Zachary Friggstad

An efficient architecture for the AES mix columns operation

In this paper, a compact architecture for the AES mix columns operation and its inverse is presented. The hardware implementation is compared with previous work done in this area. We show that our design has a lower gate count than other designs that implement both the forward and the inverse mix columns operation.

Minimizing movement in mobile facility location problems

In the mobile facility location problem, which is a variant of the classical uncapacitated facility location and k-median problems, each facility and client is assigned to a start location in a metric graph and our goal is to find a destination node for each client and facility such that every client is sent to a node which is the destination of some facility. The quality of a solution can be measured either by the total distance clients and facilities travel or by the maximum distance traveled by any client or facility. As we show in this paper (by an approximation preserving reduction), the problem of minimizing the total movement of facilities and clients generalizes the classical k-median problem. The class of movement problems was introduced by Demaine et al. in SODA 2007, where it was observed a simple 2-approximation for the minimum maximum movement mobile facility location while an approximation for the minimum total movement variant and hardness results for both were left as open problems. Our main result here is an 8-approximation algorithm for the minimum total movement mobile facility location problem. Our algorithm is obtained by rounding an LP relaxation in five phases. We also show that this problem generalizes the classical k-median problem using an approximation preserving reduction. For the minimum maximum movement mobile facility location problem, we show that we cannot have a better than a 2-approximation for the problem, unless P = NP; so the simple algorithm observed in is essentially best possible.

Approximability of packing disjoint cycles

Given a graph G, the edge-disjoint cycle packing problem is to find the largest set of cycles of which no two share an edge. For undirected graphs, the best known approximation algorithm has ratio O(√log n) [14,15]. In fact, they proved the same upper bound for the integrality gap of this problem by presenting a simple greedy algorithm. Here we show that this is almost best possible. By modifying integrality gap and hardness results for the edge-disjoint paths problem [1,9], we show that the undirected edge-disjoint cycle packing problem has an integrality gap of Ω(√log n/log log n) and furthermore it is quasi-NP-hard to approximate the edge-disjoint cycle problem within ratio of O(log1/2-Ɛ n) for any constant Ɛ > 0. The same results hold for the problem of packing vertex-disjoint cycles.

Minimizing Movement in Mobile Facility Location Problems

In the mobile facility location problem, which is a variant of the classical uncapacitated facility location and k-median problems, each facility and client is assigned to a start location in a metric graph and our goal is to find a destination node for each client and facility such that every client is sent to a node which is the destination of some facility. The quality of a solution can be measured either by the total distance clients and facilities travel or by the maximum distance traveled by any client or facility. As we show in this paper (by an approximation preserving reduction), the problem of minimizing the total movement of facilities and clients generalizes the classical k-median problem. The class of movement problems was introduced by Demaine et al. in SODA 2007, where it was observed a simple 2-approximation for the minimum maximum movement mobile facility location while an approximation for the minimum total movement variant and hardness results for both were left as open problems. Our main result here is an 8-approximation algorithm for the minimum total movement mobile facility location problem. Our algorithm is obtained by rounding an LP relaxation in five phases. We also show that this problem generalizes the classical k-median problem using an approximation preserving reduction. For the minimum maximum movement mobile facility location problem, we show that we cannot have a better than a 2-approximation for the problem, unless P = NP; so the simple algorithm observed in is essentially best possible.

Approximation algorithms for regret-bounded vehicle routing and applications to distance-constrained vehicle routing

We consider vehicle-routing problems (VRPs) that incorporate the notion of regret of a client, which is a measure of the waiting time of a client relative to its shortest-path distance from the depot. Formally, we consider both the additive and multiplicative versions of, what we call, the regret-bounded vehicle routing problem (RVRP). In these problems, we are given an undirected complete graph G = ({r} ∪ V,E) on n nodes with a distinguished root (depot) node r, edge costs {cuv} that form a metric, and a regret bound R. Given a path P rooted at r and a node v ∈ P, let cP(v) be the distance from r to v along P. The goal is to find the fewest number of paths rooted at r that cover all the nodes so that for every node v covered by (say) path P: (i) its additive regret cP(v) -- crv, with respect to P is at most R in additive-RVRP; or (ii) its multiplicative regret, cP(v)/crv, with respect to P is at most R in multiplicative-RVRP. Our main result is the first constant-factor approximation algorithm for additive-RVRP. This is a substantial improvement over the previous-best O(log n)-approximation. Additive-RVRP turns out be a rather central vehicle-routing problem, whose study reveals insights into a variety of other regret-related problems as well as the classical distance-constrained VRP (DVRP), enabling us to obtain guarantees for these various problems by leveraging our algorithm for additive-RVRP and the underlying techniques. We obtain approximation ratios of [EQUATION] for multiplicative-RVRP, and [EQUATION] for DVRP with distance bound D via reductions to additive-RVRP; the latter improves upon the previous-best approximation for DVRP. A noteworthy aspect of our results is that they are obtained by devising rounding techniques for a natural configuration-style LP. This furthers our understanding of LP-relaxations for VRPs and enriches the toolkit of techniques that have been utilized for configuration LPs.

Superabundant Numbers and the Riemann Hypothesis

For a lively exposition of this theorem and its connection to the Riemann Hypothesis see [5]. In this note, we propose a method that will establish explicit upper bounds for σ(n)/e n log log n. Our main observation is that the least number violating the inequality (2) should be a superabundant number. A positive integer n is said to be superabundant if σ(m)/m < σ(n)/n for all m < n. The first 20 superabundant numbers are 1, 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 10080. The sequence of superabundant numbers is the sequence A004394 in Sloane’s Encyclopedia [8]. The list of the first 500 superabundant numbers is available at [6], where

On Linear Programming Relaxations for Unsplittable Flow in Trees

We study some linear programming relaxations for the Unsplittable Flow problem on trees (UFP-Tree). Inspired by results obtained by Chekuri, Ene, and Korula for Unsplittable Flow on paths (UFP-Path), we present a relaxation with polynomially many constraints that has an integrality gap bound of O(log n * min(log m, log n)) where n denotes the number of tasks and m denotes the number of edges in the tree. This matches the approximation guarantee of their combinatorial algorithm and is the first demonstration of an efficiently-solvable relaxation for UFP-Tree with a sub-linear integrality gap. The new constraints in our LP relaxation are just a few of the (exponentially many) rank constraints that can be added to strengthen the natural relaxation. A side effect of how we prove our upper bound is an efficient O(1)-approximation for solving the rank LP. We also show that our techniques can be used to prove integrality gap bounds for similar LP relaxations for packing demand-weighted subtrees of an edge-capacitated tree. On the other hand, we show that the inclusion of all rank constraints does not reduce the integrality gap for UFP-Tree to a constant. Specifically, we show the integrality gap is Omega(sqrt(log n)) even in cases where all tasks share a common endpoint. In contrast, intersecting instances of UFP-Path are known to have an integrality gap of O(1) even if just a few of the rank 1 constraints are included. We also observe that applying two rounds of the Lovasz-Schrijver SDP procedure to the natural LP for UFP-Tree derives an SDP whose integrality gap is also O(log n * min(log m, log n)).

Linear Programming Hierarchies Suffice for Directed Steiner Tree

We demonstrate that l rounds of the Sherali-Adams hierarchy and 2l rounds of the Lovasz-Schrijver hierarchy suffice to reduce the integrality gap of a natural LP relaxation for Directed Steiner Tree in l-layered graphs from \(\Omega(\sqrt k)\) to O(l·logk) where k is the number of terminals. This is an improvement over Rothvoss’ result that 2l rounds of the considerably stronger Lasserre SDP hierarchy reduce the integrality gap of a similar formulation to O(l·logk).

Tight Analysis of a Multiple-Swap Heurstic for Budgeted Red-Blue Median

Budgeted Red-Blue Median is a generalization of classic

LP-Based Approximation Algorithms for Facility Location in Buy-at-Bulk Network Design

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