Log-space Algorithms for Paths and Matchings in k-trees

Reachability and shortest path problems are NL-complete for general graphs. They are known to be in L for graphs of tree-width 2 [JT07]. However, for graphs of tree-width larger than 2, no bound better than NL is known. In this paper, we improve these bounds for k-trees, where k is a constant. In particular, the main results of our paper are log-space algorithms for reachability in directed k-trees, and for computation of shortest and longest paths in directed acyclic k-trees. Besides the path problems mentioned above, we also consider the problem of deciding whether a k-tree has a perfect macthing (decision version), and if so, finding a perfect match- ing (search version), and prove that these two problems are L-complete. These problems are known to be in P and in RNC for general graphs, and in SPL for planar bipartite graphs [DKR08]. Our results settle the complexity of these problems for the class of k-trees. The results are also applicable for bounded tree-width graphs, when a tree-decomposition is given as input. The technique central to our algorithms is a careful implementation of divide-and-conquer approach in log-space, along with some ideas from [JT07] and [LMR07].Comment: Accepted in STACS 201

A Local Computation Approximation Scheme to Maximum Matching

We present a polylogarithmic local computation matching algorithm which guarantees a (1-\eps)-approximation to the maximum matching in graphs of bounded degree.Comment: Appears in Approx 201

Hardness of Exact Distance Queries in Sparse Graphs Through Hub Labeling

A distance labeling scheme is an assignment of bit-labels to the vertices of an undirected, unweighted graph such that the distance between any pair of vertices can be decoded solely from their labels. An important class of distance labeling schemes is that of hub labelings, where a node $v \in G$ stores its distance to the so-called hubs $S_v \subseteq V$, chosen so that for any $u,v \in V$ there is $w \in S_u \cap S_v$ belonging to some shortest $uv$ path. Notice that for most existing graph classes, the best distance labelling constructions existing use at some point a hub labeling scheme at least as a key building block. Our interest lies in hub labelings of sparse graphs, i.e., those with $|E(G)| = O(n)$, for which we show a lowerbound of $\frac{n}{2^{O(\sqrt{\log n})}}$ for the average size of the hubsets. Additionally, we show a hub-labeling construction for sparse graphs of average size $O(\frac{n}{RS(n)^{c}})$ for some $0 < c < 1$, where $RS(n)$ is the so-called Ruzsa-Szemer{\'e}di function, linked to structure of induced matchings in dense graphs. This implies that further improving the lower bound on hub labeling size to $\frac{n}{2^{(\log n)^{o(1)}}}$ would require a breakthrough in the study of lower bounds on $RS(n)$, which have resisted substantial improvement in the last 70 years. For general distance labeling of sparse graphs, we show a lowerbound of $\frac{1}{2^{O(\sqrt{\log n})}} SumIndex(n)$, where $SumIndex(n)$ is the communication complexity of the Sum-Index problem over $Z_n$. Our results suggest that the best achievable hub-label size and distance-label size in sparse graphs may be $\Theta(\frac{n}{2^{(\log n)^c}})$ for some $0<c < 1$

Geometry Helps to Compare Persistence Diagrams

Exploiting geometric structure to improve the asymptotic complexity of discrete assignment problems is a well-studied subject. In contrast, the practical advantages of using geometry for such problems have not been explored. We implement geometric variants of the Hopcroft--Karp algorithm for bottleneck matching (based on previous work by Efrat el al.) and of the auction algorithm by Bertsekas for Wasserstein distance computation. Both implementations use k-d trees to replace a linear scan with a geometric proximity query. Our interest in this problem stems from the desire to compute distances between persistence diagrams, a problem that comes up frequently in topological data analysis. We show that our geometric matching algorithms lead to a substantial performance gain, both in running time and in memory consumption, over their purely combinatorial counterparts. Moreover, our implementation significantly outperforms the only other implementation available for comparing persistence diagrams.Comment: 20 pages, 10 figures; extended version of paper published in ALENEX 201

Locally Optimal Load Balancing

This work studies distributed algorithms for locally optimal load-balancing: We are given a graph of maximum degree $\Delta$, and each node has up to $L$ units of load. The task is to distribute the load more evenly so that the loads of adjacent nodes differ by at most $1$. If the graph is a path ($\Delta = 2$), it is easy to solve the fractional version of the problem in $O(L)$ communication rounds, independently of the number of nodes. We show that this is tight, and we show that it is possible to solve also the discrete version of the problem in $O(L)$ rounds in paths. For the general case ($\Delta > 2$), we show that fractional load balancing can be solved in $\operatorname{poly}(L,\Delta)$ rounds and discrete load balancing in $f(L,\Delta)$ rounds for some function $f$, independently of the number of nodes.Comment: 19 pages, 11 figure

Truthful Facility Assignment with Resource Augmentation: An Exact Analysis of Serial Dictatorship

We study the truthful facility assignment problem, where a set of agents with private most-preferred points on a metric space are assigned to facilities that lie on the metric space, under capacity constraints on the facilities. The goal is to produce such an assignment that minimizes the social cost, i.e., the total distance between the most-preferred points of the agents and their corresponding facilities in the assignment, under the constraint of truthfulness, which ensures that agents do not misreport their most-preferred points. We propose a resource augmentation framework, where a truthful mechanism is evaluated by its worst-case performance on an instance with enhanced facility capacities against the optimal mechanism on the same instance with the original capacities. We study a very well-known mechanism, Serial Dictatorship, and provide an exact analysis of its performance. Although Serial Dictatorship is a purely combinatorial mechanism, our analysis uses linear programming; a linear program expresses its greedy nature as well as the structure of the input, and finds the input instance that enforces the mechanism have its worst-case performance. Bounding the objective of the linear program using duality arguments allows us to compute tight bounds on the approximation ratio. Among other results, we prove that Serial Dictatorship has approximation ratio $g/(g-2)$ when the capacities are multiplied by any integer $g \geq 3$. Our results suggest that even a limited augmentation of the resources can have wondrous effects on the performance of the mechanism and in particular, the approximation ratio goes to 1 as the augmentation factor becomes large. We complement our results with bounds on the approximation ratio of Random Serial Dictatorship, the randomized version of Serial Dictatorship, when there is no resource augmentation

Counting Euler Tours in Undirected Bounded Treewidth Graphs

We show that counting Euler tours in undirected bounded tree-width graphs is tractable even in parallel - by proving a $\#SAC^1$ upper bound. This is in stark contrast to #P-completeness of the same problem in general graphs. Our main technical contribution is to show how (an instance of) dynamic programming on bounded \emph{clique-width} graphs can be performed efficiently in parallel. Thus we show that the sequential result of Espelage, Gurski and Wanke for efficiently computing Hamiltonian paths in bounded clique-width graphs can be adapted in the parallel setting to count the number of Hamiltonian paths which in turn is a tool for counting the number of Euler tours in bounded tree-width graphs. Our technique also yields parallel algorithms for counting longest paths and bipartite perfect matchings in bounded-clique width graphs. While establishing that counting Euler tours in bounded tree-width graphs can be computed by non-uniform monotone arithmetic circuits of polynomial degree (which characterize $\#SAC^1$) is relatively easy, establishing a uniform $\#SAC^1$ bound needs a careful use of polynomial interpolation.Comment: 17 pages; There was an error in the proof of the GapL upper bound claimed in the previous version which has been subsequently remove