On Approximability of Steiner Tree in ℓp\ell_p-metrics

In the Continuous Steiner Tree problem (CST), we are given as input a set of points (called terminals) in a metric space and ask for the minimum-cost tree connecting them. Additional points (called Steiner points) from the metric space can be introduced as nodes in the solution. In the Discrete Steiner Tree problem (DST), we are given in addition to the terminals, a set of facilities, and any solution tree connecting the terminals can only contain the Steiner points from this set of facilities. Trevisan [SICOMP'00] showed that CST and DST are APX-hard when the input lies in the $\ell_1$-metric (and Hamming metric). Chleb\'ik and Chleb\'ikov\'a [TCS'08] showed that DST is NP-hard to approximate to factor of $96/95\approx 1.01$ in the graph metric (and consequently $\ell_\infty$-metric). Prior to this work, it was unclear if CST and DST are APX-hard in essentially every other popular metric! In this work, we prove that DST is APX-hard in every $\ell_p$-metric. We also prove that CST is APX-hard in the $\ell_{\infty}$-metric. Finally, we relate CST and DST, showing a general reduction from CST to DST in $\ell_p$-metrics. As an immediate consequence, this yields a $1.39$-approximation polynomial time algorithm for CST in $\ell_p$-metrics.Comment: Abstract shortened due to arxiv's requirement

Approximability of Combinatorial Optimization Problems on Power Law Networks

One of the central parts in the study of combinatorial optimization is to classify the NP-hard optimization problems in terms of their approximability. In this thesis we study the Minimum Vertex Cover (Min-VC) problem and the Minimum Dominating Set (Min-DS) problem in the context of so called power law graphs. This family of graphs is motivated by recent findings on the degree distribution of existing real-world networks such as the Internet, the World-Wide Web, biological networks and social networks. In a power law graph the number of nodes yi of a given degree i is proportional to i-ß, that is, the distribution of node degrees follows a power law. The parameter ß > 0 is the so called power law exponent. With the aim of classifying the above combinatorial optimization problems, we are pursuing two basic approaches in this thesis. One is concerned with complexity theory and the other with the theory of algorithms. As a result, our main contributions to the classification of the problems Min-VC and Min-DS in the context of power law graphs are twofold: - Firstly, we give substantial improvements on the previously known approximation lower bounds for Min-VC and Min-DS in combinatorial power law graphs. More precisely, we are going to show the APX-hardness of Min-VC and Min-DS in connected power law graphs and give constant factor lower bounds for Min-VC and the first logarithmic lower bounds for Min-DS in this setting. The results are based on new approximation-preserving embedding reductions that embed certain instances of Min-VC and Min-DS into connected power law graphs. - Secondly, we design a new approximation algorithm for the Min-VC problem in random power law graphs with an expected approximation ratio strictly less than 2. The main tool is a deterministic rounding procedure that acts on a half-integral solution for Min-VC and produces a good approximation on the subset of low degree vertices. Moreover, for the case of Min-DS, we improve on the previously best upper bounds that rely on a greedy algorithm. The improvements are based on our new techniques for determining upper and lower bounds on the size and the volume of node intervals in power law graphs

ON EXPRESSIVENESS, INFERENCE, AND PARAMETER ESTIMATION OF DISCRETE SEQUENCE MODELS

Huge neural autoregressive sequence models have achieved impressive performance across different applications, such as NLP, reinforcement learning, and bioinformatics. However, some lingering problems (e.g., consistency and coherency of generated texts) continue to exist, regardless of the parameter count. In the first part of this thesis, we chart a taxonomy of the expressiveness of various sequence model families (Ch 3). In particular, we put forth complexity-theoretic proofs that string latent-variable sequence models are strictly more expressive than energy-based sequence models, which in turn are more expressive than autoregressive sequence models. Based on these findings, we introduce residual energy-based sequence models, a family of energy-based sequence models (Ch 4) whose sequence weights can be evaluated efficiently, and also perform competitively against autoregressive models. However, we show how unrestricted energy-based sequence models can suffer from uncomputability; and how such a problem is generally unfixable without knowledge of the true sequence distribution (Ch 5). In the second part of the thesis, we study practical sequence model families and algorithms based on theoretical findings in the first part of the thesis. We introduce neural particle smoothing (Ch 6), a family of approximate sampling methods that work with conditional latent variable models. We also introduce neural finite-state transducers (Ch 7), which extend weighted finite state transducers with the introduction of mark strings, allowing scoring transduction paths in a finite state transducer with a neural network. Finally, we propose neural regular expressions (Ch 8), a family of neural sequence models that are easy to engineer, allowing a user to design flexible weighted relations using Marked FSTs, and combine these weighted relations together with various operations

O(log⁡2k/log⁡log⁡k)O(\log^2k/\log\log{k})-Approximation Algorithm for Directed Steiner Tree: A Tight Quasi-Polynomial-Time Algorithm

In the Directed Steiner Tree (DST) problem we are given an $n$-vertex directed edge-weighted graph, a root $r$, and a collection of $k$ terminal nodes. Our goal is to find a minimum-cost arborescence that contains a directed path from $r$ to every terminal. We present an $O(\log^2 k/\log\log{k})$-approximation algorithm for DST that runs in quasi-polynomial-time. By adjusting the parameters in the hardness result of Halperin and Krauthgamer, we show the matching lower bound of $\Omega(\log^2{k}/\log\log{k})$ for the class of quasi-polynomial-time algorithms. This is the first improvement on the DST problem since the classical quasi-polynomial-time $O(\log^3 k)$ approximation algorithm by Charikar et al. (The paper erroneously claims an $O(\log^2k)$ approximation due to a mistake in prior work.) Our approach is based on two main ingredients. First, we derive an approximation preserving reduction to the Label-Consistent Subtree (LCST) problem. The LCST instance has quasi-polynomial size and logarithmic height. We remark that, in contrast, Zelikovsky's heigh-reduction theorem used in all prior work on DST achieves a reduction to a tree instance of the related Group Steiner Tree (GST) problem of similar height, however losing a logarithmic factor in the approximation ratio. Our second ingredient is an LP-rounding algorithm to approximately solve LCST instances, which is inspired by the framework developed by Rothvo{\ss}. We consider a Sherali-Adams lifting of a proper LP relaxation of LCST. Our rounding algorithm proceeds level by level from the root to the leaves, rounding and conditioning each time on a proper subset of label variables. A small enough (namely, polylogarithmic) number of Sherali-Adams lifting levels is sufficient to condition up to the leaves

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum

Edge-Disjoint Paths in Planar Graphs

We study the maximum edge-disjoint paths problem (MEDP). We are given a graph G = (V,E) and a set Τ = {s1t1, s2t2, . . . , sktk} of pairs of vertices: the objective is to find the maximum number of pairs in Τ that can be connected via edge-disjoint paths. Our main result is a poly-logarithmic approximation for MEDP on undirected planar graphs if a congestion of 2 is allowed, that is, we allow up to 2 paths to share an edge. Prior to our work, for any constant congestion, only a polynomial-factor approximation was known for planar graphs although much stronger results are known for some special cases such as grids and grid-like graphs. We note that the natural multicommodity flow relaxation of the problem has an integrality gap of Ω(√|V|) even on planar graphs when no congestion is allowed. Our starting point is the same relaxation and our result implies that the integrality gap shrinks to a poly-logarithmic factor once 2 paths are allowed per edge. Our result also extends to the unsplittable flow problem and the maximum integer multicommodity flow problem. A set X ⊆ V is well-linked if for each S ⊂ V , |δ(S)| ≥ min{|S ∩ X|, |(V - S) ∩ X|}. The heart of our approach is to show that in any undirected planar graph, given any matching M on a well-linked set X, we can route Ω(|M|) pairs in M with a congestion of 2. Moreover, all pairs in M can be routed with constant congestion for a sufficiently large constant. This results also yields a different proof of a theorem of Klein, Plotkin, and Rao that shows an O(1) maxflow-mincut gap for uniform multicommodity flow instances in planar graphs. The framework developed in this paper applies to general graphs as well. If a certain graph theoretic conjecture is true, it will yield poly-logarithmic integrality gap for MEDP with constant congestion