GardenBanter.co.uk - Questions about weed/unwanted growth

I'd appreciate if anyone could identify this, it's an unwanted plant
that's growing on our property, from the property next door.

Effectively, it's a network of underground and barely-surface vines.
Diameters seem to range from reasonably slender (thinner than my
pinkie) to thicker around than my thumb (and I have thick thumbs :)).
They're sort of 'woody' looking, with a thin layer of 'bark', or outer
skin of some sort. When snipped in half, they're a very clean white on
the inside, with no rings of color. The bark/skin is similar to the
texture of a very young sapling tree (thin and easy to 'peel' off).
Very sturdy roots several inches into the ground, about every 4-5
inches along the length of the vines in/on the ground.

The vines will grow 'up' and climb around any items that they're given
the chance to cover. In that case, they will produce shiny lighish
green leaves (very small). They'll spiral around other plants (which
will kill said plants as several individual shrubs in a short hedge
have been killed by this wrapping).

The spiraling shoots seem to be a secondary product of the vines.
Really, what they do is just cross and recross the dirt
(non-driveway/concrete-covered) parts of our yard. Because they grow
just under the surface of the soil (1" or so of dirt over them), or,
_just_ along the surface, we didn't really notice they had effectively
extended an interconnected 'net' of vines over our entire yard, till
we tried to pull up 'some', and suddenly determined they were all
over.

I'm a disinterested gardener at best. I like bulbs, because they're
colorful and still grow with zero upkeep from me (I don't bother to
dig them up and separate them, for example). I am, however, determined
to get rid of these vines.

I pulled as many as I could find out about five or six summers ago,
and they've spread back from the next property and taken over our
property during that time frame.

What I'd like to know is this: if we rent a gas tiller and 'till' all
the dirt (we live on 60' by 100', and most of that is house and
driveway, so, it's not much dirt to reclaim), and then rake out the
bits of vine, will that get all the vine that's presently in our dirt?
Can we rent tillers that will cut through vines this thick? If we get
rid of all the vines in our dirt and replant our grass, how do we keep
the vines from growing back again? The property next to ours is a
poorly-maintained rental property and the owners refuse to do anything
about the vines on their side. These same vines have already decimated
three generations of dividing fences (two on their side and one on our
side). Is there _anything_ I can do to discourage the vines from
encroaching on our property? Especially since we can't afford to
replace our chain-link fence _again_, but, it's looking dangerously
saggy where the vines are twisted around it and trying to pull it
down.

Chemical warfare suggestions will be considered, although, obviously
non-toxic options are preferred.

Gwen