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Boosting: Foundations and Algorithms.

Reconfiguration is concerned with relationships among solutions to a problem instance, where the reconfiguration of one solution to another is a sequence of steps such that each step produces an intermediate feasible solution. The solution space can be represented as a reconfiguration graph, where two vertices representing solutions are adjacent if one can be formed from the other in a single step. Work in the area encompasses both structural questions (Is the reconfiguration graph connected?) and algorithmic ones (How can one find the shortest sequence of steps between two solutions?) This survey discusses techniques, results, and future directions in the area.

https://www.mdpi.com/1999-4893/11/4/52/pdf

Introduction to Reconfiguration

We present the first results on the parameterized complexity of reconfiguration problems, where a reconfiguration variant of an optimization problem $$\mathcal {Q}$$Q takes as input two feasible solutions S and T and determines if there is a sequence of reconfiguration steps, ie a reconfiguration sequence, that can be applied to transform S into T such that each step results in a feasible solution to $$\mathcal {Q}$$Q For most of the results in this paper, S and T are sets of vertices of a given graph and a reconfiguration step adds or removes a vertex Our study is motivated by results establishing that for many NP-hard problems, the classical complexity of reconfiguration is PSPACE-complete We address the question for several important graph properties under two natural parameterizations: k, a bound on the size of solutions, and $$\ell $$l, a bound on the length of reconfiguration sequences Our first general result is an algorithmic paradigm, the reconfiguration kernel, used to obtain fixed-parameter tractable algorithms for reconfiguration variants of Vertex Cover and, more generally, Bounded Hitting Set and Feedback Vertex Set, all parameterized by k In contrast, we show that reconfiguring Unbounded Hitting Set is W[2]-hard when parameterized by $$k+\ell $$k+l We also demonstrate the W[1]-hardness of reconfiguration variants of a large class of maximization problems parameterized by $$k+\ell $$k+l, and of their corresponding deletion problems parameterized by $$\ell $$l; in doing so, we show that there exist problems in FPT when parameterized by k, but whose reconfiguration variants are W[1]-hard when parameterized by $$k+\ell $$k+l

On the Parameterized Complexity of Reconfiguration Problems

We present (single-pass) streaming algorithms for maintaining extent measures of a stream S of n points in $\mathbb{R} ^{d}$  We focus on designing streaming algorithms whose working space is polynomial in d (poly(d)) and sub-linear in n For the problems of computing diameter, width and minimum enclosing ball of S, we obtain lower bounds on the worst-case approximation ratio of any streaming algorithm that uses poly(d) space On the positive side, we introduce the notion of blurred ball cover and use it for answering approximate farthest-point queries and maintaining approximate minimum enclosing ball and diameter of S We describe a streaming algorithm for maintaining a blurred ball cover whose working space is linear in d and independent of n

Streaming Algorithms for Extent Problems in High Dimensions

In this work we consider triangulations of point sets in the Euclidean plane, i.e., maximal straight-line crossing-free graphs on a finite set of points. Given a triangulation of a point set, an edge flip is the operation of removing one edge and adding another one, such that the resulting graph is again a triangulation. Flips are a major way of locally transforming triangular meshes. We show that, given a point set S in the Euclidean plane and two triangulations T"1 and T"2 of S, it is an APX-hard problem to minimize the number of edge flips to transform T"1 to T"2.

Flip distance between triangulations of a planar point set is APX-hard

Given two triangulations of a convex polygon, computing the minimum number of flips required to transform one to the other is a long-standing open problem. It is not known whether the problem is in P or NP-complete. We prove that two natural generalizations of the problem are NP-complete, namely computing the minimum number of flips between two triangulations of (1) a polygon with holes; (2) a set of points in the plane.

/pdf/flip-distance-between-two-triangulations-of-a-point-set-is-4ax0hjdygp.pdf

Flip distance between two triangulations of a point set is NP-complete

In this paper we introduce self-approaching graph drawings. A straight-line drawing of a graph is self-approaching if, for any origin vertex s and any destination vertex t, there is an st-path in the graph such that, for any point q on the path, as a point p moves continuously along the path from the origin to q, the Euclidean distance from p to q is always decreasing. This is a more stringent condition than a greedy drawing (where only the distance between vertices on the path and the destination vertex must decrease), and guarantees that the drawing is a 5.33-spanner.

We study three topics: (1) recognizing self-approaching drawings; (2) constructing self-approaching drawings of a given graph; (3)constructing a self-approaching Steiner network connecting a given set of points.

We show that: (1) there are efficient algorithms to test if a polygonal path is self-approaching in ℝ2 and ℝ3, but it is NP-hard to test if a given graph drawing in ℝ3 has a self-approaching uv-path; (2) we can characterize the trees that have self-approaching drawings; (3) for any given set of terminal points in the plane, we can find a linear sized network that has a self-approaching path between any ordered pair of terminals.

/pdf/self-approaching-graphs-28ldkkn8w9.pdf

Self-approaching graphs

/pdf/flip-distance-between-two-triangulations-of-a-point-set-is-wcj97bztus.pdf

Flip Distance Between Two Triangulations of a Point Set is NP-complete.

Given a Boolean formula and a satisfying assignment, a flip is an operation that changes the value of a variable in the assignment so that the resulting assignment remains satisfying. We study the problem of computing the shortest sequence of flips (if one exists) that transforms a given satisfying assignment $s$ to another satisfying assignment $t$ of the Boolean formula. Earlier work characterized the complexity of deciding the existence of a sequence of flips between two given satisfying assignments using Schaefer’s framework for classification of Boolean formulas. We build on it to provide a trichotomy for the complexity of finding the shortest sequence of flips and show that it is either in P, NP-complete, or PSPACE-complete. Our result adds to the small set of complexity results known for shortest reconfiguration sequence problems by providing an example where the shortest sequence can be found in polynomial time even though the path flips variables that have the same value in both $s$ and $t$. This is in contrast to all reconfiguration problems studied so far, where polynomial time algorithms for computing the shortest path were known only for cases where the path modified the symmetric difference only. Our proof uses Birkhoff’s representation theorem on a set system that we show to be a distributive lattice. The technique is insightful and can perhaps be used for other reconfiguration problems as well.

Shortest Reconfiguration Paths in the Solution Space of Boolean Formulas

At [email protected]?10, Agarwal and Sharathkumar presented a streaming algorithm for approximating the minimum enclosing ball of a set of points in d-dimensional Euclidean space. Their algorithm requires one pass, uses O(d) space, and was shown to have approximation factor at most (1+3)/[email protected]~1.3661. We prove that the same algorithm has approximation factor less than 1.22, which brings us much closer to a (1+2)/2~1.207 lower bound given by Agarwal and Sharathkumar. We also apply this technique to the dynamic version of the minimum enclosing ball problem (in the non-streaming setting). We give an O(dn)-space data structure that can maintain a 1.22-approximate minimum enclosing ball in O(dlogn) expected amortized time per insertion/deletion.

Vinayak Pathak

Papers

Flip distance between two triangulations of a point set is NP-complete

Self-approaching graphs

Flip Distance Between Two Triangulations of a Point Set is NP-complete.

Shortest Reconfiguration Paths in the Solution Space of Boolean Formulas

Streaming and dynamic algorithms for minimum enclosing balls in high dimensions