Reachability substitutes for planar digraphs

Given a digraph $G = (V,E)$ with a set $U$ of vertices marked ``interesting,'' we want to find a smaller digraph \RS{} = (V',E') with $V' \supseteq U$ in such a way that the reachabilities amongst those interesting vertices in $G$ and \RS{} are the same. So with respect to the reachability relations within $U$, the digraph \RS{} is a substitute for $G$. We show that while almost all graphs do not have reachability substitutes smaller than \Ohmega(|U|^2/\log |U|), every planar graph has a reachability substitute of size \Oh(|U| \log^2 |U|). Our result rests on two new structural results for planar dags, a separation procedure and a reachability theorem, which might be of independent interest

Experimental Analysis of Algorithms for Coflow Scheduling

Modern data centers face new scheduling challenges in optimizing job-level performance objectives, where a significant challenge is the scheduling of highly parallel data flows with a common performance goal (e.g., the shuffle operations in MapReduce applications). Chowdhury and Stoica introduced the coflow abstraction to capture these parallel communication patterns, and Chowdhury et al. proposed effective heuristics to schedule coflows efficiently. In our previous paper, we considered the strongly NP-hard problem of minimizing the total weighted completion time of coflows with release dates, and developed the first polynomial-time scheduling algorithms with O(1)-approximation ratios. In this paper, we carry out a comprehensive experimental analysis on a Facebook trace and extensive simulated instances to evaluate the practical performance of several algorithms for coflow scheduling, including the approximation algorithms developed in our previous paper. Our experiments suggest that simple algorithms provide effective approximations of the optimal, and that the performance of our approximation algorithms is relatively robust, near optimal, and always among the best compared with the other algorithms, in both the offline and online settings.Comment: 29 pages, 8 figures, 11 table

An Improved Upper Bound for the Ring Loading Problem

The Ring Loading Problem emerged in the 1990s to model an important special case of telecommunication networks (SONET rings) which gained attention from practitioners and theorists alike. Given an undirected cycle on $n$ nodes together with non-negative demands between any pair of nodes, the Ring Loading Problem asks for an unsplittable routing of the demands such that the maximum cumulated demand on any edge is minimized. Let $L$ be the value of such a solution. In the relaxed version of the problem, each demand can be split into two parts where the first part is routed clockwise while the second part is routed counter-clockwise. Denote with $L^*$ the maximum load of a minimum split routing solution. In a landmark paper, Schrijver, Seymour and Winkler [SSW98] showed that $L \leq L^* + 1.5D$, where $D$ is the maximum demand value. They also found (implicitly) an instance of the Ring Loading Problem with $L = L^* + 1.01D$. Recently, Skutella [Sku16] improved these bounds by showing that $L \leq L^* + \frac{19}{14}D$, and there exists an instance with $L = L^* + 1.1D$. We contribute to this line of research by showing that $L \leq L^* + 1.3D$. We also take a first step towards lower and upper bounds for small instances

A Robust AFPTAS for Online Bin Packing with Polynomial Migration

In this paper we develop general LP and ILP techniques to find an approximate solution with improved objective value close to an existing solution. The task of improving an approximate solution is closely related to a classical theorem of Cook et al. in the sensitivity analysis for LPs and ILPs. This result is often applied in designing robust algorithms for online problems. We apply our new techniques to the online bin packing problem, where it is allowed to reassign a certain number of items, measured by the migration factor. The migration factor is defined by the total size of reassigned items divided by the size of the arriving item. We obtain a robust asymptotic fully polynomial time approximation scheme (AFPTAS) for the online bin packing problem with migration factor bounded by a polynomial in $\frac{1}{\epsilon}$. This answers an open question stated by Epstein and Levin in the affirmative. As a byproduct we prove an approximate variant of the sensitivity theorem by Cook at el. for linear programs

On the Complexity of Conditional DAG Scheduling in Multiprocessor Systems

As parallel processing became ubiquitous in modern computing systems, parallel task models have been proposed to describe the structure of parallel applications. The workflow scheduling problem has been studied extensively over past years, focusing on multiprocessor systems and distributed environments (e.g. grids, clusters). In workflow scheduling, applications are modeled as directed acyclic graphs (DAGs). DAGs have also been introduced in the real-time scheduling community to model the execution of multi-threaded programs on a multi-core architecture. The DAG model assumes, in most cases, a fixed DAG structure capturing only straight-line code. Only recently, more general models have been proposed. In particular, the conditional DAG model allows the presence of control structures such as conditional (if-then-else) constructs. While first algorithmic results have been presented for the conditional DAG model, the complexity of schedulability analysis remains wide open. We perform a thorough analysis on the worst-case makespan (latest completion time) of a conditional DAG task under list scheduling (a.k.a. fixed-priority scheduling). We show several hardness results concerning the complexity of the optimization problem on multiple processors, even if the conditional DAG has a well-nested structure. For general conditional DAG tasks, the problem is intractable even on a single processor. Complementing these negative results, we show that certain practice-relevant DAG structures are very well tractable

The Complexity of Routing with Few Collisions

We study the computational complexity of routing multiple objects through a network in such a way that only few collisions occur: Given a graph $G$ with two distinct terminal vertices and two positive integers $p$ and $k$, the question is whether one can connect the terminals by at least $p$ routes (e.g. paths) such that at most $k$ edges are time-wise shared among them. We study three types of routes: traverse each vertex at most once (paths), each edge at most once (trails), or no such restrictions (walks). We prove that for paths and trails the problem is NP-complete on undirected and directed graphs even if $k$ is constant or the maximum vertex degree in the input graph is constant. For walks, however, it is solvable in polynomial time on undirected graphs for arbitrary $k$ and on directed graphs if $k$ is constant. We additionally study for all route types a variant of the problem where the maximum length of a route is restricted by some given upper bound. We prove that this length-restricted variant has the same complexity classification with respect to paths and trails, but for walks it becomes NP-complete on undirected graphs

A fully polynomial time approximation scheme for packing while traveling

Understanding the interactions between different combinatorial optimisation problems in real-world applications is a challenging task. Recently, the traveling thief problem (TTP), as a combination of the classical traveling salesperson problem and the knapsack problem, has been introduced to study these interactions in a systematic way. We investigate the underlying non-linear packing while traveling (PWT) problem of the TTP where items have to be selected along a fixed route. We give an exact dynamic programming approach for this problem and a fully polynomial time approximation scheme (FPTAS) when maximising the benefit that can be gained over the baseline travel cost. Our experimental investigations show that our new approaches outperform current state-of-the-art approaches on a wide range of benchmark instances

A Fully Polynomial Time Approximation Scheme for Packing While Traveling

Understanding the interaction between different combinatorial optimization problems is a challenging task of high relevance for numerous real-world applications including modern computer and memory architectures as well as high performance computing. Recently, the Traveling Thief Problem (TTP), as a combination of the classical traveling salesperson problem and the knapsack problem, has been introduced to study these interactions in a systematic way. We investigate the underlying non-linear Packing While Traveling Problem (PWTP) of the TTP where items have to be selected along a fixed route. We give an exact dynamic programming approach for this problem and a fully polynomial time approximation scheme (FPTAS) when maximizing the benefit that can be gained over the baseline travel cost. Our experimental investigations show that our new approaches outperform current state-of-the-art approaches on a wide range of benchmark instances

On the Price of Anarchy for flows over time

Dynamic network flows, or network flows over time, constitute an important model for real-world situations where steady states are unusual, such as urban traffic and the Internet. These applications immediately raise the issue of analyzing dynamic network flows from a game-theoretic perspective. In this paper we study dynamic equilibria in the deterministic fluid queuing model in single-source single-sink networks, arguably the most basic model for flows over time. In the last decade we have witnessed significant developments in the theoretical understanding of the model. However, several fundamental questions remain open. One of the most prominent ones concerns the Price of Anarchy, measured as the worst case ratio between the minimum time required to route a given amount of flow from the source to the sink, and the time a dynamic equilibrium takes to perform the same task. Our main result states that if we could reduce the inflow of the network in a dynamic equilibrium, then the Price of Anarchy is exactly e/(e − 1) ≈ 1.582. This significantly extends a result by Bhaskar, Fleischer, and Anshelevich (SODA 2011). Furthermore, our methods allow to determine that the Price of Anarchy in parallel-link networks is exactly 4/3. Finally, we argue that if a certain very natural monotonicity conjecture holds, the Price of Anarchy in the general case is exactly e/(e − 1)

Universal Sequencing on an Unreliable Machine

We consider scheduling on an unreliable machine that may experience unexpected changes in processing speed or even full breakdowns. Our objective is to minimize ∑ wjf(Cj) for any nondecreasing, nonnegative, differentiable cost function f(Cj). We aim for a universal solution that performs well without adaptation for all cost functions for any possible machine behavior. We design a deterministic algorithm that finds a universal scheduling sequence with a solution value within 4 times the value of an optimal clairvoyant algorithm that knows the machine behavior in advance. A randomized version of this algorithm attains in expectation a ratio of e. We also show that both performance guarantees are best possible for any unbounded cost function. Our algorithms can be adapted to run in polynomial time with slightly increased cost. When jobs have individual release dates, the situation changes drastically. Even if all weights are equal, there are instances for which any universal solution is a factor of Ω(log n / log log n) worse than an optimal sequence for any unbounded cost function. Motivated by this hardness, we study the special case when the processing time of each job is proportional to its weight. We present a nontrivial algorithm with a small constant performance guarantee