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Exact solutions to the Erdős-Rothschild problem
Let k := (k1,...,k2) be a sequence of natural numbers. For a graph G, let F (G;k) denote the number of colourings of the edges of G with colours 1,...,s such that, for every c ∈ {1,...,s}, the edges of colour c contain no clique of order kc. Write F (n; k) to denote the maximum of F (G;k) over all graphs G on n vertices. There are currently very few known exact (or asymptotic) results for this problem, posed by Erdős and Rothschild in 1974. We prove some new exact results for n → ∞:
(i) A sufficient condition on k which guarantees that every extremal graph is a complete multipartite graph, which systematically recovers all existing exact results.
(ii) Addressing the original question of Erdős and Rothschild, in the case k = (3,..., 3) of length 7, the unique extremal graph is the complete balanced 8-partite graph, with colourings coming from Hadamard matrices of order 8.
(iii) In the case k = (k+ 1, k), for which the sufficient condition in (i) does not hold, for 3 ≤ k ≤ 10, the unique extremal graph is complete k-partite with one part of size less than k and the other parts as equal in size as
possible
Large cliques or cocliques in hypergraphs with forbidden order-size pairs
The well-known Erdős-Hajnal conjecture states that for any graph , there exists such that every -vertex graph that contains no induced copy of has a homogeneous set of size at least . We consider a variant of the Erdős-Hajnal problem for hypergraphs where we forbid a family of hypergraphs described by their orders and sizes. For graphs, we observe that if we forbid induced subgraphs on vertices and edges for any positive and , then we obtain large homogeneous sets. For triple systems, in the first nontrivial case , for every , we give bounds on the minimum size of a homogeneous set in a triple system where the number of edges spanned by every four vertices is not in . In most cases the bounds are essentially tight. We also determine, for all , whether the growth rate is polynomial or polylogarithmic. Some open problems remain
Robust interventions in network epidemiology
Which individual should we vaccinate to minimize the spread of a disease? Designing optimal interventions of this kind can be formalized as an optimization problem on networks, in which we have to select a budgeted number of dynamically important nodes to receive treatment that optimizes a dynamical outcome. Describing this optimization problem requires specifying the network, a model of the dynamics, and an objective for the outcome of the dynamics. In real-world contexts, these inputs are vulnerable to misspecification---the network and dynamics must be inferred from data, and the decision-maker must operationalize some (potentially abstract) goal into a mathematical objective function. Moreover, the tools to make reliable inferences---on the dynamical parameters, in particular---remain limited due to computational problems and issues of identifiability. Given these challenges, models thus remain more useful for building intuition than for designing actual interventions. This thesis seeks to elevate complex dynamical models from intuition-building tools to methods for the practical design of interventions.
First, we circumvent the inference problem by searching for robust decisions that are insensitive to model misspecification.If these robust solutions work well across a broad range of structural and dynamic contexts, the issues associated with accurately specifying the problem inputs are largely moot. We explore the existence of these solutions across three facets of dynamic importance common in network epidemiology.
Second, we introduce a method for analytically calculating the expected outcome of a spreading process under various interventions. Our method is based on message passing, a technique from statistical physics that has received attention in a variety of contexts, from epidemiology to statistical inference.We combine several facets of the message-passing literature for network epidemiology.Our method allows us to test general probabilistic, temporal intervention strategies (such as seeding or vaccination). Furthermore, the method works on arbitrary networks without requiring the network to be locally tree-like .This method has the potential to improve our ability to discriminate between possible intervention outcomes.
Overall, our work builds intuition about the decision landscape of designing interventions in spreading dynamics. This work also suggests a way forward for probing the decision-making landscape of other intervention contexts. More broadly, we provide a framework for exploring the boundaries of designing robust interventions with complex systems modeling tools
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
On the Power of Threshold-Based Algorithms for Detecting Cycles in the CONGEST Model
It is known that, for every , -freeness can be decided by a
generic Monte-Carlo algorithm running in rounds in the
CONGEST model. For , faster Monte-Carlo algorithms do exist,
running in rounds, based on upper bounding the number of
messages to be forwarded, and aborting search sub-routines for which this
number exceeds certain thresholds. We investigate the possible extension of
these threshold-based algorithms, for the detection of larger cycles. We first
show that, for every , there exists an infinite family of graphs
containing a -cycle for which any threshold-based algorithm fails to detect
that cycle. Hence, in particular, neither -freeness nor
-freeness can be decided by threshold-based algorithms. Nevertheless,
we show that -freeness can still be decided by a
threshold-based algorithm, running in rounds,
which is faster than using the generic algorithm, which would run in
rounds. Moreover, we exhibit an
infinite collection of families of cycles such that threshold-based algorithms
can decide -freeness for every in this collection.Comment: to be published in SIROCCO 202
Minimal Sparsity for Second-Order Moment-SOS Relaxations of the AC-OPF Problem
AC-OPF (Alternative Current Optimal Power Flow)aims at minimizing the
operating costs of a power gridunder physical constraints on voltages and power
injections.Its mathematical formulation results in a nonconvex polynomial
optimizationproblem which is hard to solve in general,but that can be tackled
by a sequence of SDP(Semidefinite Programming) relaxationscorresponding to the
steps of the moment-SOS (Sums-Of-Squares) hierarchy.Unfortunately, the size of
these SDPs grows drastically in the hierarchy,so that even second-order
relaxationsexploiting the correlative sparsity pattern of AC-OPFare hardly
numerically tractable for largeinstances -- with thousands of power buses.Our
contribution lies in a new sparsityframework, termed minimal sparsity,
inspiredfrom the specific structure of power flowequations.Despite its
heuristic nature, numerical examples show that minimal sparsity allows the
computation ofhighly accurate second-order moment-SOS relaxationsof AC-OPF,
while requiring far less computing time and memory resources than the standard
correlative sparsity pattern. Thus, we manage to compute second-order
relaxations on test caseswith about 6000 power buses, which we believe to be
unprecedented
Computing complexity measures of degenerate graphs
We show that the VC-dimension of a graph can be computed in time , where is the degeneracy of the input graph. The core idea
of our algorithm is a data structure to efficiently query the number of
vertices that see a specific subset of vertices inside of a (small) query set.
The construction of this data structure takes time , afterwards
queries can be computed efficiently using fast M\"obius inversion.
This data structure turns out to be useful for a range of tasks, especially
for finding bipartite patterns in degenerate graphs, and we outline an
efficient algorithms for counting the number of times specific patterns occur
in a graph. The largest factor in the running time of this algorithm is
, where is a parameter of the pattern we call its left covering
number.
Concrete applications of this algorithm include counting the number of
(non-induced) bicliques in linear time, the number of co-matchings in quadratic
time, as well as a constant-factor approximation of the ladder index in linear
time.
Finally, we supplement our theoretical results with several implementations
and run experiments on more than 200 real-world datasets -- the largest of
which has 8 million edges -- where we obtain interesting insights into the
VC-dimension of real-world networks.Comment: Accepted for publication in the 18th International Symposium on
Parameterized and Exact Computation (IPEC 2023
Space-Query Tradeoffs in Range Subgraph Counting and Listing
This paper initializes the study of range subgraph counting and range subgraph listing, both of which are motivated by the significant demands in practice to perform graph analytics on subgraphs pertinent to only selected, as opposed to all, vertices. In the first problem, there is an undirected graph G where each vertex carries a real-valued attribute. Given an interval q and a pattern Q, a query counts the number of occurrences of Q in the subgraph of G induced by the vertices whose attributes fall in q. The second problem has the same setup except that a query needs to enumerate (rather than count) those occurrences with a small delay. In both problems, our goal is to understand the tradeoff between space usage and query cost, or more specifically: (i) given a target on query efficiency, how much pre-computed information about G must we store? (ii) Or conversely, given a budget on space usage, what is the best query time we can hope for? We establish a suite of upper- and lower-bound results on such tradeoffs for various query patterns
Borel versions of the Local Lemma and LOCAL algorithms for graphs of finite asymptotic separation index
Asymptotic separation index is a parameter that measures how easily a Borel
graph can be approximated by its subgraphs with finite components. In contrast
to the more classical notion of hyperfiniteness, asymptotic separation index is
well-suited for combinatorial applications in the Borel setting. The main
result of this paper is a Borel version of the Lov\'asz Local Lemma -- a
powerful general-purpose tool in probabilistic combinatorics -- under a finite
asymptotic separation index assumption. As a consequence, we show that locally
checkable labeling problems that are solvable by efficient randomized
distributed algorithms admit Borel solutions on bounded degree Borel graphs
with finite asymptotic separation index. From this we derive a number of
corollaries, for example a Borel version of Brooks's theorem for graphs with
finite asymptotic separation index
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