The Connectivity Augmentation Problem (CAP) is a fundamental problem in fault-tolerant network design and has been extensively studied in the context of approximation algorithms. In this work, we consider CAP in the online setting: given a \(k\)-edge-connected graph \(G\) and a set \(L\) of additional edges over the vertices of \(G\), called links, online requests arrive one by one, each specifying two vertices that need to be \((k + 1)\)-edge-connected. We start with the graph \(G\) and progressively add links to serve these requests. More specifically, upon the arrival of a request \(\{u, v\}\), we must immediately and irrevocably add zero or more links from \(L\) to the graph so that \(u\) and \(v\) are \((k + 1)\)-edge-connected in the resulting augmented graph. The goal is to minimize the total number of links added, and we evaluate an algorithm’s performance by its competitive ratio relative to an optimal offline solution. In this work, improving upon previous bounds, we obtain a tight competitive ratio for online CAP, along with other related results.

In the Maximum Independent Set of Hyperrectangles problem, we are given a set of $n$ (possibly overlapping) $d$-dimensional axis-aligned hyperrectangles, and the goal is to find a subset of non-overlapping hyperrectangles of maximum cardinality. For $d=1$, this corresponds to the classical Interval Scheduling problem, where a simple greedy algorithm returns an optimal solution. In the offline setting, for $d$-dimensional hyperrectangles, polynomial time $(\log n)^{O(d)}$-approximation algorithms are known. However, the problem becomes notably challenging in the online setting, where the input objects (hyperrectangles) appear one by one in an adversarial order, and on the arrival of an object, the algorithm needs to make an immediate and irrevocable decision whether or not to select the object while maintaining the feasibility. Even for interval scheduling, an $Ω(n)$ lower bound is known on the competitive ratio. To circumvent these negative results, in this work, we study the online maximum independent set of axis-aligned hyperrectangles in the random-order arrival model, where the adversary specifies the set of input objects which then arrive in a uniformly random order. Starting from the prototypical secretary problem, the random-order model has received significant attention to study algorithms beyond the worst-case competitive analysis. Surprisingly, we show that the problem in the random-order model almost matches the best-known offline approximation guarantees, up to polylogarithmic factors. In particular, we give a simple $(\log n)^{O(d)}$-competitive algorithm for $d$-dimensional hyperrectangles in this model, which runs in $\tilde{O_d}(n)$ time. Our approach also yields $(\log n)^{O(d)}$-competitive algorithms in the random-order model for more general objects such as $d$-dimensional fat objects and ellipsoids. Furthermore, our guarantees hold with high probability.

We consider the matching augmentation problem (MAP), where a matching of a graph needs to be extended into a $2$-edge-connected spanning subgraph by adding the minimum number of edges to it. We present a polynomial-time algorithm with an approximation ratio of $13/8 = 1.625$ improving upon an earlier $5/3$-approximation. The improvement builds on a new $α$-approximation preserving reduction for any $α\geq 3/2$ from arbitrary MAP instances to well-structured instances that do not contain certain forbidden structures like parallel edges, small separators, and contractible subgraphs. We further introduce, as key ingredients, the technique of repeated simultaneous contractions and provide improved lower bounds for instances that cannot be contracted.

The basic goal of survivable network design is to construct low-cost networks which preserve a sufficient level of connectivity despite the failure or removal of a few nodes or edges. One of the most basic problems in this area is the 2-Edge-Connected Spanning Subgraph problem (2-ECSS): given an undirected graph G, find a 2-edge-connected spanning subgraph H of G with the minimum number of edges (in particular, H remains connected after the removal of one arbitrary edge). 2-ECSS is NP-hard and the best-known (polynomial-time) approximation factor for this problem is 4/3. Interestingly, this factor was achieved with drastically different techniques by [Hunkenschröder, Vempala and Vetta '00,'19] and [Sebö and Vygen, '14]. In this paper we present an improved approximation for 2-ECSS. The key ingredient in our approach (which might also be helpful in future work) is a reduction to a special type of structured graphs: our reduction preserves approximation factors up to 6/5. While reducing to 2-vertex-connected graphs is trivial (and heavily used in prior work), our structured graphs are “almost” 3-vertex-connected: more precisely, given any 2-vertex-cut {u, v} of a structured graph G = (V, E), G[V \ {u, v}] has exactly 2 connected components, one of which contains exactly one node of degree 2 in G. * Partially supported by the SNSF Excellence Grant 200020B 182865/1 and the SNSF Grant 200021 200731/1.

Abstract. We study the problem of maximizing a monotone submodular function subject to a matroid constraint and present a deterministic algorithm that achieves [Formula: see text]-approximation for the problem (for some [Formula: see text]). This algorithm is the first deterministic algorithm known to improve over the [Formula: see text]-approximation ratio of the classical greedy algorithm proved by Nemhauser, Wolsey, and Fisher in 1978.

If $k

We study the problem of maximizing a monotone submodular function subject to a matroid constraint and present a deterministic algorithm that achieves (½ + ε)-approximation for the problem. This algorithm is the first deterministic algorithm known to improve over the ½-approximation ratio of the classical greedy algorithm proved by Nemhauser, Wolsely and Fisher in 1978.

We consider the non-adaptive bit-probe complexity of the set membership problem, where a set S of size at most n from a universe of size m is to be represented as a short bit vector in order to answer membership queries of the form "Is x in S?" by non-adaptively probing the bit vector at t places. Let s_N(m,n,t) be the minimum number of bits of storage needed for such a scheme. In this work, we show existence of non-adaptive and adaptive schemes for a range of t that improves an upper bound of Buhrman, Miltersen, Radhakrishnan and Srinivasan (2002) on s_N(m,n,t). For three non-adaptive probes, we improve the previous best lower bound on s_N(m,n,3) by Alon and Feige (2009).

We consider the bit-probe complexity of the set membership problem, where a set S of size at most n from a universe of size m is to be represented as a short bit vector in order to answer membership queries of the form Is x in S? by adaptively probing the bit vector at t places. Let s(m,n,t) be the minimum number of bits of storage needed for such a scheme. Several recent works investigate s(m,n,t) for various ranges of the parameter; we obtain improvements over some of the bounds shown by Buhrman, Miltersen, Radhakrishnan, and Srinivasan (2002) and Alon and Feige (2009).