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Suppose given a complete weighted graph $G=(V,E)$, with positive weight. Are there an algorithm that partition $G$ into two clusters $C_1,C_2$ such that sum of heaviest edges in $C_1,C_2$ minimized?

Note that, heaviest edge in a cluster is an edge with maximum weight. Also, i rename a partition to cluster, additionally, $G$ can partitioned to two clusters $C_1,C_2$.

If there be an approximation algorithm that running time be less than $o(n^3)$ it's helpful. I find some related paper, but all paper work on metric space, but $G$ is not a metric graph.

In essence, according this post at the first, my graph is metric, but after changing the weight of some of edges to $\infty$ then $G$ isn't remain metric.

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    $\begingroup$ Can you define "heaviest edges" in $C_1, C_2$ ? does this mean the sum of the one single largest edge in C1 and the one single largest edge in C2? Can you define what a 'cluster' is? Can $G$ be partitioned into any two $C_1$ and $C_2$? (So one could be a single edge and the other one be the rest of the graph?) Or do these partitions need to satisfy some other property to be considered a valid 'cluster' $\endgroup$
    – JimN
    Sep 10 at 7:47
  • $\begingroup$ @JimN I edit my post. $\endgroup$
    – Jut
    Sep 10 at 10:07
  • $\begingroup$ @NealYoung Thanks. Can you explain in more detail about your idea in answer section? Really, i can't correctly understand your idea. $\endgroup$
    – Jut
    Sep 10 at 16:23
  • $\begingroup$ Note: OP, you have four posts, apparently about this question: (i), (ii), (iii), and (iv). These posts are all somewhat unclear, and cross-posting is discouraged. I suggest you delete the other three posts. $\endgroup$
    – Neal Young
    Sep 14 at 15:25
  • $\begingroup$ See also OP's post (v). $\endgroup$
    – Neal Young
    Sep 14 at 15:32
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Theorem 1. The problem admits a 2-approximation algorithm that runs in $O((m+n)\log n)$ time, given a graph $G=(V,E)$ with $m$ edges and $n$ vertices.

[Caveat: The current post doesn't specify the objective-function value if one or both of the clusters contains no edges. I assume that the objective-function value only sums the maximum-weight edges within clusters that do contain edges. (If, say, the graph is bipartite then the optimal value is zero.)]

Proof. Here's the algorithm:

  1. let $e_1, e_2, \ldots, e_m$ denote the edges sorted by decreasing weight

  2. let $G_t=(V, E_t)$ where $E_t=\{e_1,e_2,\ldots, e_t\}$ denote the graph with only the heaviest $t$ edges

  3. let $t'\in\{1,\ldots, m\}$ be maximum such that $G_{t'}$ is bipartite (find $t$ using binary search)

  4. let $(C_1, C_2)$ be a bipartition of $G_{t'}$ (such that $E_{t'}\subseteq C_1\times C_2)$)

  5. return $(C_1, C_2)$

The algorithm can be implemented to run in $O((m+n)\log n)$ time, because the binary search requires $O(\log m) = O(\log n)$ rounds, and each round requires checking whether a given $G_t$ is bipartite (and finding its bipartition, if it is), which can be done in $O(n+m)$ time using, say, depth-first search (see e.g. here).

Consider any execution of the algorithm. Let $t'$ and $(C_1, C_2)$ be as computed by the algorithm. To finish we show that $(C_1, C_2)$ has objective-function value at most twice the optimum.

If the given graph $G$ is bipartite, then the algorithm returns a bipartition of $G$, which (by the caveat above) is an optimal solution. So assume that $G$ is not bipartite. So $t' < m$.

Each edge within $C_1$ or $C_2$ is not in $E_{t'}$, so has weight at most $w(e_{t'+1})$. So the algorithm's solution achieves objective-function value at most $2 w(e_{t'+1})$.

Now consider any optimal solution $(C^*_1, C^*_2)$. Because $G$ is not bipartite, there is at least one edge within one of the clusters $C^*_1$ or $C^*_2$. Let $e_{t^*}$ be the edge with maximum index (and hence maximum weight) within either cluster. The value of the optimal solution is at least $w(e_{t^*})$ (using here that the edge weights are non-negative).

Removing $e_{t^*}$ and all cheaper edges yields the graph $G_{t^*-1}=(V, E_{t^*-1})$. By the choice of $e_{t^*}$ this graph is bipartite with bipartition $(C^*_1, C^*_2)$ (as all edges within each cluster are not in $E_{t^*-1}$). Hence $t^*-1 \le t'$, and $w(e_{t^*}) \ge w(e_{t'+1})$. Hence the optimal solution value is at least $w(e_{t'+1})$. $~~~~~\Box$

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  • $\begingroup$ Thank you, also i have a greedy algorithm, but when i want find out the approximation factor i get stuck, can i add my algorithm to question and then help me to find the approximation factor of my idea? $\endgroup$
    – Jut
    Sep 10 at 21:00
  • $\begingroup$ Each cluster contains edge, because my input is as follow: exactly $n$ edges have weight $\infty$. $\endgroup$
    – Jut
    Sep 10 at 21:02
  • $\begingroup$ Excuse me, i read a paper that solve my question in with an exact algorithm in $O(n^3)$. But after reading many paper, i think, i can't do better than $o(n^3)$, so i search to find an approximation algorithm that work in $o(n^3)$ and has constant approximation factor. Are you have any comment about an exact algorithm for above problem that work in $o(n^3)$? $\endgroup$
    – Jut
    Sep 10 at 21:05
  • $\begingroup$ Specially, the first answer by @Gamow is that algorithm that unfortunately work in $O(n^3)$. $\endgroup$
    – Jut
    Sep 10 at 21:06
  • $\begingroup$ I don't have an idea for a faster exact algorithm, no. ^^^Rather than edit your post to change your question (which would invalidate the current answers), it would be better to create a new post with the new question. Also, please accept one of the current answers if it answers your current question, thanks. $\endgroup$
    – Neal Young
    Sep 10 at 22:26
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The problem even seems to be solvable in polynomial time. Let us call $C_1$ the black cluster and $C_1$ the white cluster. We test for every two edges $e_1,e_2\in E$ whether there exists a bipartition so that $e_1$ is the heaviest edge in $C_1$ and $e_2$ is the heaviest edge in $C_2$. In the end, we ouput the bipartition that minimizes the sum of the two heaviest edges.

Now the test for two edges $e_1,e_2\in E$ proceeds as follows. We use a 2-SAT formulation. There is a variable $x(v)$ for every vertex; the value TRUE corresponds to $v$ being black and the value FALSE corresponds to $v$ being white.

  • For any edge $e=\{u,v\}$ with $w(e)>w(e_1)$, we require that $u$ and $v$ are in different clusters. Hence we create the clauses $(u\lor v)(\lnot u\lor\lnot v)$.
  • For any edge $e=\{u,v\}$ with $w(e_1)\ge w(e)>w(e_2)$, the two endpoints must either both be in $C_1$, or one is in $C_1$ and the other one in $C_2$. This corresponds to the clause $(u\lor v)$.
  • Edges $e=\{u,v\}$ with $w(e_2)\ge w(e)$ are ignored for this particular test.
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  • $\begingroup$ Thanks, what is running time of your algorithm? $\endgroup$
    – Jut
    Sep 10 at 16:20
  • $\begingroup$ There is a paper that solution is identical with your solution but the running time is $O(n^3)$, so it's not good for my question, because i trying to find a solution that work in $o(n^3)$. $\endgroup$
    – Jut
    Sep 10 at 16:33
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    $\begingroup$ So, why didn't you provide us with full information on the known results around your problem (in the statement of your question)??? And PLEASE provide a link to that other paper with the $O(n^3)$ algorithm. $\endgroup$
    – Gamow
    Sep 11 at 12:03
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    $\begingroup$ FWIW I think this algorithm can be implemented (by enumerating all first edges and using binary search over the second) to run in time $O(m(m+n)\log n)$, which is $o(n^3)$ for sufficiently sparse graphs (as long as $m$ is $o(n^{3/2} /\sqrt{\log n})$). $\endgroup$
    – Neal Young
    Sep 14 at 15:23

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