In a classical covering problem, we are given a set of requests that we need
to satisfy (fully or partially), by buying a subset of items at minimum cost.
For example, in the k-MST problem we want to find the cheapest tree spanning at
least k nodes of an edge-weighted graph. Here nodes and edges represent
requests and items, respectively.
In this paper, we initiate the study of a new family of multi-layer covering
problems. Each such problem consists of a collection of h distinct instances of
a standard covering problem (layers), with the constraint that all layers share
the same set of requests. We identify two main subfamilies of these problems: -
in a union multi-layer problem, a request is satisfied if it is satisfied in at
least one layer; - in an intersection multi-layer problem, a request is
satisfied if it is satisfied in all layers. To see some natural applications,
consider both generalizations of k-MST. Union k-MST can model a problem where
we are asked to connect a set of users to at least one of two communication
networks, e.g., a wireless and a wired network. On the other hand, intersection
k-MST can formalize the problem of connecting a subset of users to both
electricity and water.
We present a number of hardness and approximation results for union and
intersection versions of several standard optimization problems: MST, Steiner
tree, set cover, facility location, TSP, and their partial covering variants.
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Wed, 02/22/2012 - 05:43 — hanse1314520really weren't interested in each other's poems at all. But remember, someone like Christy? Christy had this great reputation for poetry, and we all looked up to her for it. Even you, Ruth, you didn't dare boss Christy around. All because we thought she was great at poetry. But we didn't know a thing about poetry. We didn't care about it. It's strange."But Ruth didn't get my point--or maybe she was deliberately avoiding it. Maybe she was determined to remember us all as more sophisticated than we were. Or maybe she could sense where my talk was leading, and didn't want us to go that way. Anyway, she let out a long sigh and said: "We all thought black christian louboutin Christy's poems were so good. But I wonder how they'd look to us now. I wish we had some here, I'd love to see what we'd think." Then she laughed and said: "I have still got some poems by Peter B. But that was much later, when we were in Senior 4. I must have fancied him. I can't think why else I'd have bought his poems. They're just hysterically daft. Takes himself so seriously. But chloe leather bags Christy, she was good, I remember she christian louboutin evening was. It's funny, she went right off poems when she started her painting. And she was nowhere near as good at that.But let me get back to Tommy. What Ruth said that time in our dorm after lights-out, about how Tommy had brought all his problems on himself, probably summed up what most chloe ethel satchel people at Hailsham chloe marcie thought at that time. chloe leather bags But it was when she said what she did that it occurred to me, as I lay there, that this whole notion of his deliberately not trying was one that had been doing the rounds from as far back as the Juniors. And it came home to me, with a kind of chill, that Tommy had been chloe paraty going through what he'd been going through not black christian louboutin just for weeks or months, but for years.Tommy and I talked about all this not so long ago, and his own account of how his troubles began confirmed what I was thinking that night. According to him, it had all started one afternoon in one of Miss Geraldine's art classes. Until that day, Tommy told me, he'd always quite enjoyed painting. But then that day in Miss blue christian louboutin Geraldine's class, Tommy had done this particular watercolour--of an elephant standing in some tall grass--and that was what started it all off. He'd done it, he claimed, as a kind of joke. I quizzed him a lot on this point and I suspect the truth was that it was like a lot of things at that age: you don't have any clear reason, you just do it. You do it because you think it might get a laugh, or because you want to see if it'll cause a stir. And when you're asked to explain it afterwards, it doesn't seem to make any sense. We've all done things like that. Tommy didn't quite put it this way, but I'm sure that's how it happened.Anyway, he did his elephant, which was exactly the sort of picture a kid three years younger might have done. It took him no more than twenty minutes and it got a laugh, sure enough, though not quite the sort he'd expected. Even so, it might not have led to anything--and this is a big irony, I suppose--if Miss Geraldine hadn't been taking the class that day.Miss Geraldine was everyone's favourite guardian when we were that age. She was gentle, soft-spoken, and always comforted you when you needed it, even when you'd done something bad, or been told off by another guardian. If she ever had to tell you off herself, then for days afterwards she'd give you lots of extra attention, like she owed you something. It was unlucky for Tommy that it was Miss Geraldine taking art that day and not, say, Mr. Robert or Miss Emily herself--the head guardian--who often took art. Had it been either of those two, Tommy would have got a bit of a telling off, he could have done his smirk, and the worst the others would have thought was that it was a feeble joke. He might even have had some students think him a right clown. But Miss Geraldine being Miss Geraldine, it didn't go that way. Instead, she did her best to look at the picture with kindness and understanding. And probably guessing Tommy was in danger of getting stick from the others, she went too far the other way, actually finding things to praise, pointing them out to the class. That was how the resentment started.After we left the room," Tommy remembered, "that's when I first heard them talking. And they didn't care I could hear.My guess is that from some time before he did that elephant, Tommy had had the feeling he wasn't keeping up--that his painting in particular was like that of students much younger than him--and he'd been covering up the best he could by doing deliberately childish pictures. But after the elephant painting, the whole thing had been brought into the open, and now everyone was watching to see what he did next. It seems he did make an effort for a while, but he'd no sooner have started on something, there'd be sneers and giggles all around him. In fact, the harder he tried, the more laughable his efforts turned out. So before long Tommy had gone back to his original defence, producing work that seemed deliberately childish, work that said he couldn't care less. From there, the thing had got deeper and deeper.For a while he'd only had to suffer during art lessons--though that was often enough, because we did a lot of art in the Juniors. But then it grew bigger. He got left out of games, boys refused to sit next to him at dinner, or pretended not to hear if he said anything in his dorm after lights-out. At first it wasn't so relentless. Months could go by without incident, he'd think the whole thing was behind him, then something he did--or one of his enemies, like Arthur H.--would get it all going again.I'm not sure when the big temper tantrums started. My own memory of it is that Tommy was always known for his temper, even in the Infants, but he claimed to me they only began after the teasing got bad. Anyway, it was those temper tantrums that really got people going, escalating everything, and around the time I'm talking about--the summer of our Senior 2, when we were thirteen--that was when the persecution reached its peak.Then it all stopped, not overnight, but rapidly enough. red christian louboutin I was, as I say, watching the situation closely around then, so I saw the signs before most of the others. It started with a period--it might have been a month, maybe longer--when the pranks went on pretty steadily, but Tommy failed to lose his temper. Sometimes I could see he was close to it, but he somehow controlled himself; other times, he'd quietly shrug, or react like he hadn't noticed a thing. At first these responses caused disappointment; maybe blue christian louboutin people were resentful, even, like he'd let them down. Then gradually, people got bored and the pranks became more halfhearted, until one day it struck me there hadn't been any for over a week.This wouldn't necessarily have been so significant by itself, but I'd spotted other changes. Little things, like Alexander men christian louboutin sneakers J. and Peter N. walking across the courtyard with him towards the fields, the three of them chatting quite naturally; a subtle but clear difference in people's voices when his name got mentioned. Then once, towards the end of an afternoon break, a group of us were sitting on the grass quite close to the South Playing Field where the boys, as usual, were playing their football. I was joining in our conversation, but keeping an eye on Tommy, who I noticed was right at the heart of the game. At one point he got tripped, and picking himself up, placed the ball on the ground to take the free kick himself. As the boys spread out in anticipation, I saw Arthur H.--one of his biggest tormentors--a few yards behind Tommy's back, begin mimicking him, doing a daft version of the way Tommy was standing over the ball, hands on hips. I watched carefully, but none red christian louboutin of the others took up Arthur's cue. They must all have seen, because all eyes were looking towards Tommy, waiting for his kick, and Arthur was right behind him--but no one was interested. Tommy christian louboutin wedding floated the ball across christian louboutin sandals the grass, the game went on, and Arthur H. didn't try anything else.
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