to the edge-weighted version of the problems.

Bounded-Distance network creation games

A network creation game simulates a decentralized and non-cooperative building of a communication network. Informally, there are n players sitting on the network nodes, which attempt to establish a reciprocal communication by activating, incurring a certain cost, any of their incident links. The goal of each player is to have all the other nodes as close as possible in the resulting network, while buying as few links as possible. According to this intuition, any model of the game must then appropriately address a balance between these two conflicting objectives. Motivated by the fact that a player might have a strong requirement about its centrality in the network, in this paper we introduce a new setting in which if a player maintains its (either maximum or average) distance to the other nodes within a given bound, then its cost is simply equal to the number of activated edges, otherwise its cost is unbounded. We study the problem of understanding the structure of pure Nash equilibria of the resulting games, that we call MaxBD and SumBD, respectively. For both games, we show that when distance bounds associated with players are non-uniform, then equilibria can be arbitrarily bad. On the other hand, for MaxBD, we show that when nodes have a uniform bound R on the maximum distance, then the Price of Anarchy (PoA) is lower and upper bounded by 2 and

Computational Aspects of a 2-Player Stackelberg Shortest Paths Tree Game

O\left(n^{\frac{1}{\lfloor\log_3 R\rfloor+1}}\right)

Improved Purely Additive Fault-Tolerant Spanners

for R≥3 (i.e., the PoA is constant as soon as R is Ω(ne), for some e>0), while for the interesting case R=2, we are able to prove that the PoA is

NEW REOPTIMIZATION TECHNIQUES APPLIED TO STEINER TREE PROBLEM

\Omega(\sqrt{n})

Fault-Tolerant Approximate Shortest-Path Trees

and