873 research outputs found
Collapsing Superstring Conjecture
In the Shortest Common Superstring (SCS) problem, one is given a collection of strings, and needs to find a shortest string containing each of them as a substring. SCS admits 2 11/23-approximation in polynomial time (Mucha, SODA\u2713). While this algorithm and its analysis are technically involved, the 30 years old Greedy Conjecture claims that the trivial and efficient Greedy Algorithm gives a 2-approximation for SCS.
We develop a graph-theoretic framework for studying approximation algorithms for SCS. The framework is reminiscent of the classical 2-approximation for Traveling Salesman: take two copies of an optimal solution, apply a trivial edge-collapsing procedure, and get an approximate solution. In this framework, we observe two surprising properties of SCS solutions, and we conjecture that they hold for all input instances. The first conjecture, that we call Collapsing Superstring conjecture, claims that there is an elementary way to transform any solution repeated twice into the same graph G. This conjecture would give an elementary 2-approximate algorithm for SCS. The second conjecture claims that not only the resulting graph G is the same for all solutions, but that G can be computed by an elementary greedy procedure called Greedy Hierarchical Algorithm.
While the second conjecture clearly implies the first one, perhaps surprisingly we prove their equivalence. We support these equivalent conjectures by giving a proof for the special case where all input strings have length at most 3 (which until recently had been the only case where the Greedy Conjecture was proven). We also tested our conjectures on millions of instances of SCS.
We prove that the standard Greedy Conjecture implies Greedy Hierarchical Conjecture, while the latter is sufficient for an efficient greedy 2-approximate approximation of SCS. Except for its (conjectured) good approximation ratio, the Greedy Hierarchical Algorithm provably finds a 3.5-approximation, and finds exact solutions for the special cases where we know polynomial time (not greedy) exact algorithms: (1) when the input strings form a spectrum of a string (2) when all input strings have length at most 2
Menger Path Systems
https://digitalcommons.memphis.edu/speccoll-faudreerj/1235/thumbnail.jp
l-connectivity, l-edge-connectivity and spectral radius of graphs
Let G be a connected graph. The toughness of G is defined as
t(G)=min{\frac{|S|}{c(G-S)}}, in which the minimum is taken over all proper
subsets S\subset V(G) such that c(G-S)\geq 2 where c(G-S) denotes the number of
components of G-S. Confirming a conjecture of Brouwer, Gu [SIAM J. Discrete
Math. 35 (2021) 948--952] proved a tight lower bound on toughness of regular
graphs in terms of the second largest absolute eigenvalue. Fan, Lin and Lu
[European J. Combin. 110 (2023) 103701] then studied the toughness of simple
graphs from the spectral radius perspective. While the toughness is an
important concept in graph theory, it is also very interesting to study |S| for
which c(G-S)\geq l for a given integer l\geq 2. This leads to the concept of
the l-connectivity, which is defined to be the minimum number of vertices of G
whose removal produces a disconnected graph with at least l components or a
graph with fewer than l vertices. Gu [European J. Combin. 92 (2021) 103255]
discovered a lower bound on the l-connectivity of regular graphs via the second
largest absolute eigenvalue. As a counterpart, we discover the connection
between the l-connectivity of simple graphs and the spectral radius. We also
study similar problems for digraphs and an edge version
Constructing brain connectivity group graphs from EEG time series
Graphical analysis of complex brain networks is a fundamental area of modern neuroscience. Functional connectivity is important since many neurological and psychiatric disorders, including schizophrenia, are described as ‘dys-connectivity’ syndromes. Using electroencephalogram time series collected on each of a group of 15 individuals with a common medical diagnosis of positive syndrome schizophrenia we seek to build a single, representative, brain functional connectivity group graph. Disparity/distance measures between spectral matrices are identified and used to define the normalized graph Laplacian enabling clustering of the spectral matrices for detecting ‘outlying’ individuals. Two such individuals are identified. For each remaining individual, we derive a test for each edge in the connectivity graph based on average estimated partial coherence over frequencies, and associated p-values are found. For each edge these are used in a multiple hypothesis test across individuals and the proportion rejecting the hypothesis of no edge is used to construct a connectivity group graph. This study provides a framework for integrating results on multiple individuals into a single overall connectivity structure
A stochastic intra-ring synchronous optimal network design problem
We develop a stochastic programming approach to solving an intra-ring Synchronous Optical Network (SONET) design problem. This research differs from pioneering SONET design studies in two fundamental ways. First, while traditional approaches to solving this problem assume that all data are deterministic, we observe that for practical planning situations, network demand levels are stochastic. Second, while most models disallow demand shortages and focus only on the minimization of capital Add-Drop Multiplexer (ADM) equipment expenditure, our model minimizes a mix of ADM installations and expected penalties arising from the failure to satisfy some or all of the actual telecommunication demand. We propose an L-shaped algorithm to solve this design problem, and demonstrate how a nonlinear reformulation of the problem may improve the strength of the generated optimality cuts. We next enhance the ba-sic algorithm by implementing powerful lower and upper bounding techniques via an assortment of modeling, valid inequality, and heuristic strategies. Our computational results conclusively demonstrate the efficacy of our proposed algorithm as opposed to standard L-shaped and extensive form approaches to solving the problem
Strong Connectivity in Real Directed Networks
In many real, directed networks, the strongly connected component of nodes
which are mutually reachable is very small. This does not fit with current
theory, based on random graphs, according to which strong connectivity depends
on mean degree and degree-degree correlations. And it has important
implications for other properties of real networks and the dynamical behaviour
of many complex systems. We find that strong connectivity depends crucially on
the extent to which the network has an overall direction or hierarchical
ordering -- a property measured by trophic coherence. Using percolation theory,
we find the critical point separating weakly and strongly connected regimes,
and confirm our results on many real-world networks, including ecological,
neural, trade and social networks. We show that the connectivity structure can
be disrupted with minimal effort by a targeted attack on edges which run
counter to the overall direction. And we illustrate with example dynamics --
the SIS model, majority vote, Kuramoto oscillators and the voter model -- how a
small number of edge deletions can utterly change dynamical processes in a wide
range of systems.Comment: 16 pages, 6 figure
Positroids are 3-colorable
We show that every positroid of rank has a positive coline. Using
the definition of the chromatic number of oriented matroid introduced by J.\
Ne\v{s}et\v{r}il, R.\ Nickel, and W.~Hochst\"{a}ttler, this shows that every
orientation of a positroid is 3-colorable
Correctness of Multiplicative (and Exponential) Proof Structures is NL-Complete
15 pagesInternational audienceWe provide a new correctness criterion for unit-free MLL proof structures and MELL proof structures with units. We prove that deciding the correctness of a MLL and of a MELL proof structure is NL-complete. We also prove that deciding the correctness of an intuitionistic multiplicative essential net is NL-complete
- …