Parameterized approximation algorithms for bidirected Steiner network problems

The Directed Steiner Network (DSN) problem takes as input a directed graph G=(V, E) with non-negative edge-weights and a set D⊆ V × V of k demand pairs. The aim is to compute the cheapest network N⊆ G for which there is an s\rightarrow t path for each (s, t)∈ D. It is known that this problem is notoriously hard, as there is no k1/4−o(1)-approximation algorithm under Gap-ETH, even when parametrizing the runtime by k [Dinur & Manurangsi, ITCS 2018]. In light of this, we systematically study several special cases of DSN and determine their parameterized approximability for the parameter k. For the bi-DSNPlanar problem, the aim is to compute a solution N⊆ G whose cost is at most that of an optimum planar solution in a bidirected graph G, i.e., for every edge uv of G the reverse edge vu exists and has the same weight. This problem is a generalization of several well-studied special cases. Our main result is that this problem admits a parameterized approximation scheme (PAS) for k. We also prove that our result is tight in the sense that (a) the runtime of our PAS cannot be significantly improved, and (b) no PAS exists for any generalization of bi-DSNPlanar, under standard complexity assumptions. The techniques we use also imply a polynomial-sized approximate kernelization scheme (PSAKS). Additionally, we study several generalizations of bi-DSNPlanar and obtain upper and lower bounds on obtainable runtimes parameterized by k. One important special case of DSN is the Strongly Connected Steiner Subgraph (SCSS) problem, for which the solution network N⊆ G needs to strongly connect a given set of k terminals. It has been observed before that for SCSS a parameterized 2-approximation exists for parameter k [Chitnis et al., IPEC 2013]. We give a tight inapproximability result by showing that for k no parameterized (2 − ε)-approximation algorithm exists under Gap-ETH. Additionally, we show that when restricting the input of SCSS to bidirected graphs, the problem remains NP-hard but becomes FPT for k

Asymptotic existence of fair divisions for groups

The problem of dividing resources fairly occurs in many practical situations and is therefore an important topic of study in economics. In this paper, we investigate envy-free divisions in the setting where there are multiple players in each interested party. While all players in a party share the same set of resources, each player has her own preferences. Under additive valuations drawn randomly from probability distributions, we show that when all groups contain an equal number of players, a welfare-maximizing allocation is likely to be envy-free if the number of items exceeds the total number of players by a logarithmic factor. On the other hand, an envy-free allocation is unlikely to exist if the number of items is less than the total number of players. In addition, we show that a simple truthful mechanism, namely the random assignment mechanism, yields an allocation that satisfies the weaker notion of approximate envy-freeness with high probability

Computing a small agreeable set of indivisible items

We study the problem of assigning a small subset of indivisible items to a group of agents so that the subset is agreeable to all agents, meaning that all agents value the subset as least as much as its complement. For an arbitrary number of agents and items, we derive a tight worst-case bound on the number of items that may need to be included in such a set. We then present polynomial-time algorithms that find an agreeable set whose size matches the worst-case bound when there are two or three agents. We also show that finding small agreeable sets is possible even when we only have access to the agents' preferences on single items. Furthermore, we investigate the problem of efficiently computing an agreeable set whose size approximates the size of the smallest agreeable set for any given instance. We consider two well-known models for representing the preferences of the agents—the value oracle model and additive utilities—and establish tight bounds on the approximation ratio that can be obtained by algorithms running in polynomial time in each of these models

Asymptotic existence of fair divisions for groups

The problem of dividing resources fairly occurs in many practical situations and is therefore an important topic of study in economics. In this paper, we investigate envy-free divisions in the setting where there are multiple players in each interested party. While all players in a party share the same set of resources, each player has her own preferences. Under additive valuations drawn randomly from probability distributions, we show that when all groups contain an equal number of players, a welfare-maximizing allocation is likely to be envy-free if the number of items exceeds the total number of players by a logarithmic factor. On the other hand, an envy-free allocation is unlikely to exist if the number of items is less than the total number of players. In addition, we show that a simple truthful mechanism, namely the random assignment mechanism, yields an allocation that satisfies the weaker notion of approximate envy-freeness with high probability

The price of fairness for indivisible goods

We investigate the efficiency of fair allocations of indivisible goods using the well-studied price of fairness concept. Previous work has focused on classical fairness notions such as envy-freeness, proportionality, and equitability. However, these notions cannot always be satisfied for indivisible goods, leading to certain instances being ignored in the analysis. In this paper, we focus instead on notions with guaranteed existence, including envy-freeness up to one good (EF1), balancedness, maximum Nash welfare (MNW), and leximin. We mostly provide tight or asymptotically tight bounds on the worst-case efficiency loss for allocations satisfying these notions

The price of fairness for indivisible goods

We investigate the efficiency of fair allocations of indivisible goods using the well-studied price of fairness concept. Previous work has focused on classical fairness notions such as envy-freeness, proportionality, and equitability. However, these notions cannot always be satisfied for indivisible goods, leading to certain instances being ignored in the analysis. In this paper, we focus instead on notions with guaranteed existence, including envy-freeness up to one good (EF1), balancedness, maximum Nash welfare (MNW), and leximin. We mostly provide tight or asymptotically tight bounds on the worst-case efficiency loss for allocations satisfying these notions

Consensus halving for sets of items

Consensus halving refers to the problem of dividing a resource into two parts so that every agent values both parts equally. Prior work shows that, when the resource is represented by an interval, a consensus halving with at most n cuts always exists but is hard to compute even for agents with simple valuation functions. In this paper, we study consensus halving in a natural setting in which the resource consists of a set of items without a linear ordering. For agents with linear and additively separable utilities, we present a polynomial-time algorithm that computes a consensus halving with at most n cuts and show that n cuts are almost surely necessary when the agents’ utilities are randomly generated. On the other hand, we show that, for a simple class of monotonic utilities, the problem already becomes polynomial parity argument, directed version–hard. Furthermore, we compare and contrast consensus halving with the more general problem of consensus k-splitting, with which we wish to divide the resource into k parts in possibly unequal ratios and provide some consequences of our results on the problem of computing small agreeable sets

Average Whenever You Meet: Opportunistic Protocols for Community Detection

Consider the following asynchronous, opportunistic communication model over a graph G: in each round, one edge is activated uniformly and independently at random and (only) its two endpoints can exchange messages and perform local computations. Under this model, we study the following random process: The first time a vertex is an endpoint of an active edge, it chooses a random number, say +/- 1 with probability 1/2; then, in each round, the two endpoints of the currently active edge update their values to their average. We provide a rigorous analysis of the above process showing that, if G exhibits a two-community structure (for example, two expanders connected by a sparse cut), the values held by the nodes will collectively reflect the underlying community structure over a suitable phase of the above process. Our analysis requires new concentration bounds on the product of certain random matrices that are technically challenging and possibly of independent interest. We then exploit our analysis to design the first opportunistic protocols that approximately recover community structure using only logarithmic (or polylogarithmic, depending on the sparsity of the cut) work per node

Robust secret sharing with almost optimal share size and security against rushing adversaries

We show a robust secret sharing scheme for a maximal threshold t < n/2 that features an optimal overhead in share size, offers security against a rushing adversary, and runs in polynomial time. Previous robust secret sharing schemes for t < n/2 either suffered from a suboptimal overhead, offered no (provable) security against a rushing adversary, or ran in superpolynomial time