Gwen Spencer

Improved lower bounds for the universal and a priori TSP

We consider two partial-information generalizations of the metric traveling salesman problem (TSP) in which the task is to produce a total ordering of a given metric space that performs well for a subset of the space that is not known in advance. In the universal TSP, the subset is chosen adversarially, and in the a priori TSP it is chosen probabilistically. Both the universal and a priori TSP have been studied since the mid-80s, starting with the work of Bartholdi & Platzman and Jaillet, respectively. We prove a lower bound of Ω(log n) for the universal TSP by bounding the competitive ratio of shortest-path metrics on Ramanujan graphs, which improves on the previous best bound of Hajiaghayi, Kleinberg & Leighton, who showed that the competitive ratio of the n × n grid is Ω(√6log n/log log n). Furthermore, we show that for a large class of combinatorial optimization problems that includes TSP, a bound for the universal problem implies a matching bound on the approximation ratio achievable by deterministic algorithms for the corresponding black-box a priori problem. As a consequence, our lower bound of Ω(log n) for the universal TSP implies a matching lower bound for the black-box a priori TSP.

Approximation algorithms for fragmenting a graph against a stochastically-located threat

Motivated by issues in allocating limited preventative resources to protect a landscape against the spread of a wildfire from a stochastic ignition point, we give approximation algorithms for a new family of stochastic optimization problems.

Graph-Coloring Ideals: Nullstellensatz Certificates, Gröbner Bases for Chordal Graphs, and Hardness of Gröbner Bases

We consider a well-known family of polynomial ideals encoding the problem of graph-k-colorability. Our paper describes how the inherent combinatorial structure of the ideals implies several interesting algebraic properties. Specifically, we provide lower bounds on the difficulty of computing Gröbner bases and Nullstellensatz certificates for the coloring ideals of general graphs. We revisit the fact that computing a Gröbner basis is NP-hard and prove a robust notion of hardness derived from the inapproximability of coloring problems. For chordal graphs, however, we explicitly describe a Gröbner basis for the coloring ideal and provide a polynomial-time algorithm to construct it.

The LSB Theorem Implies the KKM Lemma

Let Sd be the unit d-sphere, the set of all points of unit Euclidean distance from the origin in Rd+l. Any pair of points in Sd of the form x, -x is a pair of antipodes in Sd. Let Ad be the d-simplex formed by the convex hull of the standard unit vectors in Rd+l. Equivalent^, Ad = {(xu ..., xd+x) : ?/ xt = 1, x{ > 0}. The following are two classical results about closed covers of these topological spaces (for the first see [6] or [3], for the second see [5]):

Clustered networks protect cooperation against catastrophic collapse

Assuming a society of conditional cooperators (or moody conditional cooperators), this computational study proposes a new perspective on the structural advantage of social network clustering. Previous work focused on how clustered structure might encourage initial outbreaks of cooperation or defend against invasion by a few defectors. Instead, we explore the ability of a societal structure to retain cooperative norms in the face of widespread disturbances. Such disturbances may abstractly describe hardships like famine and economic recession, or the random spatial placement of a substantial numbers of pure defectors (or round-1 defectors ) among a spatially structured population of players in a laboratory game, etc. As links in tightly clustered societies are reallocated to distant contacts, we observe that a society becomes increasingly susceptible to catastrophic cascades of defection : mutually-beneficial cooperative norms can be destroyed completely by modest shocks of defection. In contrast, networks with higher clustering coefficients can withstand larger shocks of defection before being forced to catastrophically low levels of cooperation. We observe a remarkably linear protective effect of clustering coefficient that becomes active above a critical level of clustering . Notably, both the critical level and the slope of this dependence is higher for decision-rule parameterizations that correspond to higher costs of cooperation . Our modeling framework provides a simple way to reinterpret the counter-intuitive and widely cited human experiments of Suri and Watts (2011) while also affirming the classical intuition that network clustering and higher levels of cooperation should be positively associated.

Measuring the value of accurate link prediction for network seeding

Merging two classic questionsThe influence-maximization literature seeks small sets of individuals whose structural placement in the social network can drive large cascades of behavior. Optimization efforts to find the best seed set often assume perfect knowledge of the network topology. Unfortunately, social network links are rarely known in an exact way. When do seeding strategies based on less-than-accurate link prediction provide valuable insight?Our contributionWe introduce optimized-against-a-sample (

Gr\"obner Bases and Nullstellens\"atze for Graph-Coloring Ideals

\text{OAS}

Limits of Approximation Algorithms: PCPs and Unique Games (DIMACS Tutorial Lecture Notes)

OAS) performance to measure the value of optimizing seeding based on a noisy observation of a network. Our computational study investigates