Mixing
Find $k$-clique with max min weight.
$begingroup$
I have a problem with the following equivalent formulation in graph theory:
Take $G(V, E, W)$ a complete weighted graph, where $w_{ij} > 0$ is the weight of edge $e_{ij}$. For a given $k$, find the clique $Q$ with $k$ vertices, such that its minimum edge weight is maximized over the graph.
$$Q = text{argmax}_{I} min_{i,j in I} w_{i,j}\
text{such that } |I| = k$$
I have seen similar problems such as finding cliques where the sum of weights are maximized, but not a maxmin problem like above.
I have also found the following paper [1], which addresses a similar problem, but again different than what I have here.
Based on what I have seen in similar problems, I suspect that this problem is also NP-hard. I was wondering if this is the case, and whether anyone knows an approximation to this problem.
[1] T. F. Gonzalez, Clustering to minimize the maximum intercluster distance, Theoretical Computer Science, 38(2-3):293-306, 1985.
graph-theory optimization algorithms clustering
edited Jan 21 at 20:09
Alex Ravsky
42.4k32383
asked Jan 20 at 12:11
e-gole-gol
303
$endgroup$
add a comment |
$begingroup$
I have a problem with the following equivalent formulation in graph theory:
Take $G(V, E, W)$ a complete weighted graph, where $w_{ij} > 0$ is the weight of edge $e_{ij}$. For a given $k$, find the clique $Q$ with $k$ vertices, such that its minimum edge weight is maximized over the graph.
$$Q = text{argmax}_{I} min_{i,j in I} w_{i,j}\
text{such that } |I| = k$$
I have seen similar problems such as finding cliques where the sum of weights are maximized, but not a maxmin problem like above.
I have also found the following paper [1], which addresses a similar problem, but again different than what I have here.
Based on what I have seen in similar problems, I suspect that this problem is also NP-hard. I was wondering if this is the case, and whether anyone knows an approximation to this problem.
[1] T. F. Gonzalez, Clustering to minimize the maximum intercluster distance, Theoretical Computer Science, 38(2-3):293-306, 1985.
graph-theory optimization algorithms clustering
edited Jan 21 at 20:09
Alex Ravsky
42.4k32383
asked Jan 20 at 12:11
e-gole-gol
303
$endgroup$
add a comment |
$begingroup$
I have a problem with the following equivalent formulation in graph theory:
Take $G(V, E, W)$ a complete weighted graph, where $w_{ij} > 0$ is the weight of edge $e_{ij}$. For a given $k$, find the clique $Q$ with $k$ vertices, such that its minimum edge weight is maximized over the graph.
$$Q = text{argmax}_{I} min_{i,j in I} w_{i,j}\
text{such that } |I| = k$$
I have seen similar problems such as finding cliques where the sum of weights are maximized, but not a maxmin problem like above.
I have also found the following paper [1], which addresses a similar problem, but again different than what I have here.
Based on what I have seen in similar problems, I suspect that this problem is also NP-hard. I was wondering if this is the case, and whether anyone knows an approximation to this problem.
[1] T. F. Gonzalez, Clustering to minimize the maximum intercluster distance, Theoretical Computer Science, 38(2-3):293-306, 1985.
graph-theory optimization algorithms clustering
edited Jan 21 at 20:09
Alex Ravsky
42.4k32383
asked Jan 20 at 12:11
e-gole-gol
303
$endgroup$
I have a problem with the following equivalent formulation in graph theory:
Take $G(V, E, W)$ a complete weighted graph, where $w_{ij} > 0$ is the weight of edge $e_{ij}$. For a given $k$, find the clique $Q$ with $k$ vertices, such that its minimum edge weight is maximized over the graph.
$$Q = text{argmax}_{I} min_{i,j in I} w_{i,j}\
text{such that } |I| = k$$
I have seen similar problems such as finding cliques where the sum of weights are maximized, but not a maxmin problem like above.
I have also found the following paper [1], which addresses a similar problem, but again different than what I have here.
Based on what I have seen in similar problems, I suspect that this problem is also NP-hard. I was wondering if this is the case, and whether anyone knows an approximation to this problem.
[1] T. F. Gonzalez, Clustering to minimize the maximum intercluster distance, Theoretical Computer Science, 38(2-3):293-306, 1985.
graph-theory optimization algorithms clustering
graph-theory optimization algorithms clustering
edited Jan 21 at 20:09
Alex Ravsky
42.4k32383
asked Jan 20 at 12:11
e-gole-gol
303
edited Jan 21 at 20:09
Alex Ravsky
42.4k32383
asked Jan 20 at 12:11
e-gole-gol
303
edited Jan 21 at 20:09
Alex Ravsky
42.4k32383
edited Jan 21 at 20:09
Alex Ravsky
42.4k32383
edited Jan 21 at 20:09
Alex Ravsky
42.4k32383
42.4k32383
asked Jan 20 at 12:11
e-gole-gol
303
asked Jan 20 at 12:11
e-gole-gol
303
asked Jan 20 at 12:11
e-gole-gol
303
303
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
This problem is easy and can be solved by the following algorithm. Remove all the edges from the graph and order them with respect to its weight, from the biggest to the smallest. Then reinsert the edges of the graph consecutively with respect to the order (if several edges have the same weight then we reinsert them simultaneously). A first $k$-clique appeared in the graph will be the required $operatorname{argmax}$.
answered Jan 21 at 20:09
Alex RavskyAlex Ravsky
42.4k32383
$endgroup$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3080504%2ffind-k-clique-with-max-min-weight%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
This problem is easy and can be solved by the following algorithm. Remove all the edges from the graph and order them with respect to its weight, from the biggest to the smallest. Then reinsert the edges of the graph consecutively with respect to the order (if several edges have the same weight then we reinsert them simultaneously). A first $k$-clique appeared in the graph will be the required $operatorname{argmax}$.
answered Jan 21 at 20:09
Alex RavskyAlex Ravsky
42.4k32383
$endgroup$
add a comment |
$begingroup$
This problem is easy and can be solved by the following algorithm. Remove all the edges from the graph and order them with respect to its weight, from the biggest to the smallest. Then reinsert the edges of the graph consecutively with respect to the order (if several edges have the same weight then we reinsert them simultaneously). A first $k$-clique appeared in the graph will be the required $operatorname{argmax}$.
answered Jan 21 at 20:09
Alex RavskyAlex Ravsky
42.4k32383
$endgroup$
add a comment |
$begingroup$
This problem is easy and can be solved by the following algorithm. Remove all the edges from the graph and order them with respect to its weight, from the biggest to the smallest. Then reinsert the edges of the graph consecutively with respect to the order (if several edges have the same weight then we reinsert them simultaneously). A first $k$-clique appeared in the graph will be the required $operatorname{argmax}$.
answered Jan 21 at 20:09
Alex RavskyAlex Ravsky
42.4k32383
$endgroup$
This problem is easy and can be solved by the following algorithm. Remove all the edges from the graph and order them with respect to its weight, from the biggest to the smallest. Then reinsert the edges of the graph consecutively with respect to the order (if several edges have the same weight then we reinsert them simultaneously). A first $k$-clique appeared in the graph will be the required $operatorname{argmax}$.
answered Jan 21 at 20:09
Alex RavskyAlex Ravsky
42.4k32383
answered Jan 21 at 20:09
Alex RavskyAlex Ravsky
42.4k32383
answered Jan 21 at 20:09
Alex RavskyAlex Ravsky
42.4k32383
answered Jan 21 at 20:09
Alex RavskyAlex Ravsky
42.4k32383
42.4k32383
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3080504%2ffind-k-clique-with-max-min-weight%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
