A $k$-spanner of a graph $G$ is a sparse subgraph that preserves its shortest path distances up to a multiplicative stretch factor of $k$, and a $k$-emulator is similar but not required to be a subgraph of $G$. A classic theorem by Thorup and Zwick [
JACM 05] shows that, despite the extra flexibility available to emulators, the size/stretch tradeoffs for spanners and emulators are equivalent. Our main result is that this equivalence in tradeoffs no longer holds in the commonly-studied setting of graphs with vertex failures. That is: we introduce a natural definition of vertex fault-tolerant emulators, and then we show a three-way tradeoff between size, stretch, and fault-tolerance for these emulators that polynomially surpasses the tradeoff known to be optimal for spanners. We complement our emulator upper bound with a lower bound construction that is essentially tight (within $log n$ factors of the upper bound) when the stretch is $2k-1$ and $k$ is either a fixed odd integer or $2$. We also show constructions of fault-tolerant emulators with additive error, demonstrating that these also enjoy significantly improved tradeoffs over those available for fault-tolerant additive spanners.
We study the optimization version of the equal cardinality set partition problem (where the absolute difference between the equal sized partitions sums are minimized). While this problem is NP-hard and requires exponential complexity to solve in gene
ral, we have formulated a weaker version of this NP-hard problem, where the goal is to find a locally optimal solution. The local optimality considered in our work is under any swap between the opposing partitions element pairs. To this end, we designed an algorithm which can produce such a locally optimal solution in $O(N^2)$ time and $O(N)$ space. Our approach does not require positive or integer inputs and works equally well under arbitrary input precisions. Thus, it is widely applicable in different problem scenarios.
We study the problem of approximating the eigenspectrum of a symmetric matrix $A in mathbb{R}^{n times n}$ with bounded entries (i.e., $|A|_{infty} leq 1$). We present a simple sublinear time algorithm that approximates all eigenvalues of $A$ up to a
dditive error $pm epsilon n$ using those of a randomly sampled $tilde{O}(frac{1}{epsilon^4}) times tilde O(frac{1}{epsilon^4})$ principal submatrix. Our result can be viewed as a concentration bound on the full eigenspectrum of a random principal submatrix. It significantly extends existing work which shows concentration of just the spectral norm [Tro08]. It also extends work on sublinear time algorithms for testing the presence of large negative eigenvalues in the spectrum [BCJ20]. To complement our theoretical results, we provide numerical simulations, which demonstrate the effectiveness of our algorithm in approximating the eigenvalues of a wide range of matrices.
Given two sets $S$ and $T$ of points in the plane, of total size $n$, a {many-to-many} matching between $S$ and $T$ is a set of pairs $(p,q)$ such that $pin S$, $qin T$ and for each $rin Scup T$, $r$ appears in at least one such pair. The {cost of a
pair} $(p,q)$ is the (Euclidean) distance between $p$ and $q$. In the {minimum-cost many-to-many matching} problem, the goal is to compute a many-to-many matching such that the sum of the costs of the pairs is minimized. This problem is a restricted version of minimum-weight edge cover in a bipartite graph, and hence can be solved in $O(n^3)$ time. In a more restricted setting where all the points are on a line, the problem can be solved in $O(nlog n)$ time [Colannino, Damian, Hurtado, Langerman, Meijer, Ramaswami, Souvaine, Toussaint; Graphs Comb., 2007]. However, no progress has been made in the general planar case in improving the cubic time bound. In this paper, we obtain an $O(n^2cdot poly(log n))$ time exact algorithm and an $O( n^{3/2}cdot poly(log n))$ time $(1+epsilon)$-approximation in the planar case. Our results affirmatively address an open problem posed in [Colannino et al., Graphs Comb., 2007].
Individualization-Refinement (IR) algorithms form the standard method and currently the only practical method for symmetry computations of graphs and combinatorial objects in general. Through backtracking, on each graph an IR-algorithm implicitly cre
ates an IR-tree whose order is the determining factor of the running time of the algorithm. We give a precise and constructive characterization which trees are IR-trees. This characterization is applicable both when the tree is regarded as an uncolored object but also when regarded as a colored object where vertex colors stem from a node invariant. We also provide a construction that given a tree produces a corresponding graph whenever possible. This provides a constructive proof that our necessary conditions are also sufficient for the characterization.
Tens of thousands of parent companies control millions of subsidiaries through long chains of intermediary entities in global corporate networks. Conversely, tens of millions of entities are directly held by sole owners. We propose an algorithm for i
dentification of ultimate controlling entities in such networks that unifies direct and indirect control and allows for continuous interpolation between the two concepts via a factor damping long paths. By exploiting onion-like properties of ownership networks the algorithm scales linearly with the network size and handles circular ownership by design. We apply it to the universe of 4.2 mln UK companies and 4 mln of their holders to understand the distribution of control in the country. Furthermore, we provide the first independent evaluation of the control identification results. We reveal that the proposed $alpha$-ICON algorithm identifies more than 96% of beneficiary entities from the evaluation set and supersedes the existing approaches reported in the literature. We refer the superiority of $alpha$-ICON algorithm to its ability to correctly identify the parents long away from their subsidiaries in the network.
Fluid human-agent communication is essential for the future of human-in-the-loop reinforcement learning. An agent must respond appropriately to feedback from its human trainer even before they have significant experience working together. Therefore,
it is important that learning agents respond well to various feedback schemes human trainers are likely to provide. This work analyzes the COnvergent Actor-Critic by Humans (COACH) algorithm under three different types of feedback-policy feedback, reward feedback, and advantage feedback. For these three feedback types, we find that COACH can behave sub-optimally. We propose a variant of COACH, episodic COACH (E-COACH), which we prove converges for all three types. We compare our COACH variant with two other reinforcement-learning algorithms: Q-learning and TAMER.
We give a probabilistic analysis of the unit-demand Euclidean capacitated vehicle routing problem in the random setting, where the input distribution consists of $n$ unit-demand customers modeled as independent, identically distributed uniform random
points in the two-dimensional plane. The objective is to visit every customer using a set of routes of minimum total length, such that each route visits at most $k$ customers, where $k$ is the capacity of a vehicle. All of the following results are in the random setting and hold asymptotically almost surely. The best known polynomial-time approximation for this problem is the iterated tour partitioning (ITP) algorithm, introduced in 1985 by Haimovich and Rinnooy Kan. They showed that the ITP algorithm is near-optimal when $k$ is either $o(sqrt{n})$ or $omega(sqrt{n})$, and they asked whether the ITP algorithm was also effective in the intermediate range. In this work, we show that when $k=sqrt{n}$, the ITP algorithm is at best a $(1+c_0)$-approximation for some positive constant $c_0$. On the other hand, the approximation ratio of the ITP algorithm was known to be at most $0.995+alpha$ due to Bompadre, Dror, and Orlin, where $alpha$ is the approximation ratio of an algorithm for the traveling salesman problem. In this work, we improve the upper bound on the approximation ratio of the ITP algorithm to $0.915+alpha$. Our analysis is based on a new lower bound on the optimal cost for the metric capacitated vehicle routing problem, which may be of independent interest.
Shortest path algorithms have played a key role in the past century, paving the way for modern day GPS systems to find optimal routes along static systems in fractions of a second. One application of these algorithms includes optimizing the total dis
tance of power lines (specifically in star topological configurations). Due to the relevancy of discovering well-connected electrical systems in certain areas, finding a minimum path that is able to account for geological features would have far-reaching consequences in lowering the cost of electric power transmission. We initialize our research by proving the convex hull as an effective bounding mechanism for star topological minimum path algorithms. Building off this bounding, we propose novel algorithms to manage certain cases that lack existing methods (weighted regions and obstacles) by discretizing Euclidean space into squares and combining pre-existing algorithms that calculate local minimums that we believe have a possibility of being the absolute minimum. We further designate ways to evaluate iterations necessary to reach some level of accuracy. Both of these novel algorithms fulfill certain niches that past literature does not cover.
To overcome incompatibility issues, kidney patients may swap their donors. In international kidney exchange programmes (IKEPs), countries merge their national patient-donor pools. We consider a recent credit system where in each round, countries are
given an initial kidney transplant allocation which is adjusted by a credit function yielding a target allocation. The goal is to find a solution in the patient-donor compatibility graph that approaches the target allocation as closely as possible, to ensure long-term stability of the international pool. As solutions, we use maximum matchings that lexicographically minimize the country deviations from the target allocation. We first give a polynomial-time algorithm for computing such matchings. We then perform, for the first time, a computational study for a large number of countries. For the initial allocations we use, besides two easy-to-compute solution concepts, two classical concepts: the Shapley value and nucleolus. These are hard to compute, but by using state-of-the-art software we show that they are now within reach for IKEPs of up to fifteen countries. Our experiments show that using lexicographically minimal maximum matchings instead of ones that only minimize the largest deviation from the target allocation (as previously done) may make an IKEP up to 52% more balanced.
