Finding kk Simple Shortest Paths and Cycles

The problem of finding multiple simple shortest paths in a weighted directed graph $G=(V,E)$ has many applications, and is considerably more difficult than the corresponding problem when cycles are allowed in the paths. Even for a single source-sink pair, it is known that two simple shortest paths cannot be found in time polynomially smaller than $n^3$ (where $n=|V|$) unless the All-Pairs Shortest Paths problem can be solved in a similar time bound. The latter is a well-known open problem in algorithm design. We consider the all-pairs version of the problem, and we give a new algorithm to find $k$ simple shortest paths for all pairs of vertices. For $k=2$, our algorithm runs in $O(mn + n^2 \log n)$ time (where $m=|E|$), which is almost the same bound as for the single pair case, and for $k=3$ we improve earlier bounds. Our approach is based on forming suitable path extensions to find simple shortest paths; this method is different from the `detour finding' technique used in most of the prior work on simple shortest paths, replacement paths, and distance sensitivity oracles. Enumerating simple cycles is a well-studied classical problem. We present new algorithms for generating simple cycles and simple paths in $G$ in non-decreasing order of their weights; the algorithm for generating simple paths is much faster, and uses another variant of path extensions. We also give hardness results for sparse graphs, relative to the complexity of computing a minimum weight cycle in a graph, for several variants of problems related to finding $k$ simple paths and cycles.Comment: The current version includes new results for undirected graphs. In Section 4, the notion of an (m,n) reduction is generalized to an f(m,n) reductio

A Graph Theoretic Method for Determining Generating Sets of Prime Ideals in Quantum Matrices

We take a graph theoretic approach to the problem of finding generators for those prime ideals of $\mathcal{O}_q(\mathcal{M}_{m,n}(\mathbb{K}))$ which are invariant under the torus action ($\mathbb{K}^*)^{m+n}$. Launois \cite{launois3} has shown that the generators consist of certain quantum minors of the matrix of canonical generators of $\mathcal{O}_q(\mathcal{M}_{m,n}(\mathbb{K}))$ and in \cite{launois2} gives an algorithm to find them. In this paper we modify a classic result of Lindstr\"{o}m \cite{lind} and Gessel-Viennot~\cite{gv} to show that a quantum minor is in the generating set for a particular ideal if and only if we can find a particular set of vertex-disjoint directed paths in an associated directed graph.Comment: 29 pages, 9 figure

On the Approximability of Digraph Ordering

Given an n-vertex digraph D = (V, A) the Max-k-Ordering problem is to compute a labeling $\ell : V \to [k]$ maximizing the number of forward edges, i.e. edges (u,v) such that $\ell$(u) < $\ell$(v). For different values of k, this reduces to Maximum Acyclic Subgraph (k=n), and Max-Dicut (k=2). This work studies the approximability of Max-k-Ordering and its generalizations, motivated by their applications to job scheduling with soft precedence constraints. We give an LP rounding based 2-approximation algorithm for Max-k-Ordering for any k={2,..., n}, improving on the known 2k/(k-1)-approximation obtained via random assignment. The tightness of this rounding is shown by proving that for any k={2,..., n} and constant $\varepsilon > 0$, Max-k-Ordering has an LP integrality gap of 2 - $\varepsilon$ for $n^{\Omega\left(1/\log\log k\right)}$ rounds of the Sherali-Adams hierarchy. A further generalization of Max-k-Ordering is the restricted maximum acyclic subgraph problem or RMAS, where each vertex v has a finite set of allowable labels $S_v \subseteq \mathbb{Z}^+$. We prove an LP rounding based $4\sqrt{2}/(\sqrt{2}+1) \approx 2.344$ approximation for it, improving on the $2\sqrt{2} \approx 2.828$ approximation recently given by Grandoni et al. (Information Processing Letters, Vol. 115(2), Pages 182-185, 2015). In fact, our approximation algorithm also works for a general version where the objective counts the edges which go forward by at least a positive offset specific to each edge. The minimization formulation of digraph ordering is DAG edge deletion or DED(k), which requires deleting the minimum number of edges from an n-vertex directed acyclic graph (DAG) to remove all paths of length k. We show that both, the LP relaxation and a local ratio approach for DED(k) yield k-approximation for any $k\in [n]$.Comment: 21 pages, Conference version to appear in ESA 201

Fast Monotone Summation over Disjoint Sets

We study the problem of computing an ensemble of multiple sums where the summands in each sum are indexed by subsets of size $p$ of an $n$-element ground set. More precisely, the task is to compute, for each subset of size $q$ of the ground set, the sum over the values of all subsets of size $p$ that are disjoint from the subset of size $q$. We present an arithmetic circuit that, without subtraction, solves the problem using $O((n^p+n^q)\log n)$ arithmetic gates, all monotone; for constant $p$, $q$ this is within the factor $\log n$ of the optimal. The circuit design is based on viewing the summation as a "set nucleation" task and using a tree-projection approach to implement the nucleation. Applications include improved algorithms for counting heaviest $k$-paths in a weighted graph, computing permanents of rectangular matrices, and dynamic feature selection in machine learning

Efficiently computing Top-K shortest path join

© 2015, Copyright is with the authors. Driven by many applications, in this paper we study the problem of computing the top-k shortest paths from one set of target nodes to another set of target nodes in a graph, namely the top-k shortest path join (KPJ) between two sets of target nodes. While KPJ is an extension of the problem of computing the top-k shortest paths (KSP) between two target nodes, the existing technique by converting KPJ to KSP has several deficiencies in conducting the computation. To resolve these, we propose to use the best-first paradigm to recursively divide search subspaces into smaller subspaces, and to compute the shortest path in each of the subspaces in a prioritized order based on their lower bounds. Consequently, we only compute shortest paths in subspaces whose lower bounds are larger than the length of the current k-th shortest path. To improve the efficiency, we further propose an iteratively bounding approach to tightening lower bounds of subspaces. Moreover, we propose two index structures which can be used to reduce the exploration area of a graph dramatically; these greatly speed up the computation. Extensive performance studies based on real road networks demonstrate the scalability of our approaches and that our approaches outperform the existing approach by several orders of magnitude. Furthermore, our approaches can be immediately used to compute KSP. Our experiment also demonstrates that our techniques outperform the state-of-the-art algorithm for KSP by several orders of magnitude

Call Control in Rings

The call control problem is an important optimization problem encountered in the design and operation of communication networks. The goal of the call control problem in rings is to compute, for a given ring network with edge capacities and a set of paths in the ring, a maximum cardinality subset of the paths such that no edge capacity is violated. We give a polynomial-time algorithm to solve the problem optimally. The algorithm is based on a decision procedure that checks whether a solution with at least k paths exists, which is in turn implemented by an iterative greedy approach operating in rounds. We show that the algorithm can be implemented efficiently and, as a by-product, obtain a linear-time algorithm to solve the problem in chains optimally. For the weighted version of call control in rings, where each path is associated with a weight and the goal is to maximize the total weight of the paths in the solution, we present a simple 2-approximation algorithm and a polynomial-time approximation scheme. While the complexity of the weighted version remains open, we show that it is at least as hard as the bipartite exact matching problem, which has not been resolved to be in P or NP-hard. This latter result follows from recent work by Hochbaum and Levi

Metric Embedding via Shortest Path Decompositions

We study the problem of embedding shortest-path metrics of weighted graphs into $\ell_p$ spaces. We introduce a new embedding technique based on low-depth decompositions of a graph via shortest paths. The notion of Shortest Path Decomposition depth is inductively defined: A (weighed) path graph has shortest path decomposition (SPD) depth $1$. General graph has an SPD of depth $k$ if it contains a shortest path whose deletion leads to a graph, each of whose components has SPD depth at most $k-1$. In this paper we give an $O(k^{\min\{\frac{1}{p},\frac{1}{2}\}})$-distortion embedding for graphs of SPD depth at most $k$. This result is asymptotically tight for any fixed $p>1$, while for $p=1$ it is tight up to second order terms. As a corollary of this result, we show that graphs having pathwidth $k$ embed into $\ell_p$ with distortion $O(k^{\min\{\frac{1}{p},\frac{1}{2}\}})$. For $p=1$, this improves over the best previous bound of Lee and Sidiropoulos that was exponential in $k$; moreover, for other values of $p$ it gives the first embeddings whose distortion is independent of the graph size $n$. Furthermore, we use the fact that planar graphs have SPD depth $O(\log n)$ to give a new proof that any planar graph embeds into $\ell_1$ with distortion $O(\sqrt{\log n})$. Our approach also gives new results for graphs with bounded treewidth, and for graphs excluding a fixed minor