Generalized river routing — Algorithms and performance bounds

With the increasing density of VLSI circuits, the interconnection wires are getting packed even closer. This has increased the effect of interaction between these wires on circuit performance and hence, the importance of controlling crosstalk. In this paper, we consider river routing with crosstalk constraints. Given the positions of the pins in a single-layer routing channel and the maximum tolerable crosstalk between each pair of nets, we give a polynomial time algorithm to decide whether there is a feasible river routing solution and produce one with minimum crosstalk whenever the problem is feasible.

River routing in VLSI

The single-layer rectilinear river routing model with no restriction on the number of jogs per wire is considered. In particular, the author studies the model in which there is a fixed constant upper bound J on the number of jogs each wire can have. The author proposes an optimal O(n) time algorithm for the feasibility problem. This leads to an O(n log n) time algorithm for the minimum separation problem. Both algorithms take O(n) space are quite practical. This is a significant improvement over T.C. Tuan and S.L. Hakimi's (1987) minimum separation algorithm, which is designed for J=2 and takes O(n/sup 3/) time and O(n/sup 2/) space. >

This article reviews the history and theory of dynamic programming (DP), a recursive method of solving sequential decision problems under uncertainty. It discusses computational algorithms for the numerical solution of DP problems, and an important limitation in our ability to solve realistic large-scale dynamic programming problems, the ‘curse of dimensionality’. It also summarizes recent research in complexity theory that delineates situations where the curse can be broken (allowing us to solve DPs using fast polynomial time algorithms), and situations where it is insuperable. The literature on econometric estimation and testing of DP models is reviewed, as is another ‘scientific limit to knowledge’, namely, the identification problem.

Two P-complete problems in the theory of the reals

We address the problem of efficient circuit switching in wide area optical networks. The solution provided is based on finding optimal routes for lightpaths and the new concept of semilightpaths. A lightpath is a fully optical transmission path, while a semilightpath is a transmission path constructed by chaining together several lightpaths, using wavelength conversion at their junctions. A fast and practical algorithm is presented to optimally route lightpaths and semilightpaths taking into account both the cost of using the wavelengths on links and the cost of wavelength conversion. We prove that the running time of the algorithm is the best possible in the wide class of algorithms allowing linear algebraic operations on weights. This class encompasses all known related practical methods. Additionally, our method works for any physical realization of wavelength conversion, independently whether it is done via optoelectronic conversion or in a fully optical way.

This paper examines the problem of determining the threshold dimension of a graph. There exists numerous characterizations of threshold graphs, those graphs of threshold dimension one, as well as fast polynomial time algorithms to test if a graph is a threshold graph. Yannakakis [1982] proved that, in general, determining the threshold dimension is a hard problem by proving that for fixed $k\geqq 3$, determining if the threshold dimension of a graph is less than or equal to k is an NP-complete problem. In this paper we compute the threshold dimension of several classes of graphs and obtain upper and lower bounds on the threshold dimension of a graph. Counterexamples to a conjecture on threshold dimension are provided.

Transitive orientations in bull-reducible Berge graphs

The current problem of digital logistics is investigated - the calculation of optimal routes for freight transportation by computer means to reduce time and distance. Heuristic methods used in logistics for constructing optimal routes are considered. A comparative analysis of ten methods for solving the optimization problem of the “nondeterministic polynomial time” complexity class traveling salesman is carried out. The study performs a comparative analysis of the following methods: “convex hull, cheapest insertion and angle selection”, “greedy”, “greedy-cycle”, “integer-linear-programming”, “or-opt”, “or-zweig”, “remove crossings”, “space filling curve”, “simulated annealing”, “two-opt”. A computational experiment is performed, on the basis of which the accuracy and computational complexity of the considered methods are estimated. The results of the computational experiment show the construction of the optimal route by the “integer-linear-programming” method and the highest computation speed for the “greedy” method. Application of the “integer-linear-programming” method in logistics is the most accurate at the optimal time for calculating efficient routes of freight traffic.

Public bus users have to wait at the bus stops that have been provided to take the bus. Failure to find the nearest bus stop according to the user's location may cause them to miss the bus. Therefore, a GIS-based system is developed to support the bus users in finding the nearest bus stop with an optimum route by utilizing Haversine and Tabu Search. The Haversine method is used to find the nearest bus stop from the user's location and the Tabu Search is used to find an optimum route from the user's location to the nearest bus stop. The system testing showed that the Haversine method gives the closest distance of bus stop to the user's location while the Tabu Search algorithm generates the optimum route with an average running time of 0.526 ms.

This paper initiates the study of algorithms (minimizing cache misses) from the perspective of fine-grained complexity (conditional polynomial lower bounds). Specifically, we aim to answer why sparse graph problems are so hard, and why the Longest Common Subsequence problem gets a savings of a factor of the size of cache times the length of a cache line, but no more. We take the reductions and techniques from complexity and fine-grained complexity and apply them to the model to generate new (conditional) lower bounds as well as new faster algorithms. We also prove the existence of a hierarchy for the model, which motivates the fine-grained reductions. - Using fine-grained reductions, we give an algorithm for distinguishing 2 vs. 3 diameter and radius that runs in O(|E|^2/(MB)) cache misses, which for sparse graphs improves over the previous O(|V|^2/B) running time. - We give new reductions from radius and diameter to Wiener index and median. These reductions are new in both the RAM and models. - We show meaningful reductions between problems that have linear-time solutions in the RAM model. The reductions use low complexity (typically O(n/B)), and thus help to finely capture between I/O linear time O(n/B) and RAM linear O(n). - We generate new assumptions based on the difficulty of improving sparse graph problem running times in the model. We create conjectures that the current best known algorithms for Single Source Shortest Paths (SSSP), diameter, and radius are optimal. - From these I/O-model assumptions, we show that many of the known reductions in the word-RAM model can naturally extend to hold in the model as well (e.g., a lower bound on the complexity of Longest Common Subsequence that matches the best known running time). - We prove an analog of the Time Hierarchy Theorem in the model, further motivating the study of fine-grained algorithmic differences.

We develop fast approximation algorithms for the minimum-cost version of the Bounded-Degree MST problem (BD-MST) and its generalization the Crossing Spanning Tree problem (Crossing-ST). We solve the underlying LP to within a $(1+\epsilon)$ approximation factor in near-linear time via the multiplicative weight update (MWU) technique. This yields, in particular, a near-linear time algorithm that outputs an estimate $B$ such that $B \le B^* \le \lceil (1+\epsilon)B \rceil +1$ where $B^*$ is the minimum-degree of a spanning tree of a given graph. To round the fractional solution, in our main technical contribution, we describe a fast near-linear time implementation of swap-rounding in the spanning tree polytope of a graph. The fractional solution can also be used to sparsify the input graph that can in turn be used to speed up existing combinatorial algorithms. Together, these ideas lead to significantly faster approximation algorithms than known before for the two problems of interest. In addition, a fast algorithm for swap rounding in the graphic matroid is a generic tool that has other applications, including to TSP and submodular function maximization.

Fast Dimension-based Partitioning and Merging clustering algorithm

In this paper, we address the problem of incentive based routing in Dynamic Spectrum Access (DSA) networks, where each secondary node incurs a cost for routing traffic from a flow and also has a certain capacity that it can provide to the flow. We propose a path auction based routing mechanism in which each secondary node announces its privately known cost and capacity, based on which a route is chosen and payments are made to the nodes. We design the route selection mechanism and the pricing function in a manner that can induce nodes to reveal their cost and capacity honestly, while minimizing the payment that needs to be given to the nodes that relay traffic, thereby making our path auction optimal. In our proposed mechanism, the optimal route over which traffic should be routed and the payment that each node should receive can be computed in polynomial time. Simulation results suggest that our mechanism can ensure truthful reporting of both cost and capacity while making a payment less than that required for VCG based least cost path routing.

Resource assignment problems occur in a vast variety of applications, from scheduling problems over image recognition to communication networks. Often these problems can be modeled by a maximum weight matching problem in (bipartite) graphs or generalizations thereof, and efficient and practical algorithms are known for these problems. Although in some of the applications an assignment of the resources may be needed only once, in many of these applications, the assignment has to be computed more often for different scenarios. In that case it is often essential that the assignments can be computed very fast. Moreover, implementing different assignments in different scenarios may come with a certain cost for the reconfiguration of the system. In this paper, we consider the problem of determining optimal assignments sequentially over a given time horizon, where consecutive assignments are coupled by constraints that control the cost of reconfiguration. We develop fast approximation and online algorithms for this problem with provable approximation guarantees and competitive ratios. Moreover, we present an extensive computational study about the applicability of our model and our algorithms in the context of orthogonal frequency division multiple access (OFDMA) wireless networks, finding a significant performance improvement for the total bandwidth of the system using our algorithms. For this application (the downlink of an OFDMA wireless cell) , the run time of matching algorithms is extremely important, having an acceptable range of a few milliseconds only. For the considered realistic instances, our algorithms perform extremely well: the solution quality is, on average, within a factor of 0.8–0.9 of optimal off-line solutions, and the running times are at most 5 ms per phase even in the worst case. Thus, our algorithms are well suited to be applied in the context of OFDMA systems. Data, as supplemental material, are available at http://dx.doi.org/10.1287/mnsc.2015.2221 . This paper was accepted by Teck-Hua Ho, optimization.