Non-approximability and Polylogarithmic Approximations of the Single-Sink Unsplittable and Confluent Dynamic Flow Problems

Dynamic Flows were introduced by Ford and Fulkerson in 1958 to model flows over time. They define edge capacities to be the total amount of flow that can enter an edge in one time unit. Each edge also has a length, representing the time needed to traverse it. Dynamic Flows have been used to model many problems including traffic congestion, hop-routing of packets and evacuation protocols in buildings. While the basic problem of moving the maximal amount of supplies from sources to sinks is polynomial time solvable, natural minor modifications can make it NP-hard. One such modification is that flows be confluent, i.e., all flows leaving a vertex must leave along the same edge. This corresponds to natural conditions in, e.g., evacuation planning and hop routing. We investigate the single-sink Confluent Quickest Flow problem. The input is a graph with edge capacities and lengths, sources with supplies and a sink. The problem is to find a confluent flow minimizing the time required to send supplies to the sink. Our main results include: a) Logarithmic Non-Approximability: Directed Confluent Quickest Flows cannot be approximated in polynomial time with an O(log n) approximation factor, unless P=NP. b) Polylogarithmic Bicriteria Approximations: Polynomial time (O(log^8 n), O(log^2 kappa)) bicritera approximation algorithms for the Confluent Quickest Flow problem where kappa is the number of sinks, in both directed and undirected graphs. Corresponding results are also developed for the Confluent Maximum Flow over time problem. The techniques developed also improve recent approximation algorithms for static confluent flows

A Linear Kernel for Planar Total Dominating Set

A total dominating set of a graph $G=(V,E)$ is a subset $D \subseteq V$ such that every vertex in $V$ is adjacent to some vertex in $D$. Finding a total dominating set of minimum size is NP-hard on planar graphs and W[2]-complete on general graphs when parameterized by the solution size. By the meta-theorem of Bodlaender et al. [J. ACM, 2016], there exists a linear kernel for Total Dominating Set on graphs of bounded genus. Nevertheless, it is not clear how such a kernel can be effectively constructed, and how to obtain explicit reduction rules with reasonably small constants. Following the approach of Alber et al. [J. ACM, 2004], we provide an explicit kernel for Total Dominating Set on planar graphs with at most $410k$ vertices, where $k$ is the size of the solution. This result complements several known constructive linear kernels on planar graphs for other domination problems such as Dominating Set, Edge Dominating Set, Efficient Dominating Set, Connected Dominating Set, or Red-Blue Dominating Set.Comment: 33 pages, 13 figure

Complexity of Scheduling in Synthesizing Hardware from Concurrent Action Oriented Specifications

Concurrent Action Oriented Specifications (CAOS) formalism such as Bluespec Inc.\u27s Bluespec System Verilog (BSV) has been recently shown to be effective for hardware modeling and synthesis. This formalism offers the benefits of automatic handling of concurrency issues in highly concurrent system descriptions, and the associated synthesis algorithms have been shown to produce efficient hardware comparable to those generated from hand-written Verilog/VHDL. These benefits which are inherent in such a synthesis process also aid in faster architectural exploration. This is because CAOS allows a high-level description (above RTL) of a design in terms of atomic transactions, where each transaction corresponds to a collection of operations. Optimal scheduling of such actions in CAOS-based synthesis process is crucial in order to generate hardware that is efficient in terms of area, latency and power. In this paper, we analyze the complexity of the scheduling problems associated with CAOS-based synthesis and discuss several heuristics for meeting the peak power goals of designs generated from CAOS. We also discuss approximability of these problems as appropriate