arXiv subjects

Jennifer Iglesias

Publications and source records attributed to Jennifer Iglesias.

8 recordsLinked to original sources

arXiv2020

Symmetric Linear Programming Formulations for Minimum Cut with Applications to TSP

We introduce multiple symmetric LP relaxations for minimum cut problems. The relaxations give optimal and approximate solutions when the input is a Hamiltonian cycle. We show that this leads to one of two interesting results. In one case, these LPs always give optimal and near optimal solutions, and then they would be the smallest known symmetric LPs for the problems considered. Otherwise, these LP formulations give strictly better LP relaxations for the traveling salesperson problem than the subtour relaxation. We have the smallest known LP formulation that is a 9/8-approximation or better for min-cut. In addition, the LP relaxation of min-cut investigated in this paper has interesting constraints; the LP contains only a single typical min-cut constraint and all other constraints are typically only used for max-cut relaxations.

cs.DS↗

arXiv2017

Egalitarian Graph Orientations

Given an undirected graph, one can assign directions to each of the edges of the graph, thus orienting the graph. To be as egalitarian as possible, one may wish to find an orientation such that no vertex is unfairly hit with too many arcs directed into it. We discuss how this objective arises in problems resulting from telecommunications. We give optimal, polynomial-time algorithms for: finding an orientation that minimizes the lexicographic order of the indegrees and finding a strongly-connected orientation that minimizes the maximum indegree. We show that minimizing the lexicographic order of the indegrees is NP-hard when the resulting orientation is required to be acyclic.

cs.DM↗

arXiv2017

Coloring Down: $3/2$-approximation for special cases of the weighted tree augmentation problem

In this paper, we investigate the weighted tree augmentation problem (TAP), where the goal is to augment a tree with a minimum cost set of edges such that the graph becomes two edge connected. First we show that in weighted TAP, we can restrict our attention to trees which are binary and where all the non-tree edges go between two leaves of the tree. We then give two different top-down coloring algorithms. Both algorithms differ from known techniques for a 3/2-approximation in unweighted TAP and current attempts to reach a 3/2-approximation for weighted TAP. The first algorithm we describe always gives a 2-approximation for any feasible fractional solution to the natural edge cover LP. When the fractional solution is such that all the edges with non-zero weight are at least $α$, then this algorithm achieves a $2/(1+α)$-approximation. We propose a new conjecture on extreme points of LP relaxations for the problem, which if true, will lead to a constructive proof of an integrality gap of at most 3/2 for weighted TAP. In the second algorithm, we introduce simple valid constraints to the edge cover LP. In this algorithm, we focus on deficient edges, edges covered to an extent less than 4/3 in the fractional solution. We show that for fractional feasible solutions, deficient edges occur in node-disjoint paths in the tree. When the number of such paths is at most two, we give a top-down coloring algorithm which decomposes 3/2 times the fractional solution into a convex combination of integer solutions. We believe our algorithms will be useful in eventually resolving the integrality gap of linear programming formulations for TAP. We also investigate a variant of TAP where each edge in the solution must be covered by a cycle of length three. We give a $Θ(\log n)$-approximation algorithm for this problem in the weighted case and a 4-approximation in the unweighted case.

cs.DS↗

arXiv2017

Plane Gossip: Approximating rumor spread in planar graphs

We study the design of schedules for multi-commodity multicast; we are given an undirected graph $G$ and a collection of source destination pairs, and the goal is to schedule a minimum-length sequence of matchings that connects every source with its respective destination. Multi-commodity multicast models a classic information dissemination problem in networks where the primary communication constraint is the number of connections that a node can make, not link bandwidth. Multi-commodity multicast is closely related to the problem of finding a subgraph, $H$, of optimal poise, where the poise is defined as the sum of the maximum degree of $H$ and the maximum distance between any source-destination pair in $H$. We first show that the minimum poise subgraph for single-commodity multicast can be approximated to within a factor of $O(\log k)$ with respect to the value of a natural LP relaxation in an instance with $k$ terminals. This is the first upper bound on the integrality gap of the natural LP. Using this poise result and shortest-path separators in planar graphs, we obtain a $O(\log^3 k\log n/(\log\log n))$-approximation for multi-commodity multicast for planar graphs. We also study the minimum-time radio gossip problem in planar graphs where a message from each node must be transmitted to all other nodes under a model where nodes can broadcast to all neighbors in a single step but only nodes with a single broadcasting neighbor get a message. We give an $O(\log^2 n)$-approximation for radio gossip in planar graphs breaking previous barriers. This is the first bound for radio gossip that does not rely on the maximum degree of the graph. Finally, we show that our techniques for planar graphs extend to graphs with excluded minors. We establish polylogarithmic-approximation algorithms for both multi-commodity multicast and radio gossip problems in minor-free graphs.

cs.DS↗

arXiv2017

Single-sink Fractionally Subadditive Network Design

We study a generalization of the Steiner tree problem, where we are given a weighted network $G$ together with a collection of $k$ subsets of its vertices and a root $r$. We wish to construct a minimum cost network such that the network supports one unit of flow to the root from every node in a subset simultaneously. The network constructed does not need to support flows from all the subsets simultaneously. We settle an open question regarding the complexity of this problem for $k=2$, and give a $\frac{3}{2}$-approximation algorithm that improves over a (trivial) known 2-approximation. Furthermore, we prove some structural results that prevent many well-known techniques from doing better than the known $O(\log n)$-approximation. Despite these obstacles, we conjecture that this problem should have an $O(1)$-approximation. We also give an approximation result for a variant of the problem where the solution is required to be a path.

cs.DS↗

arXiv2016

Balls and Funnels: Energy Efficient Group-to-Group Anycasts

We introduce group-to-group anycast (g2g-anycast), a network design problem of substantial practical importance and considerable generality. Given a collection of groups and requirements for directed connectivity from source groups to destination groups, the solution network must contain, for each requirement, an omni-directional down-link broadcast, centered at any node of the source group, called the ball; the ball must contain some node from the destination group in the requirement and all such destination nodes in the ball must aggregate into a tree directed towards the source, called the funnel-tree. The solution network is a collection of balls along with the funnel-trees they contain. g2g-anycast models DBS (Digital Broadcast Satellite), Cable TV systems and drone swarms. It generalizes several well known network design problems including minimum energy unicast, multicast, broadcast, Steiner-tree, Steiner-forest and Group-Steiner tree. Our main achievement is an $O(\log^4 n)$ approximation, counterbalanced by an $\log^{(2-ε)}n$ hardness of approximation, for general weights. Given the applicability to wireless communication, we present a scalable and easily implemented $O(\log n)$ approximation algorithm, Cover-and-Grow for fixed-dimensional Euclidean space with path-loss exponent at least 2.

cs.NI↗

arXiv2012

Computing with voting trees

The classical paradox of social choice theory asserts that there is no fair way to deterministically select a winner in an election among more than two candidates; the only definite collective preferences are between individual pairs of candidates. Combinatorially, one may summarize this information with a graph-theoretic tournament on N vertices (one per candidate), placing an edge from U to V if U would beat V in an election between only those two candidates (no ties are permitted). One well-studied procedure for selecting a winner is to specify a complete binary tree whose leaves are labeled by the candidates, and evaluate it by running pairwise elections between the pairs of leaves, sending the winners to successive rounds of pairwise elections which ultimately terminate with a single winner. This structure is called a voting tree. Much research has investigated which functions on tournaments are computable in this way. Fischer, Procaccia, and Samorodnitsky quantitatively studied the computability of the Copeland rule, which returns a vertex of maximum out-degree in the given tournament. Perhaps surprisingly, the best previously known voting tree could only guarantee a returned out-degree of at least log_2 N, despite the fact that every tournament has a vertex of degree at least (N-1)/2. In this paper, we present three constructions, the first of which substantially improves this guarantee to Θ(sqrt{N}). The other two demonstrate the richness of the voting tree universe, with a tree that resists manipulation, and a tree which implements arithmetic modulo three.

math.CO↗

arXiv2012

Length 3 Edge-Disjoint Paths and Partial Orientation

In 2003, it was claimed that the following problem was solvable in polynomial time: do there exist k edge-disjoint paths of length exactly 3 between vertices s and t in a given graph? The proof was flawed, and we show that this problem is NP-hard even if we disallow multiple edges. We use a reduction from Partial Orientation, a problem recently shown by Pálvölgyi to be NP-hard.

cs.CC↗