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One of the most famous conjectures in combinatorial optimization is the four-thirds conjecture, which states that the integrality gap of the subtour LP relaxation of the TSP is equal to $\frac43$. For 40 years, the best known upper bound was 1.5, due to Wolsey (1980). Recently, Karlin, Klein, and Oveis Gharan (2022) showed that the max entropy algorithm for the TSP gives an improved bound of $1.5 - 10^{-36}$. In this paper, we show that the approximation ratio of the max entropy algorithm is at least 1.375, even for graphic TSP. Thus the max entropy algorithm does not appear to be the algorithm that will ultimately resolve the four-thirds conjecture in the affirmative, should that be possible.

This paper revisits the 2-approximation algorithm for $k$-MST presented by Garg in light of a recent paper of Paul et al.. In the $k$-MST problem, the goal is to return a tree spanning $k$ vertices of minimum total edge cost. Paul et al. extend Garg's primal-dual subroutine to improve the approximation ratios for the budgeted prize-collecting traveling salesman and minimum spanning tree problems. We follow their algorithm and analysis to provide a cleaner version of Garg's result. Additionally, we introduce the novel concept of a kernel which allows an easier visualization of the stages of the algorithm and a clearer understanding of the pruning phase. Other notable updates include presenting a linear programming formulation of the $k$-MST problem, including pseudocode, replacing the coloring scheme used by Garg with the simpler concept of neutral sets, and providing an explicit potential function., Comment: Proceedings of SIAM Symposium on Simplicity in Algorithms (SOSA) 2023

A long-standing conjecture for the traveling salesman problem (TSP) states that the integrality gap of the standard linear programming relaxation of the TSP is at most 4/3. Despite significant efforts, the conjecture remains open. We consider the half-integral case, in which the LP has solution values in $\{0, 1/2, 1\}$. Such instances have been conjectured to be the most difficult instances for the overall four-thirds conjecture. Karlin, Klein, and Oveis Gharan, in a breakthrough result, were able to show that in the half-integral case, the integrality gap is at most 1.49993. This result led to the first significant progress on the overall conjecture in decades; the same authors showed the integrality gap is at most $1.5- 10^{-36}$ in the non-half-integral case. For the half-integral case, the current best-known ratio is 1.4983, a result by Gupta et al. With the improvements on the 3/2 bound remaining very incremental even in the half-integral case, we turn the question around and look for a large class of half-integral instances for which we can prove that the 4/3 conjecture is correct. The previous works on the half-integral case perform induction on a hierarchy of critical tight sets in the support graph of the LP solution, in which some of the sets correspond to "cycle cuts" and the others to "degree cuts". We show that if all the sets in the hierarchy correspond to cycle cuts, then we can find a distribution of tours whose expected cost is at most 4/3 times the value of the half-integral LP solution; sampling from the distribution gives us a randomized 4/3-approximation algorithm. We note that the known bad cases for the integrality gap have a gap of 4/3 and have a half-integral LP solution in which all the critical tight sets in the hierarchy are cycle cuts; thus our result is tight., Comment: Comments, questions, and suggestions are welcome!

The Simplex algorithm for solving linear programs-one of Computing in Science & Engineering's top 10 most influential algorithms of the 20th century-is an important topic in many algorithms courses. While the Simplex algorithm relies on intuitive geometric ideas, the computationally-involved mechanics of the algorithm can obfuscate a geometric understanding. In this paper, we present gilp, an easy-to-use Simplex algorithm visualization tool designed to explicitly connect the mechanical steps of the algorithm with their geometric interpretation. We provide an extensive library with example visualizations, and our tool allows an instructor to quickly produce custom interactive HTML files for students to experiment with the algorithm (without requiring students to install anything!). The tool can also be used for interactive assignments in Jupyter notebooks, and has been incorporated into a forthcoming Data Science and Decision Making interactive textbook. In this paper, we first describe how the tool fits into the existing literature on algorithm visualizations: how it was designed to facilitate student engagement and instructor adoption, and how it substantially extends existing algorithm visualization tools for Simplex. We then describe the development and usage of the tool, and report feedback from its use in a course with roughly 100 students. Student feedback was overwhelmingly positive, with students finding the tool easy to use: it effectively helped them link the algebraic and geometrical views of the Simplex algorithm and understand its nuances. Finally, gilp is open-source, includes an extension to visualizing linear programming-based branch and bound, and is readily amenable to further extensions., Comment: ACM SIGCSE 2023 Manuscript, 12 pages, 6 figures

The symmetric circulant TSP is a special case of the traveling salesman problem in which edge costs are symmetric and obey circulant symmetry. Despite the substantial symmetry of the input, remarkably little is known about the symmetric circulant TSP, and the complexity of the problem has been an often-cited open question. Considerable effort has been made to understand the case in which only edges of two lengths are allowed to have finite cost: the two-stripe symmetric circulant TSP. In this paper, we resolve the complexity of the two-stripe symmetric circulant TSP. To do so, we reduce two-stripe symmetric circulant TSP to the problem of finding certain minimum-cost Hamiltonian paths on cylindrical graphs. We then solve this Hamiltonian path problem. Our results show that the two-stripe symmetric circulant TSP is in P. Note that a two-stripe symmetric circulant TSP instance consists of a constant number of inputs (including $n$, the number of cities), so that a polynomial-time algorithm for the decision problem must run in time polylogarithmic in $n$, and a polynomial-time algorithm for the optimization problem cannot output the tour. We address this latter difficulty by showing that the optimal tour must fall into one of two parameterized classes of tours, and that we can output the class and the parameters in polynomial time. Thus we make a substantial contribution to the set of polynomial-time solvable special cases of the TSP, and take an important step towards resolving the complexity of the general symmetric circulant TSP., Comment: 72 pages, 26 figures. A preliminary version appeared in IPCO 2022

This paper considers the interplay between semidefinite programming, matrix rank, and graph coloring. Karger, Motwani, and Sudan \cite{KMS98} give a vector program for which a coloring of the graph can be encoded as a semidefinite matrix of low rank. By complementary slackness conditions of semidefinite programming, if an optimal dual solution has sufficiently high rank, any optimal primal solution must have low rank. We attempt to characterize graphs for which we can show that the corresponding dual optimal solution must have sufficiently high rank. In the case of the original Karger, Motwani, and Sudan vector program, we show that any graph which is a $k$-tree has sufficiently high dual rank, and we can extract the coloring from the corresponding low-rank primal solution. We can also show that if the graph is not uniquely colorable, then no sufficiently high rank dual optimal solution can exist. This allows us to completely characterize the planar graphs for which dual optimal solutions have sufficiently high dual rank. We then modify the semidefinite program to have an objective function with costs, and explore when we can create a cost function whose optimal dual solution has sufficiently high rank. We show that it is always possible to construct such a cost function given the graph coloring. The construction of the cost function gives rise to a heuristic for graph coloring which we show works well in the case of planar graphs. Our research was motivated by the Colin de Verdi\`ere graph invariant \cite{CDV90}(and a corresponding conjecture of Colin de Verdi\`ere), in which matrices that have some similarities to the dual feasible matrices must have high rank in the case that graphs are of a certain type. We explore the connection between the conjecture and the rank of the dual solutions., Comment: 21 pages, 4 figures

We produce the first set of radiation hydrodynamics simulations of binary AGNs at parsec-scale separation in scale-model simulations. We use SPH for hydrodynamics, and raytracing to calculate optical depths and radiation pressure from the two AGNs. We confirm that, without radiation pressure, the sign of gravitational torque is sensitive to the binary parameters, although in one of our two orbital configurations the binary should coalesce in a time-scale of $<10^9$ yr. However, radiation pressure quickly destroys the 'minitori' around each SMBH, drastically reducing gravitational torques and accretion, and greatly increasing the coalescence time-scale. Our simulations suggest a new 'minitorus' duty cycle with a time-scale of ~10 binary periods (~$10^6$ yr when scaling our models to a total binary mass of $2\times10^7\,M_\odot$). The growth and blow-out phases of the 'minitori' are of similar time-scales, and thus we expect about half of observed binary SMBHs to be active, in at least one component. The 'minitorus' structure provides asymmetries that could be observed by infrared interferometry., Comment: 12 pages, 8 Figures, 3 Tables; Accepted for publication in MNRAS

We study the problem of finding flows in undirected graphs so as to minimize the weighted $p$-norm of the flow for any $p > 1$. When $p=2$, the problem is that of finding an electrical flow, and its dual is equivalent to solving a Laplacian linear system. The case $p = \infty$ corresponds to finding a min-congestion flow, which is equivalent to max-flows. A typical algorithmic construction for such problems considers vertex potentials corresponding to the flow conservation constraints, and has two simple types of update steps: cycle toggling, which modifies the flow along a cycle, and cut toggling, which modifies all potentials on one side of a cut. Both types of steps are typically performed relative to a spanning tree $T$; then the cycle is a fundamental cycle of $T$, and the cut is a fundamental cut of $T$. In this paper, we show that these simple steps can be used to give a novel efficient implementation for the $p = 2$ case and to find near-optimal $p$-norm flows in a low number of iterations for all values of $p > 1$. Compared to known faster algorithms for these problems, our algorithms are simpler, more combinatorial, and also expose several underlying connections between these algorithms and dynamic graph data structures that have not been formalized previously., Comment: arXiv admin note: text overlap with arXiv:2010.16316

Photoactivated localization microscopy (PALM) is a powerful imaging technique for characterization of protein organization in biological cells. Due to the stochastic blinking of fluorescent probes, and camera discretization effects, each protein gives rise to a cluster of artificial observations. These blinking artifacts are an obstacle for quantitative analysis of PALM data, and tools for their correction are in high demand. We develop the Independent Blinking Cluster point process (IBCpp) family of models, which is suited for modeling of data from single-molecule localization microscopy modalities, and we present results on the mark correlation function. We then construct the PALM-IBCpp - a semiparametric IBCpp tailored for PALM data, and we describe a procedure for estimation of parameters, which can be used without parametric assumptions on the spatial organization of proteins. Our model is validated on nuclear pore complex reference data, where the ground truth was accurately recovered, and we demonstrate how the estimated blinking parameters can be used to perform a blinking corrected test for protein clustering in a cell expressing the adaptor protein LAT. Finally, we consider simulations with varying degrees of blinking and protein clustering to shed light on the expected performance in a range of realistic settings., Comment: 36 pages, 8 figures

Facet-defining inequalities of the symmetric Traveling Salesman Problem (TSP) polytope play a prominent role in both polyhedral TSP research and state-of-the-art TSP solvers. In this paper, we introduce a new class of facet-defining inequalities, the \emph{circlet inequalities}. These inequalities were first conjectured in Gutekunst and Williamson \cite{Gut19b} when studying Circulant TSP, and they provide a bridge between polyhedral TSP research and number-theoretic investigations of Hamiltonian cycles stemming from a conjecture due to Marco Buratti in 2017. The circlet inequalities exhibit circulant symmetry by placing the same weight on all edges of a given length; our main proof exploits this symmetry to prove the validity of the circlet inequalities. We then show that the circlet inequalities are facet-defining and compute their strength following Goemans \cite{Goe95b}; they achieve the same worst-case strength as the similarly circulant crown inequalities of Naddef and Rinaldi \cite{Nad92}, but are generally stronger.

We present a series of chemodynamical simulations of Magellanic-like systems consisting of two interacting, equal-mass dwarf galaxies orbiting a massive host galaxy, including feedback and star formation, tides, and ram pressure. We study the star formation and chemical enrichment history of the dwarfs, and the production of a Magellanic Stream analogue. The dwarfs interact with each other through tidal forces, distorting their morphologies and triggering star formation. A stream is naturally produced as outflows, induced by feedback and interactions, are stretched by tidal forces. Counter to some recent simulations, we find that the leading arm persists even in the presence of ram pressure from the host galaxy. Interactions between the dwarfs and the host galaxies produce multiple kinematic components in the stream, as observed. A combination of ongoing star-formation and entrained low-metallicity gas causes the stream to have a complex chemical structure, with an average metallicity that is generally lower than that of the dwarfs., Comment: Accepted to ApJ