Finding Almost Tight Witness Trees

This paper addresses a graph optimization problem, called the Witness Tree problem, which seeks a spanning tree of a graph minimizing a certain non-linear objective function. This problem is of interest because it plays a crucial role in the analysis of the best approximation algorithms for two fundamental network design problems: Steiner Tree and Node-Tree Augmentation. We will show how a wiser choice of witness trees leads to an improved approximation for Node-Tree Augmentation, and for Steiner Tree in special classes of graphs

Approximation Algorithms for Demand Strip Packing

In the Demand Strip Packing problem (DSP), we are given a time interval and a collection of tasks, each characterized by a processing time and a demand for a given resource (such as electricity, computational power, etc.). A feasible solution consists of a schedule of the tasks within the mentioned time interval. Our goal is to minimize the peak resource consumption, i.e. the maximum total demand of tasks executed at any point in time. It is known that DSP is NP-hard to approximate below a factor 3/2, and standard techniques for related problems imply a (polynomial-time) 2-approximation. Our main result is a (5/3+?)-approximation algorithm for any constant ? > 0. We also achieve best-possible approximation factors for some relevant special cases

Choose your witnesses wisely

This paper addresses a graph optimization problem, called the Witness Tree problem, which seeks a spanning tree of a graph minimizing a certain non-linear objective function. This problem is of interest because it plays a crucial role in the analysis of the best approximation algorithms for two fundamental network design problems: Steiner Tree and Node-Tree Augmentation. We will show how a wiser choice of witness trees leads to an improved approximation for Node-Tree Augmentation, and for Steiner Tree in special classes of graphs.Comment: 33 pages, 7 figures, submitted to IPCO 202

Breaching the 2-Approximation Barrier for Connectivity Augmentation: a Reduction to Steiner Tree

The basic goal of survivable network design is to build a cheap network that maintains the connectivity between given sets of nodes despite the failure of a few edges/nodes. The Connectivity Augmentation Problem (CAP) is arguably one of the most basic problems in this area: given a $k$(-edge)-connected graph $G$ and a set of extra edges (links), select a minimum cardinality subset $A$ of links such that adding $A$ to $G$ increases its edge connectivity to $k+1$. Intuitively, one wants to make an existing network more reliable by augmenting it with extra edges. The best known approximation factor for this NP-hard problem is $2$, and this can be achieved with multiple approaches (the first such result is in [Frederickson and J\'aj\'a'81]). It is known [Dinitz et al.'76] that CAP can be reduced to the case $k=1$, a.k.a. the Tree Augmentation Problem (TAP), for odd $k$, and to the case $k=2$, a.k.a. the Cactus Augmentation Problem (CacAP), for even $k$. Several better than $2$ approximation algorithms are known for TAP, culminating with a recent $1.458$ approximation [Grandoni et al.'18]. However, for CacAP the best known approximation is $2$. In this paper we breach the $2$ approximation barrier for CacAP, hence for CAP, by presenting a polynomial-time $2\ln(4)-\frac{967}{1120}+\epsilon<1.91$ approximation. Previous approaches exploit properties of TAP that do not seem to generalize to CacAP. We instead use a reduction to the Steiner tree problem which was previously used in parameterized algorithms [Basavaraju et al.'14]. This reduction is not approximation preserving, and using the current best approximation factor for Steiner tree [Byrka et al.'13] as a black-box would not be good enough to improve on $2$. To achieve the latter goal, we ``open the box'' and exploit the specific properties of the instances of Steiner tree arising from CacAP.Comment: Corrected a typo in the abstract (in metadata

A 4/3 Approximation for 2-Vertex-Connectivity

The 2-Vertex-Connected Spanning Subgraph problem (2VCSS) is among the most basic NP-hard (Survivable) Network Design problems: we are given an (unweighted) undirected graph G. Our goal is to find a subgraph S of G with the minimum number of edges which is 2-vertex-connected, namely S remains connected after the deletion of an arbitrary node. 2VCSS is well-studied in terms of approximation algorithms, and the current best (polynomial-time) approximation factor is 10/7 by Heeger and Vygen [SIDMA\u2717] (improving on earlier results by Khuller and Vishkin [STOC\u2792] and Garg, Vempala and Singla [SODA\u2793]). Here we present an improved 4/3 approximation. Our main technical ingredient is an approximation preserving reduction to a conveniently structured subset of instances which are "almost" 3-vertex-connected. The latter reduction might be helpful in future work

Approximation algorithms for survivable network design

Many relevant discrete optimization problems are believed to be hard to solve efficiently (i.e. they cannot be solved in polynomial time unless P=NP). An approximation algorithm is one of the ways to tackle these hard optimization problems. These algorithms have polynomial running time and compute a feasible solution whose value is within a proven factor (approximation factor) of the optimal solution value. The field of approximation algorithms has grown quickly over the last few decades, leading to the development of several algorithmic and analytical techniques. In this doctoral dissertation, we focus on Survivable Network Design problems, where the goal is to construct low-cost networks that are resilient to a few edge/node faults. More specifically, we consider a basic problem in this area, that is the Connectivity Augmentation problem (CAP). In this problem, we are given a k-edge-connected graph (namely, a graph in which removing any k -1 edges preserves the connectivity of the graph) and a collection of extra edges (links). Our goal is to identify a minimum cardinality subset of links whose addition to the graph makes it (k+1)-edge connected. This problem is NP-hard and has many interesting real- world applications; For this reason it has been studied through the lens of approximation algorithms in the past. Despite the efforts of several researchers, no progress was made on this problem after the 2-approximation algorithm by Frederickson and JáJá [1981]. We remark that a 2 approximation is known even for wide generalizations of CAP. The main contribution of this thesis is breaching the 2 approximation barrier for CAP by presenting a 1.91 approximation algorithm. Our result is based on a non-trivial reduction to another fundamental problem, Steiner Tree. Along the way to this main achievement, we studied a special case of CAP, the Cycle Augmentation problem (CycAP) for which 2 was the best-known approximation factor. Here we are given a cycle plus additional links, and the goal is to find a subset of links with minimum size whose addition to G makes it 3-edge-connected. We show that CycAP is APX-hard, in particular it does not admit an approximation factor arbitrarily close to 1 (even the NP-hardness of this problem was not known earlier). Furthermore, we present a 3/2+ε approximation algorithm for any constant ε > 0

Breaching the 2-Approximation Barrier for the Forest Augmentation Problem

The basic goal of survivable network design is to build cheap networks that guarantee the connectivity of certain pairs of nodes despite the failure of a few edges or nodes. A celebrated result by Jain [Combinatorica'01] provides a 2-approximation for a wide class of these problems. However nothing better is known even for very basic special cases, raising the natural question whether any improved approximation factor is possible at all. In this paper we address one of the most basic problems in this family for which 2 is still the best-known approximation factor, the Forest Augmentation Problem (FAP): given an undirected unweighted graph (that w.l.o.g. is a forest) and a collection of extra edges (links), compute a minimum cardinality subset of links whose addition to the graph makes it 2-edge-connected. Several better-than-2 approximation algorithms are known for the special case where the input graph is a tree, a.k.a. the Tree Augmentation Problem (TAP). Recently this was achieved also for the weighted version of TAP, and for the k-edge-connectivity generalization of TAP. These results heavily exploit the fact that the input graph is connected, a condition that does not hold in FAP. In this paper we breach the 2-approximation barrier for FAP. Our result is based on two main ingredients. First, we describe a reduction to the Path Augmentation Problem (PAP), the special case of FAP where the input graph is a collection of disjoint paths. Our reduction is not approximation preserving, however it is sufficiently accurate to improve on a factor 2 approximation. Second, we present a better-than-2 approximation algorithm for PAP, an open problem on its own. Here we exploit a novel notion of implicit credits which might turn out to be helpful in future related work

Breaching the 2-approximation barrier for the forest augmentation problem

The basic goal of survivable network design is to build cheap networks that guarantee the connectivity of certain pairs of nodes despite the failure of a few edges or nodes. A celebrated result by Jain [Combinatorica'01] provides a 2-approximation for a wide class of these problems. However nothing better is known even for very basic special cases, raising the natural question whether any improved approximation factor is possible at all. In this paper we address one of the most basic problems in this family for which 2 is still the best-known approximation factor, the Forest Augmentation Problem (FAP): given an undirected unweighted graph (that w.l.o.g. we can assume to be a forest) and a collection of extra edges (links), compute a minimum cardinality subset of links whose addition to the graph makes it 2-edge-connected. Several better-than-2 approximation algorithms are known for the special case where the input graph is a tree, a.k.a. the Tree Augmentation Problem (TAP), see e.g. [Grandoni, Kalaitzis, Zenklusen-STOC'18; Cecchetto, Traub, Zenklusen-STOC'21] and references therein. Recently this was achieved also for the weighted version of TAP [Traub, Zenklusen-FOCS'21], and for the k-connectivity generalization of TAP [Byrka, Grandoni, Jabal-Ameli-STOC'20; Cecchetto, Traub, Zenklusen-STOC'21]. These results heavily exploit the fact that the input graph is connected, a condition that does not hold in FAP. In this paper we breach the 2-approximation barrier for FAP. Our result is based on two main ingredients. First, we describe a reduction to the Path Augmentation Problem (PAP), the special case of FAP where the input graph is a collection of disjoint paths. Our reduction is not approximation preserving, however it is sufficiently accurate to improve on a factor 2 approximation. Second, we present a better-than-2 approximation algorithm for PAP, an open problem on its own. Here we exploit a novel notion of implicit credits which might turn out to be helpful in future related work

A Tight (3/2+ε) Approximation for Skewed Strip Packing

In the Strip Packing problem, we are given a vertical half-strip [0 , W] × [0 , + ∞) and a collection of open rectangles of width at most W. Our goal is to find an axis-aligned (non-overlapping) packing of such rectangles into the strip such that the maximum height OPT spanned by the packing is as small as possible. It is NP-hard to approximate this problem within a factor (3 / 2 - ε) for any constant ε&gt; 0 by a simple reduction from the Partition problem, while the current best approximation factor for it is (5 / 3 + ε) . It seems plausible that Strip Packing admits a (3 / 2 + ε) -approximation. We make progress in that direction by achieving such tight approximation guarantees for a special family of instances, which we call skewed instances. As standard in the area, for a given constant parameter δ&gt; 0 , we call large the rectangles with width at least δW and height at least δOPT , and skewed the remaining rectangles. If all the rectangles in the input are large, then one can easily compute the optimal packing in polynomial time (since the input can contain only a constant number of rectangles). We consider the complementary case where all the rectangles are skewed. This second case retains a large part of the complexity of the original problem; in particular, the skewed case is still NP-hard to approximate within a factor (3 / 2 - ε) , and we provide an (almost) tight (3 / 2 + ε) -approximation algorithm.</p