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main.tex

@@ -179,11 +179,16 @@ Consider the Lagrangian dual,
 We have shown that the budget $B$ in normalized min-cut does not really matter as long as $B>b$. Note that $L(\lambda)$ and the normalized min-cut look similar to the principal sequence of partitions of a graph and the graph strength problem.
 
 \subsection{graph strength}
-Given a graph $G=(V,E)$ and a cost function $c:V\to \Z_+$, the strength $\sigma(G)$ is defined as $\sigma(G)=\min_{\Pi}\frac{c(\delta(\Pi))}{|\Pi|-1}$, where $\Pi$ is any partition of $V$, $|\Pi|$ is the number of parts in the partition and $\delta(\Pi)$ is the set of edges between parts.
+Assume that the graph $G$ is connected (otherwise add dummy edges). 
+Given a graph $G=(V,E)$ and a cost function $c:V\to \Z_+$, the strength $\sigma(G)$ is defined as $\sigma(G)=\min_{\Pi}\frac{c(\delta(\Pi))}{|\Pi|-1}$, where $\Pi$ is any partition of $V$, $|\Pi|$ is the number of parts in the partition and $\delta(\Pi)$ is the set of edges between parts. Note that an alternative formulation of strength (using graphic matroid rank) is $\sigma(G)=\min_{F\subset E} \frac{|E-F|}{r(E)-r(F)}$, which in general is the fractional optimum of matroid base packing.
 
-The principal sequence of partitions of $G$ is a piecewise linear concave curve $L(\lambda)=\min_\Pi c(\delta(\Pi))-\lambda |\Pi|$.
+The principal sequence of partitions of $G$ is a piecewise linear concave curve $L(\lambda)=\min_\Pi c(\delta(\Pi))-\lambda |\Pi|$. Cunningham used principal partition to computed graph strength\cite{cunningham_optimal_1985}. There is a list of good properties mentioned in \cite[Section 6]{chekuri_lp_2020}.
+\begin{itemize}
+    \item $L(\lambda)$ is piecewise linear concave since it is the lower envelope of some line arrangement.
+    \item Consider two adjacent breakpoints on $L$...
+\end{itemize}
 
-(there is a $\pm1$ difference...)
+(there is a $\pm1$ difference between principal partition and graph strength... but we dont care those $c\lambda$ terms since the difficult part is minimize $L(\lambda)$ for fixed $\lambda$)
 
 
 \section{Random Stuff}

ref.bib

@@ -33,13 +33,6 @@
 	pages = {37:1--37:20},
 }
 
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 @article{garg_faster_nodate,
 author = {Garg, Naveen and K\"{o}nemann, Jochen},
 title = {Faster and Simpler Algorithms for Multicommodity Flow and Other Fractional Packing Problems},
@@ -54,6 +47,41 @@ eprint = {https://doi.org/10.1137/S0097539704446232},
 abstract = { This paper considers the problem of designing fast, approximate, combinatorial algorithms for multicommodity flows and other fractional packing problems. We present new, faster, and much simpler algorithms for these problems. }
 }
 
+@article{cunningham_optimal_1985,
+	title = {Optimal attack and reinforcement of a network},
+	volume = {32},
+	issn = {0004-5411, 1557-735X},
+	url = {https://dl.acm.org/doi/10.1145/3828.3829},
+	doi = {10.1145/3828.3829},
+	abstract = {In a nonnegative edge-weighted network, the weight of an edge represents the effort required by an attacker to destroy the edge, and the attacker derives a benefit for each new component created by destroying edges.The attacker may want to minimize over subsetsof edgesthe difference between (or the ratio of) the effort incurred and the benefit received. This idea leads to the definition of the “strength” of the network, a measure of the resistanceof the network to such attacks. Efficient algorithms for the optimal attack problem, the problem of computing the strength, and the problem of finding a minimum cost “reinforcement” to achieve a desired strength are given. These problems are also solved for a different model, in which the attacker wants to separate vertices from a fixed central vertex.},
+	language = {en},
+	number = {3},
+	urldate = {2025-04-11},
+	journal = {Journal of the ACM},
+	author = {Cunningham, William H.},
+	month = jul,
+	year = {1985},
+	pages = {549--561},
+}
+
+
+@article{chekuri_lp_2020,
+	title = {{LP} {Relaxation} and {Tree} {Packing} for {Minimum} \$k\$-{Cut}},
+	volume = {34},
+	issn = {0895-4801, 1095-7146},
+	url = {https://epubs.siam.org/doi/10.1137/19M1299359},
+	doi = {10.1137/19M1299359},
+	abstract = {Karger used spanning tree packings [D. R. Karger, J. ACM, 47 (2000), pp. 46-76] to derive a near linear-time randomized algorithm for the global minimum cut problem as well as a bound on the number of approximate minimum cuts. This is a different approach from his well-known random contraction algorithm [D. R. Karger, Random Sampling in Graph Optimization Problems, Ph.D. thesis, Stanford University, Stanford, CA, 1995, D. R. Karger and C. Stein, J. ACM, 43 (1996), pp. 601--640]. Thorup developed a fast deterministic algorithm for the minimum k-cut problem via greedy recursive tree packings [M. Thorup, Minimum k-way cuts via deterministic greedy tree packing, in Proceedings of the Fortieth Annual ACM Symposium on Theory of Computing, ACM, 2008, pp. 159--166]. In this paper we revisit properties of an LP relaxation for k-Cut proposed by Naor and Rabani [Tree packing and approximating k-cuts, in Proceedings of the Twelfth Annual ACM-SIAM Symposium on Discrete Algorithms, Vol. 103, SIAM, Philadelphia, 2001, pp. 26--27], and analyzed in [C. Chekuri, S. Guha, and J. Naor, SIAM J. Discrete Math., 20 (2006), pp. 261--271]. We show that the dual of the LP yields a tree packing that, when combined with an upper bound on the integrality gap for the LP, easily and transparently extends Karger's analysis for mincut to the k-cut problem. In addition to the simplicity of the algorithm and its analysis, this allows us to improve the running time of Thorup's algorithm by a factor of n. We also improve the bound on the number of {\textbackslash}alpha -approximate k-cuts. Second, we give a simple proof that the integrality gap of the LP is 2(1  -  1/n). Third, we show that an optimum solution to the LP relaxation, for all values of k, is fully determined by the principal sequence of partitions of the input graph. This allows us to relate the LP relaxation to the Lagrangean relaxation approach of Barahona [Oper. Res. Lett., 26 (2000), pp. 99--105] and Ravi and Sinha [European J. Oper. Res., 186 (2008), pp. 77--90]; it also shows that the idealized recursive tree packing considered by Thorup gives an optimum dual solution to the LP.},
+	language = {en},
+	number = {2},
+	urldate = {2022-04-10},
+	journal = {SIAM Journal on Discrete Mathematics},
+	author = {Chekuri, Chandra and Quanrud, Kent and Xu, Chao},
+	month = jan,
+	year = {2020},
+	keywords = {Approximation, K-cut, Minimum cut, Tree packing},
+	pages = {1334--1353},
+}