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arXiv cs.CC bot

Holant* Dichotomy on Domain Size 3: A Geometric Perspective

Jin-Yi Cai, Jin Soo Ihm
arxiv.org/abs/2504.14074 arxiv.org/pdf/2504.14074 arxiv.org/html/2504.14074

arXiv:2504.14074v1 Announce Type: new
Abstract: Holant problems are a general framework to study the computational complexity of counting problems. It is a more expressive framework than counting constraint satisfaction problems (CSP) which are in turn more expressive than counting graph homomorphisms (GH). In this paper, we prove the first complexity dichotomy of $\mathrm{Holant}_3(\mathcal{F})$ where $\mathcal{F}$ is an arbitrary set of symmetric, real valued constraint functions on domain size $3$. We give an explicit tractability criterion and prove that, if $\mathcal{F}$ satisfies this criterion then $\mathrm{Holant}_3(\mathcal{F})$ is polynomial time computable, and otherwise it is \#P-hard, with no intermediate cases. We show that the geometry of the tensor decomposition of the constraint functions plays a central role in the formulation as well as the structural internal logic of the dichotomy.

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arXiv.orgHolant* Dichotomy on Domain Size 3: A Geometric PerspectiveHolant problems are a general framework to study the computational complexity of counting problems. It is a more expressive framework than counting constraint satisfaction problems (CSP) which are in turn more expressive than counting graph homomorphisms (GH). In this paper, we prove the first complexity dichotomy of $\mathrm{Holant}_3(\mathcal{F})$ where $\mathcal{F}$ is an arbitrary set of symmetric, real valued constraint functions on domain size $3$. We give an explicit tractability criterion and prove that, if $\mathcal{F}$ satisfies this criterion then $\mathrm{Holant}_3(\mathcal{F})$ is polynomial time computable, and otherwise it is \#P-hard, with no intermediate cases. We show that the geometry of the tensor decomposition of the constraint functions plays a central role in the formulation as well as the structural internal logic of the dichotomy.
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Tau {&}

@OrionKidder

Viiuan:

i did already have my eyes on LibreOffice (and OpenOffice), and i don't need to use photoshop as i use clip studio paint… which isn't developed natively for linux, but i'm sure i would be able to get it to work somehow


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arXiv math.CO bot

A sharp upper bound for the number of connected sets in any grid graph

Hongxia Ma, Xian'an Jin, Weiling Yang, Meiqiao Zhang
arxiv.org/abs/2504.02309 arxiv.org/pdf/2504.02309 arxiv.org/html/2504.02309

arXiv:2504.02309v1 Announce Type: new
Abstract: A connected set in a graph is a subset of vertices whose induced subgraph is connected. Although counting the number of connected sets in a graph is generally a \#P-complete problem, it remains an active area of research. In 2020, Vince posed the problem of finding a formula for the number of connected sets in the $(n\times n)$-grid graph. In this paper, we establish a sharp upper bound for the number of connected sets in any grid graph by using multistep recurrence formulas, which further derives enumeration formulas for the numbers of connected sets in $(3\times n)$- and $(4\times n)$-grid graphs, thus solving a special case of the general problem posed by Vince. In the process, we also determine the number of connected sets of $K_{m}\times P_{n}$ by employing the transfer matrix method, where $K_{m}\times P_{n}$ is the Cartesian product of the complete graph of order $m$ and the path of order $n$.

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arXiv.orgA sharp upper bound for the number of connected sets in any grid graphA connected set in a graph is a subset of vertices whose induced subgraph is connected. Although counting the number of connected sets in a graph is generally a \#P-complete problem, it remains an active area of research. In 2020, Vince posed the problem of finding a formula for the number of connected sets in the $(n\times n)$-grid graph. In this paper, we establish a sharp upper bound for the number of connected sets in any grid graph by using multistep recurrence formulas, which further derives enumeration formulas for the numbers of connected sets in $(3\times n)$- and $(4\times n)$-grid graphs, thus solving a special case of the general problem posed by Vince. In the process, we also determine the number of connected sets of $K_{m}\times P_{n}$ by employing the transfer matrix method, where $K_{m}\times P_{n}$ is the Cartesian product of the complete graph of order $m$ and the path of order $n$.
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Bones In Jars

My Little Emos!

A project I had in the works for a good like while, finally fully realized in all its silly glory. I was given this set of 3 little ponies & wanted to give them a complete do-over inspired by a few of my favorite bands.

A lot of tlc went into this one, so be sure to check for all the little details !

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#spookyspotlight #bonesinjars #wisconsinsmallbusiness #wisconsinartist #wisconsinartshop
#mlp #mylittlepony #emo #paramore #mcr #mychemicalromance #panicatthedisco #p!atd

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arXiv cs.DS bot

Network Unreliability in Almost-Linear Time

Ruoxu Cen, Jason Li, Debmalya Panigrahi
arxiv.org/abs/2503.23526 arxiv.org/pdf/2503.23526 arxiv.org/html/2503.23526

arXiv:2503.23526v1 Announce Type: new
Abstract: The network unreliability problem asks for the probability that a given undirected graph gets disconnected when every edge independently fails with a given probability $p$. Valiant (1979) showed that this problem is \#P-hard; therefore, the best we can hope for are approximation algorithms. In a classic result, Karger (1995) obtained the first FPTAS for this problem by leveraging the fact that when a graph disconnects, it almost always does so at a near-minimum cut, and there are only a small (polynomial) number of near-minimum cuts. Since then, a series of results have obtained progressively faster algorithms to the current bound of $m^{1+o(1)} + \tilde{O}(n^{3/2})$ (Cen, He, Li, and Panigrahi, 2024). In this paper, we obtain an $m^{1+o(1)}$-time algorithm for the network unreliability problem. This is essentially optimal, since we need $O(m)$ time to read the input graph. Our main new ingredient is relating network unreliability to an {\em ideal} tree packing of spanning trees (Thorup, 2001).

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arXiv.orgNetwork Unreliability in Almost-Linear TimeThe network unreliability problem asks for the probability that a given undirected graph gets disconnected when every edge independently fails with a given probability $p$. Valiant (1979) showed that this problem is \#P-hard; therefore, the best we can hope for are approximation algorithms. In a classic result, Karger (1995) obtained the first FPTAS for this problem by leveraging the fact that when a graph disconnects, it almost always does so at a near-minimum cut, and there are only a small (polynomial) number of near-minimum cuts. Since then, a series of results have obtained progressively faster algorithms to the current bound of $m^{1+o(1)} + \tilde{O}(n^{3/2})$ (Cen, He, Li, and Panigrahi, 2024). In this paper, we obtain an $m^{1+o(1)}$-time algorithm for the network unreliability problem. This is essentially optimal, since we need $O(m)$ time to read the input graph. Our main new ingredient is relating network unreliability to an {\em ideal} tree packing of spanning trees (Thorup, 2001).
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TCS blog aggregator

Network Unreliability in Almost-Linear Time arxiv.org/abs/2503.23526v1

Authors: Ruoxu Cen, Jason Li, Debmalya PanigrahiThe network unreliability problem asks for the probability that a given
undirected graph gets disconnected when every edge independently fails with a
given probability $p$. Valiant (1979) showed that this problem is -hard;
therefore, the best we can hope for are approximation algorithms. In a classic
result, Karger (1995) obtained the first FPTAS

arXiv logo
arXiv.orgNetwork Unreliability in Almost-Linear TimeThe network unreliability problem asks for the probability that a given undirected graph gets disconnected when every edge independently fails with a given probability $p$. Valiant (1979) showed that this problem is \#P-hard; therefore, the best we can hope for are approximation algorithms. In a classic result, Karger (1995) obtained the first FPTAS for this problem by leveraging the fact that when a graph disconnects, it almost always does so at a near-minimum cut, and there are only a small (polynomial) number of near-minimum cuts. Since then, a series of results have obtained progressively faster algorithms to the current bound of $m^{1+o(1)} + \tilde{O}(n^{3/2})$ (Cen, He, Li, and Panigrahi, 2024). In this paper, we obtain an $m^{1+o(1)}$-time algorithm for the network unreliability problem. This is essentially optimal, since we need $O(m)$ time to read the input graph. Our main new ingredient is relating network unreliability to an {\em ideal} tree packing of spanning trees (Thorup, 2001).
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