〜ても構わない
it's fine if, I don't mind if
Verb て-form + も構わない
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ても構わない gives permission, like てもいい, but with more of a sense of "I don't mind": 窓を開けても構いません, "you're welcome to open the window".
構う means "to mind" or "care about", so the negative says there's nothing to mind. It's a little more formal than てもいい, which makes 構いません common at work and in shops: カードで払っても構いませんか, "is it all right if I pay by card?"
The negative version works too: 来なくても構わない, "it's fine if you don't come".
On its own, 構いません is a polite "that's fine", while 構わないで means "leave me alone" or "don't go to any trouble".
Examples
暑かったら、窓を開けても構いません。
If it's hot, feel free to open the window.
カードで払っても構いませんか。
Is it all right if I pay by card?
忙しかったら、来なくても構わないよ。
If you're busy, it's fine if you don't come.
この部屋のパソコンは誰が使っても構いません。
Anyone can use the computers in this room.
日本語でも英語でも構いません。
Either Japanese or English is fine.