Hardness of Token Swapping on Trees

Given a graph where every vertex has exactly one labeled token, how can we most quickly execute a given permutation on the tokens? In (sequential) token swapping, the goal is to use the shortest possible sequence of swaps, each of which exchanges the tokens at the two endpoints of an edge of the graph. In parallel token swapping, the goal is to use the fewest rounds, each of which consists of one or more swaps on the edges of a matching. We prove that both of these problems remain NP-hard when the graph is restricted to be a tree. These token swapping problems have been studied by disparate groups of researchers in discrete mathematics, theoretical computer science, robot motion planning, game theory, and engineering. Previous work establishes NP-completeness on general graphs (for both problems), constant-factor approximation algorithms, and some poly-time exact algorithms for simple graph classes such as cliques, stars, paths, and cycles. Sequential and parallel token swapping on trees were first studied over thirty years ago (as "sorting with a transposition tree") and over twenty-five years ago (as "routing permutations via matchings"), yet their complexities were previously unknown. We also show limitations on approximation of sequential token swapping on trees: we identify a broad class of algorithms that encompass all three known polynomial-time algorithms that achieve the best known approximation factor (which is 2) and show that no such algorithm can achieve an approximation factor less than 2

Locality-Aware Qubit Routing for the Grid Architecture

Due to the short decohorence time of qubits available in the NISQ-era, it is essential to pack (minimize the size and or the depth of) a logical quantum circuit as efficiently as possible given a sparsely coupled physical architecture. In this work we introduce a locality-aware qubit routing algorithm based on a graph theoretic framework. Our algorithm is designed for the grid and certain \u27grid-like\u27 architectures. We experimentally show the competitiveness of algorithm by comparing it against the approximate token swapping algorithm, which is used as a primitive in many state-of-the-art quantum trans pilers. Our algorithm produces circuits of comparable depth (better on random permutations) while being an order of magnitude faster than a typical implementation of the approximate token swapping algorithm

IST Austria Thesis

This thesis considers two examples of reconfiguration problems: flipping edges in edge-labelled triangulations of planar point sets and swapping labelled tokens placed on vertices of a graph. In both cases the studied structures – all the triangulations of a given point set or all token placements on a given graph – can be thought of as vertices of the so-called reconfiguration graph, in which two vertices are adjacent if the corresponding structures differ by a single elementary operation – by a flip of a diagonal in a triangulation or by a swap of tokens on adjacent vertices, respectively. We study the reconfiguration of one instance of a structure into another via (shortest) paths in the reconfiguration graph. For triangulations of point sets in which each edge has a unique label and a flip transfers the label from the removed edge to the new edge, we prove a polynomial-time testable condition, called the Orbit Theorem, that characterizes when two triangulations of the same point set lie in the same connected component of the reconfiguration graph. The condition was first conjectured by Bose, Lubiw, Pathak and Verdonschot. We additionally provide a polynomial time algorithm that computes a reconfiguring flip sequence, if it exists. Our proof of the Orbit Theorem uses topological properties of a certain high-dimensional cell complex that has the usual reconfiguration graph as its 1-skeleton. In the context of token swapping on a tree graph, we make partial progress on the problem of finding shortest reconfiguration sequences. We disprove the so-called Happy Leaf Conjecture and demonstrate the importance of swapping tokens that are already placed at the correct vertices. We also prove that a generalization of the problem to weighted coloured token swapping is NP-hard on trees but solvable in polynomial time on paths and stars

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum

Fixed-Parameter Tractability of Maximum Colored Path and Beyond

We introduce a general method for obtaining fixed-parameter algorithms for problems about finding paths in undirected graphs, where the length of the path could be unbounded in the parameter. The first application of our method is as follows. We give a randomized algorithm, that given a colored $n$-vertex undirected graph, vertices $s$ and $t$, and an integer $k$, finds an $(s,t)$-path containing at least $k$ different colors in time $2^k n^{O(1)}$. This is the first FPT algorithm for this problem, and it generalizes the algorithm of Bj\"orklund, Husfeldt, and Taslaman [SODA 2012] on finding a path through $k$ specified vertices. It also implies the first $2^k n^{O(1)}$ time algorithm for finding an $(s,t)$-path of length at least $k$. Our method yields FPT algorithms for even more general problems. For example, we consider the problem where the input consists of an $n$-vertex undirected graph $G$, a matroid $M$ whose elements correspond to the vertices of $G$ and which is represented over a finite field of order $q$, a positive integer weight function on the vertices of $G$, two sets of vertices $S,T \subseteq V(G)$, and integers $p,k,w$, and the task is to find $p$ vertex-disjoint paths from $S$ to $T$ so that the union of the vertices of these paths contains an independent set of $M$ of cardinality $k$ and weight $w$, while minimizing the sum of the lengths of the paths. We give a $2^{p+O(k^2 \log (q+k))} n^{O(1)} w$ time randomized algorithm for this problem.Comment: 50 pages, 16 figure

Involution factorizations of Ewens random permutations

An involution is a bijection that is its own inverse. Given a permutation $\sigma$ of $[n],$ let $\mathsf{invol}(\sigma)$ denote the number of ways to express $\sigma$ as a composition of two involutions of $[n].$ The statistic $\mathsf{invol}$ is asymptotically lognormal when the symmetric groups $\mathfrak{S}_n$ are each equipped with Ewens Sampling Formula probability measures of some fixed positive parameter $\theta.$ This paper strengthens and generalizes previously determined results on the limiting distribution of $\log(\mathsf{invol})$ for uniform random permutations, i.e. the specific case of $\theta = 1$. We also investigate the first two moments of $\mathsf{invol}$ itself.Comment: 23 pages, no figures. Some minor edits. Extra material added to sections 2 and 4 and concluding remark

Algorithms and Generalizations for the Lovasz Local Lemma

The Lovasz Local Lemma (LLL) is a cornerstone principle of the probabilistic method for combinatorics. This shows that one can avoid a large of set of “bad-events” (forbidden configurations of variables), provided the local conditions are satisfied. The original probabilistic formulation of this principle did not give efficient algorithms. A breakthrough result of Moser & Tardos led to an framework based on resampling variables which turns nearly all applications of the LLL into efficient algorithms. We extend and generalize the algorithm of Moser & Tardos in a variety of ways. We show tighter bounds on the complexity of the Moser-Tardos algorithm, particularly its parallel form. We also give a new, faster parallel algorithm for the LLL. We show that in some cases, the Moser-Tardos algorithm can converge even thoughthe LLL itself does not apply; we give a new criterion (comparable to the LLL) for determining when this occurs. This leads to improved bounds for k-SAT and hypergraph coloring among other applications. We describe an extension of the Moser-Tardos algorithm based on partial resampling, and use this to obtain better bounds for problems involving sums of independent random variables, such as column-sparse packing and packet-routing. We describe a variant of the partial resampling algorithm specialized to approximating column-sparse covering integer programs, a generalization of set-cover. We also give hardness reductions and integrality gaps, showing that our partial resampling based algorithm obtains nearly optimal approximation factors. We give a variant of the Moser-Tardos algorithm for random permutations, one of the few cases of the LLL not covered by the original algorithm of Moser & Tardos. We use this to develop the first constructive algorithms for Latin transversals and hypergraph packing, including parallel algorithms. We analyze the distribution of variables induced by the Moser-Tardos algorithm. We show it has a random-like structure, which can be used to accelerate the Moser-Tardos algorithm itself as well as to cover problems such as MAX k-SAT in which we only partially avoid bad-events

Content addressable memory project

A parameterized version of the tree processor was designed and tested (by simulation). The leaf processor design is 90 percent complete. We expect to complete and test a combination of tree and leaf cell designs in the next period. Work is proceeding on algorithms for the computer aided manufacturing (CAM), and once the design is complete we will begin simulating algorithms for large problems. The following topics are covered: (1) the practical implementation of content addressable memory; (2) design of a LEAF cell for the Rutgers CAM architecture; (3) a circuit design tool user's manual; and (4) design and analysis of efficient hierarchical interconnection networks