Computer Science

Milgrom (2017) has proposed a heuristic for determining a maximum weight basis of an independence system I given that we want an approximation guarantee only for sets in a prescribed O⊆I . This O reflects prior knowledge of the designer about the location of the optimal basis. The heuristic is based on finding an `inner matroid', one contained in the independence system. We show that even in the case O=I of zero additional knowledge the worst-case performance of this new heuristic can be better than that of the classical greedy algorithm.

Unusually large prize pools in lotteries like Mega Millions and Powerball attract additional bettors, which increases the likelihood that multiple winners will have to share the pool. Thus, the expected value of a lottery ticket decreases as the probability of collisions (two or more bettors with identical winning tickets) increase. We propose a way to increase the expected value of lottery tickets by minimizing collisions, while preserving the independent generation necessary in a distributed point-of-sales environment. Our approach involves partitioning the ticket space among different vendors and pairing them off to ensure no collisions among pairs. Our analysis demonstrates that this approach increases the expected value each ticket, without increasing the size of the prize pool. We also analyze when ticket sales have maximal expected value, and show that they provide positive returns when the jackpot is between $775.2 million and $1.67 billion dollars.

We study the Maximum Bipartite Subgraph (MBS) problem, which is defined as follows. Given a set S of n geometric objects in the plane, we want to compute a maximum-size subset S ′ ⊆S such that the intersection graph of the objects in S ′ is bipartite. We first give a simple O(n) -time algorithm that solves the MBS problem on a set of n intervals. We also give an O( n 2 ) -time algorithm that computes a near-optimal solution for the problem on circular-arc graphs. We show that the MBS problem is NP-hard on geometric graphs for which the maximum independent set is NP-hard (hence, it is NP-hard even on unit squares and unit disks). On the other hand, we give a PTAS for the problem on unit squares and unit disks. Moreover, we show fast approximation algorithms with small-constant factors for the problem on unit squares, unit disks and unit-height rectangles. Finally, we study a closely related geometric problem, called Maximum Triangle-free Subgraph (TFS), where the objective is the same as that of MBS except the intersection graph induced by the set S ′ needs to be triangle-free only (instead of being bipartite).

The median of a set of vertices P of a graph G is the set of all vertices x of G minimizing the sum of distances from x to all vertices of P . In this paper, we present a linear time algorithm to compute medians in median graphs, improving over the existing quadratic time algorithm. We also present a linear time algorithm to compute medians in the ℓ 1 -cube complexes associated with median graphs. Median graphs constitute the principal class of graphs investigated in metric graph theory and have a rich geometric and combinatorial structure, due to their bijections with CAT(0) cube complexes and domains of event structures. Our algorithm is based on the majority rule characterization of medians in median graphs and on a fast computation of parallelism classes of edges ( Θ -classes or hyperplanes) via Lexicographic Breadth First Search (LexBFS). To prove the correctness of our algorithm, we show that any LexBFS ordering of the vertices of G satisfies the following fellow traveler property of independent interest: the parents of any two adjacent vertices of G are also adjacent. Using the fast computation of the Θ -classes, we also compute the Wiener index (total distance) of G in linear time and the distance matrix in optimal quadratic time.

Recently, random walks on dynamic graphs have been studied because of its adaptivity to dynamical settings including real network analysis. However, previous works showed a tremendous gap between static and dynamic networks for the cover time of a lazy simple random walk: Although O( n 3 ) cover time was shown for any static graphs of n vertices, there is an edge-changing dynamic graph with an exponential cover time. We study a lazy Metropolis walk of Nonaka, Ono, Sadakane, and Yamashita (2010), which is a weighted random walk using local degree information. We show that this walk is robust to an edge-changing in dynamic networks: For any connected edge-changing graphs of n vertices, the lazy Metropolis walk has the O( n 2 ) hitting time, the O( n 2 logn) cover time, and the O( n 2 ) coalescing time, while those times can be exponential for lazy simple random walks. All of these bounds are tight up to a constant factor. At the heart of the proof, we give upper bounds of those times for any reversible random walks with a time-homogeneous stationary distribution.

A tree with n vertices has at most 95 n/13 minimal dominating sets. The growth constant λ= 95 − − √ 13 ≈1.4194908 is best possible. It is obtained in a semi-automatic way as a kind of "dominant eigenvalue" of a bilinear operation on sixtuples that is derived from the dynamic-programming recursion for computing the number of minimal dominating sets of a tree. We also derive an output-sensitive algorithm for listing all minimal dominating sets with linear set-up time and linear delay between successive solutions.

The Known Menger's theorem states that in a finite graph, the size of a minimum separator set of any pair of vertices is equal to the maximum number of disjoint paths that can be found between these two vertices. In this paper, we study the minimal separators of two non-adjacent vertices in a finite graph, and we give a new elementary proof of Menger's theorem.

For a metric μ on a finite set T , the minimum 0-extension problem 0-Ext [μ] is defined as follows: Given V⊇T and c:( V 2 )→ Q + , minimize ∑c(xy)μ(γ(x),γ(y)) subject to γ:V→T, γ(t)=t (∀t∈T) , where the sum is taken over all unordered pairs in V . This problem generalizes several classical combinatorial optimization problems such as the minimum cut problem or the multiterminal cut problem. The complexity dichotomy of 0-Ext [μ] was established by Karzanov and Hirai, which is viewed as a manifestation of the dichotomy theorem for finite-valued CSPs due to Thapper and Živný. In this paper, we consider a directed version 0 → -Ext [μ] of the minimum 0-extension problem, where μ and c are not assumed to be symmetric. We extend the NP-hardness condition of 0-Ext [μ] to 0 → -Ext [μ] : If μ cannot be represented as the shortest path metric of an orientable modular graph with an orbit-invariant ``directed'' edge-length, then 0 → -Ext [μ] is NP-hard. We also show a partial converse: If μ is a directed metric of a modular lattice with an orbit-invariant directed edge-length, then 0 → -Ext [μ] is tractable. We further provide a new NP-hardness condition characteristic of 0 → -Ext [μ] , and establish a dichotomy for the case where μ is a directed metric of a star.

Consider a connected graph G and let T be a spanning tree of G . Every edge e?�G?�T induces a cycle in T?�{e} . The intersection of two distinct such cycles is the set of edges of T that belong to both cycles. We consider the problem of finding a spanning tree that has the least number of such non-empty intersections.

The minimum status of a graph is the minimum of statuses of all vertices of this graph. We give a sharp upper bound for the minimum status of a connected graph with fixed order and matching number (domination number, respectively), and characterize the unique trees achieving the bound. We also determine the unique tree such that its minimum status is as small as possible when order and matching number (domination number, respectively) are fixed.