Scheduling Maintenance Jobs in Networks

0.32 MB·

by  Fidaa Abed

⋆
Scheduling Maintenance Jobs in Networks
Fidaa Abed1, Lin Chen2, Yann Disser3, Martin Groß4, Nicole Megow5, Julie
Meißner6, Alexander T. Richter7, and Roman Rischke8
1 Universityof Jeddah, Jeddah,Saudi Arabia.
[email protected]
2 University of Houston, Texas, USA.
[email protected]
3 TU Darmstadt, Darmstadt, Germany.
7 [email protected]
1 4 Universityof Waterloo, Waterloo, Ontario, Canada.
0 [email protected]
2 5 University of Bremen, Bremen, Germany.
[email protected]
n
a 6 TU Berlin, Berlin, Germany.
J [email protected]
0 7 TU Braunschweig, Braunschweig, Germany.
3 [email protected]
8 TU München,München,Germany.

] [email protected]
S
D
.
s
c
[ Abstract. We investigate the problem of scheduling the maintenance

of edges in a network, motivated by the goal of minimizing outages in
1
transportationortelecommunicationnetworks.Wefocusonmaintaining
v
connectivity between two nodes over time; for the special case of path
9
0 networks, this is related to the problem of minimizing the busy time of
8 machines.
8 Weshowthattheproblemcanbesolvedinpolynomialtimeinarbitrary
0 networksifpreemption isallowed. Ifpreemption isrestricted tointegral
.
1 time points, the problem is NP-hard and in the non-preemptive case
0 we give strong non-approximability results. Furthermore, we give tight
7 bounds on the power of preemption, that is, the maximum ratio of the
1 values of non-preemptiveand preemptiveoptimal solutions.
:
v Interestingly, the preemptive and the non-preemptive problem can be
i solvedefficientlyonpaths,whereasweshowthatmixingbothleadstoa
X
weakly NP-hard problem that allows for a simple 2-approximation.
r
a
Keywords: Scheduling, Maintenance, Connectivity, Complexity The-
ory, Approximation Algorithm
⋆
This work is supported by the German Research Foundation (DFG) under project
ME 3825/1 and within project A07 of CRC TRR 154. It is partially funded in the
framework of Matheon supported by the Einstein Foundation Berlin and by the
Alexandervon Humboldt Foundation.
1 Introduction
Transportation and telecommunication networks are important backbones of
moderninfrastructureandhavebeen amajorfocus of researchincombinatorial
optimization and other areas. Research on such networks usually concentrates
onoptimizingtheirusage,forexamplebymaximizingthroughputorminimizing
costs. In the majorityof the studied optimization models it is assumedthat the
networkispermanentlyavailable,andourchoicesonlyconsistindecidingwhich
parts of the network to use at each point in time.
Practicaltransportationandtelecommunicationnetworks,however,cangen-
erally not be used non-stop. Be it due to wear-and-tear, repairs, or moderniza-
tions of the network,there are times when parts ofthe networkare unavailable.
We studyhowto scheduleandcoordinatesuchmaintenanceindifferentpartsof
the network to ensure connectivity.
Whilenetworkproblemsandschedulingproblemsindividually arefairlywell
understood,the combinationofbothareasthatresults fromschedulingnetwork
maintenance has only recently received some attention [2,4,17,1,11] and is theo-
retically hardly understood.
Problem Definition.In this paper, we study connectivity problems which are
fundamental in this context. In these problems, we aim to schedule the mainte-
nance of edges in a network in such a way as to preserve connectivity between
two designatedvertices.Given a network and maintenance jobs with processing
times and feasible time windows, we need to decide on the temporal allocation
of the maintenance jobs. While a maintenance on an edge is performed, the
edge is not available. We distinguish between MINCONNECTIVITY, the prob-
lem in which we minimize the total time in which the network is disconnected,
and MAXCONNECTIVITY, the problem in which we maximize the total time in
which it is connected.
Inbothoftheseproblems,wearegivenanundirectedgraphG=(V,E)with
two distinguished vertices s+,s− ∈ V. We assume w.l.o.g. that the graph is
simple; we can replace a parallel edge {u,w} by a new node v and two edges
{u,v},{v,w}. Every edge e ∈ E needs to undergo p ∈ Z time units of
e ≥0
maintenance within the time window [r ,d ] with r ,d ∈ Z , where r is
e e e e ≥0 e
called the release date and d is called the deadline of the maintenance job for
e
edgee.Anedgee={u,v}∈E thatis maintainedattime t,isnotavailableatt
in the graph G. We consider preemptive and non-preemptive maintenance jobs.
Ifajobmustbe schedulednon-preemptivelythen,onceitisstarted,itmustrun
until completion without any interruption. If a job is allowed to be preempted,
then its processing can be interrupted at any time and may resume at any later
time without incurring extra cost.
A schedule S for G assigns the maintenance job of every edge e ∈ E to
a single time interval (if non-preemptive) or a set of disjoint time intervals (if
preemptive) S(e):={[a ,b ],...,[a ,b ]} with
1 1 k k
r ≤a ≤b ≤d , for i∈[k] and (b−a)=p .
e i i e e
[a,bX]∈S(e)
If not specified differently, we define T := max d as our time horizon. We
e∈E e
do not limit the number of simultaneously maintained edges.
Foragivenmaintenanceschedule,wesaythatthenetworkGisdisconnected
at time t if there is no path from s+ to s− in G at time t, otherwise we call
the network G connected at time t. The goal is to find a maintenance schedule
for the network G so that the total time where G is disconnected is minimized
(MINCONNECTIVITY).Wealsostudythe maximizationvariantofthe problem,
in which we want to find a schedule that maximizes the total time where G is
connected (MAXCONNECTIVITY).
Our Results. For preemptive maintenance jobs, we show that we can solve
both problems, MAXCONNECTIVITY and MINCONNECTIVITY, efficiently in
arbitrary networks (Theorem 1). This result crucially requires that we are free
to preempt jobs at arbitrary points in time. Under the restriction that we can
preempt jobs only at integral points in time, the problem becomes NP-hard.
More specifically, MAXCONNECTIVITY does notadmit a (2−ǫ)-approximation
algorithmforanyǫ>0inthiscase,andMINCONNECTIVITYisinapproximable
(Theorem2),unlessP=NP.Byinapproximable,wemeanthatitisNP-complete
to decide whether the optimal objective value is zero or positive, leading to
unbounded approximationfactors.
This is true even for unit-size jobs. This complexity result is interesting and
may be surprising, as it is in contrast to results for standard scheduling prob-
lems,withoutanunderlyingnetwork.Here,therestrictiontointegralpreemption
typicallydoesnotincreasetheproblemcomplexitywhenallotherinputparame-
ters areintegral.However,the samequestionremainsopenin arelatedproblem
concerning the busy-time in scheduling, studied in [8,7].
For non-preemptive instances, we establish that there is no (c3 |E|)-ap-
proximation algorithm for MAXCONNECTIVITY for some constant c > 0 and
p
thatMINCONNECTIVITYisinapproximableevenondisjointpathsbetweentwo
nodes s and t, unless P = NP (Theorems 3,4). On the positive side, we pro-
vide an (ℓ + 1)-approximation algorithm for MAXCONNECTIVITY in general
graphs (Theorem 6), where ℓ is the number of distinct latest start times (dead-
line minus processing time) for jobs.
We use the notion power of preemption to capture the benefit of allowing
arbitraryjobpreemption.Thepowerofpreemptionisacommonlyusedmeasure
fortheimpactofpreemptioninscheduling[6,10,19,20].Othertermsusedinthis
context include price of non-preemption [9], benefit of preemption [18] and gain
of preemption [12]. It is defined as the maximum ratio of the objective values
of an optimal non-preemptive and an optimal preemptive solution. We show
that the power of preemption is Θ(log|E|) for MINCONNECTIVITY on a path
(Theorem 7) and unbounded for MAXCONNECTIVITY on a path (Theorem 8).
Thisisincontrastto otherschedulingproblems,wherethepowerofpreemption
is constant, e.g.[10,19].
On paths, we show that mixed instances, which have both preemptive and
non-preemptive jobs, are weakly NP-hard (Theorem 9). This hardness result
is of particular interest, as both purely non-preemptive and purely preemptive
instances can be solved efficiently on a path (see Theorem 1 and [14]). Fur-
thermore, we give a simple 2-approximation algorithm for mixed instances of
MINCONNECTIVITY (Theorem 10).
Related Work. The concept of combining scheduling with network problems
has been consideredby different communities lately.However,the specific prob-
lem of only maintaining connectivity over time between two designated nodes
has not been studied to our knowledge.Bolandet al. [2,3,4] study the combina-
tion of non-preemptive arc maintenance in a transport network, motivated by
annualmaintenanceplanningfortheHunterValleyCoalChain[5].Theirgoalis
to schedule maintenance such that the maximum s-t-flow over time in the net-
work with zero transit times is maximized. They show strong NP-hardness for
their problem and describe various heuristics and IP based methods to address
it. Also, they show in [3] that in their non-preemptive setting, if the input is
integer, there is always an optimal solution that starts all jobs at integer time
points. In [2], they consider a variant of their problem, where the number of
concurrently performable maintenances is bounded by a constant.
Their model generalizes ours in two ways – it has capacities and the objec-
tiveis tomaximizethe totalflowvalue.Asaconsequenceofthis,theirIP-based
methods carryoverto oursetting, but these methods are ofcoursenot efficient.
Their hardness results do not carry over, since they rely on the capacities and
the different objective. However, our hardness results – in particular our ap-
proximationhardness results – carryoverto their setting, illustrating why their
IP-based models are a good approach for some of these problems.
Bley,KarchandD’Andreagiovanni[1]study howto upgradeatelecommuni-
cationnetworktoanewtechnologyemployingaboundednumberoftechnicians.
Their goal is to minimize the total service disruption caused by downtimes. A
majordifferencetoourproblemisthatthereisasetofgivenpathsthatshallbe
upgradedand a path can only be used if it is either completely upgradedor not
upgraded. They give ILP-based approaches for solving this problem and show
strong NP-hardness for a non-constant number of paths by reduction from the
linear arrangementproblem.
Nurre et al. [17] consider the problem of restoring arcs in a network after a
majordisruption,withrestorationpertime stepbeing boundedby theavailable
work force. Such network design problems over time have also been considered
by Kalinowski, Matsypura and Savelsbergh[13].
In scheduling, minimizing the busy time refers to minimizing the amount
of time for which a machine is used. Such problems have applications for in-
stance inthe contextofenergymanagement[16]orfiber managementin optical
networks [11]. They have been studied from the complexity and approximation
point of view in [7,11,14,16]. The problem ofminimizing the busy time is equiv-
alent to our problem in the case of a path, because there we have connectivity
atatimepointwhennoedgeinthepathismaintained,i.e.,nomachineisbusy.
Thus, the results of Khandekar et al. [14] and Chang, Khuller and Mukher-
jee[7]havedirectimplicationsforus.Theyshowthatminimizingbusytimecan
be done efficiently for purely non-preemptive and purely preemptive instances,
respectively.
2 Preemptive Scheduling
In this section, we consider problem instances where all maintenance jobs can
be preempted.
Theorem 1. Both MAXCONNECTIVITY and MINCONNECTIVITY with pre-
emptive jobs can be solved optimally in polynomial time on arbitrary graphs.
Proof. We establisha linearprogram(LP) for MAXCONNECTIVITY.LetTP =
{0}∪{r ,d :e∈E}={t ,t ,...,t } be the setofallrelevant time points with
e e 0 1 k
t < t < ··· < t . We define I := [t ,t ] and w := |I | to be the length of
0 1 k i i−1 i i i
interval I for i=1,...,k.
i
InourlinearprogramwemodelconnectivityduringintervalI byan(s+,s−)-
i
flow x(i), i ∈ {1,...,k}. To do so, we add for every undirected edge e = {u,v}
two directed arcs (u,v) and (v,u). Let A be the resulting arc set. With each
edge/arc we associate a capacity variable y(i), which represents the fraction of
e
availability of edge e in interval I . Hence, 1−y(i) gives the relative amount
i e
of time spent on the maintenance of edge e in I . Additionally, the variable f
i i
expresses the fraction of availability for interval I .
i
k
max w ·f (1)
i i
i=1
X
f ∀i∈[k], v =s+,
i
s.t. x(i) − x(i) = 0 ∀i∈[k], v ∈V \{s+,s−},
(v,u) (u,v) 
u:(vX,u)∈A u:(uX,v)∈A −fi ∀i∈[k], v =s−,
(2)

(1−y(i))w ≥p ∀e∈E, (3)
e i e
i:Ii⊆X[re,de]
x(i) ,x(i) ≤y(i) ∀i∈[k], {u,v}∈E, (4)
(u,v) (v,u) {u,v}
f ≤1 ∀i∈[k], (5)
i
x(i) ,x(i) ,y(i) ∈[0,1] ∀i∈[k], {u,v}∈E. (6)
(u,v) (v,u) {u,v}
Notice that the LP is polynomial in the input size, since k ≤ 2|E|. We show in
Lemma 1 that this LP is a relaxation of preemptive MAXCONNECTIVITY on
generalgraphsandinLemma2thatanyoptimalsolutiontoitcanbeturnedinto
a feasible schedule with the same objective function value in polynomial time,
whichprovesthe claimfor MAXCONNECTIVITY. ForMINCONNECTIVITY, no-
tice that any solution that maximizes the time in which s and t are connected
also minimizes the time in which s and t are disconnected – thus, we can use
the above LP there as well. ⊓⊔
Next, we need to prove the two lemmas that we used in the proof of Theo-
rem 1. We begin by showing that the LP is indeed a relaxation of our problem.
Lemma 1. The given LP is a relaxation of preemptive MAXCONNECTIVITY
on general graphs.
Proof. Given a feasible maintenance schedule, consider an arbitrary interval I ,
i
i ∈ {1,...,k}, and let [ai,bi]∪˙ ...∪˙ [ai ,bi ] ⊆ I be all intervals where s+
1 1 mi mi i
and s− are connected in interval I . We set f = mi (bi −ai)/w ≤ 1 and set
i i ℓ=1 ℓ ℓ i
y(i) ∈[0,1] to the fraction of time where edge e is not maintained in interval I .
e P i
Note that (3) is automatically fulfilled, since we consider a feasible schedule. It
is left to construct a feasible flow x(i) for the fixed variables f and y(i) for all
i
i=1,...,k.
Whenever the given schedule admits connectivity we can send one unit of
flow from s+ to s− along some directed path in G. Moreover,in intervals where
the setofprocessededgesdoesnotchangewecanusethe samepathforsending
the flow. Let [a,b] ⊆ I be an interval where the set of processed edges does
i
not change and in which we have connectivity. Let C be the collection of all
i
such intervals in I . Then, we send a flow x(i) from s+ to s− along any path
i [a,b]
of totalvalue (b−a)/w using only arcs for whichthe correspondingedge is not
i
processed in [a,b]. The flow x(i) = x(i) , which is a sum of vectors,
[a,b]∈Ci [a,b]
gives the desired flow. The constructed flow x(i) respects the flow conservation
P
(2) andnon-negativityconstraints(6), uses no arcmore thanthe corresponding
y(i), since flow x(i) is driven by the schedule. ⊓⊔
e
Lemma 2. Any feasible LP solution can be turned into a feasible maintenance
schedule at no loss in the objective function value in polynomial time.
Proof. Let(x,y,f)beafeasiblesolutionofthegivenLP.LetPi :=(Pi,...,Pi )
1 λi
be a path decomposition [15] of the (s+,s−)-flow x(i) for an arbitrary interval
I := [a ,b ], i ∈ {1,...,k}, after deleting all flow from possible circulations.
i i i
Furthermore, let x(Pi) be the value of the (s+,s−)-flow x(i) sent along the
ℓ
directed path Pi. For each arc a∈A we have that x(Pi)=x(i) by
ℓ ℓ∈[λi]:a∈Pℓi ℓ a
the definition of Pi. Hence, we get x(Pi)=f ≤1 by using (5).We now
ℓ∈[λi] ℓ Pi
divide the interval I into disjoint subintervals to allocate connectivity time for
i
P
each path in our path decomposition. More precisely, we do not maintain any
arc (u,v) (resp. edge {u,v}) contained in Pi, ℓ=1,...,λ , in the time interval
ℓ i
ℓ−1 ℓ
a + w ·x(Pi ),a + w ·x(Pi ) of length w ·x(Pi). (7)
i i m i i m i ℓ
" #
m=1 m=1
X X
Inequality(4)and x(Pi)=x(i) therebyensurethatbynowthetotal
ℓ∈[λi]:a∈Pℓi ℓ a
time where edge e does not undergo maintenance in interval I equals at most
P i
w ·y(i) time units.ByInequality (3),wecanthusdistribute theprocessingtime
i e
of the job for edge e among the remaining slots of all intervals I , i = 1,...,k.
i
For instance, we could greedily process the job for edge e as early as possible in
available intervals. Note that arbitrary preemption of the processing is allowed.
By construction, we have connectivity on path Pi, ℓ = 1,...,λ , for at least
ℓ i
w · x(Pi) time units in interval I . Thus, the constructed schedule has total
i ℓ i
connectivity time of at least k w λi x(Pi)= k w ·f . Since the path
i=1 i ℓ=1 ℓ i=1 i i
decomposition can be computed in polynomial-time and the resulting number
P P P
of paths is bounded by the number of edges [15], we can obtain the feasible
schedule in polynomial-time. ⊓⊔
For unit-size jobs we can simplify the given LP by restricting to the first
|E| slots within every interval I . This, in turn, allows to consider intervals of
i
unit-size, i.e., we have w = 1 for all intervals I , which affects constraint (3).
i i
However, one can show that the constraint matrix of this LP is generally not
totally unimodular. We illustrate the behaviour of the LP with the help of the
following exemplary instance in Figure 1, in which all edges have unit-size jobs
associatedand the label of an edge e represents (r ,d ). It is easy to verify that
e e
a schedule that preempts jobs only at integral time points, has maximum con-
nectivitytimeofone.However,thefollowingschedulewitharbitrarypreemption
has connectivitytime oftwo.We process{s+,v }in [0,0.5]∪[1,1.5],{s+,v } in
2 3
[0.5,1]∪[1.5,2], {v ,s−} in [0,0.5]∪[1.5,2], {v ,s−} in [0.5,1.5], and the other
4 5
edges are fixed by their time window. This instance shows that the integrality
gap of the LP is at least two.
(1,2)
v2 v4
2) (
(0, (0, 0,2)
1)
s+ s−
(0,2) (0,1) (0,2)
(1,2)
v3 v5
Fig.1.Exampleforthedifferencebetweenarbitrarypreemptionandpreemptiononly
at integral time points.
ThestatementofTheorem1cruciallyreliesonthefactthatwemaypreempt
jobs arbitrarily. However, if preemption is only possible at integral time points,
theproblembecomesNP-hardevenforunit-sizejobs.Thisfollowsfromtheproof
of Theorem 3 for t =0, t =1, and T =2.
1 2
Theorem 2. MAXCONNECTIVITYwithpreemptiononlyatintegraltimepoints
is NP-hard and does not admit a (2−ǫ)-approximation algorithm for any ǫ >
0, unless P = NP. Furthermore, MINCONNECTIVITY with preemption only at
integral time points is inapproximable.
3 Non-Preemptive Scheduling
We consider problem instances in which no job can be preempted. We show
that there is no (c3 |E|)-approximationalgorithmfor MAXCONNECTIVITY for
some c > 0. We also show that MINCONNECTIVITY is inapproximable, unless
p
P = NP. Furthermore, we give an (ℓ+1)-approximation algorithm, where ℓ :=
|{d −p | e∈E}| is the number of distinct latest start times for jobs.
e e
To showthe strong hardnessof approximationfor MAXCONNECTIVITY,we
begin with a weaker result which provides us with a crucial gadget.
Theorem 3. Non-preemptive MAXCONNECTIVITY does not admit a (2−ǫ)-
approximation algorithm, for ǫ>0,and non-preemptive MINCONNECTIVITY is
inapproximable, unless P=NP. This holds even for unit-size jobs.
Proof. Weshowthattheexistenceofa(2−ǫ)-approximationalgorithmfornon-
preemptive MAXCONNECTIVITY allows to distinguish between YES- and NO-
instancesof3SATinpolynomialtime.Givenaninstanceof3SATconsistingofm
clauses C ,C ,...,C each of exactly three variables in X = {x ,x ,...,x },
1 2 m 1 2 n
we construct the following instance of non-preemptive MAXCONNECTIVITY.
We pick two arbitrary but distinct time points t +1 ≤ t and a polynomially
1 2
bounded time horizon T ≥t +1. We construct our instance such that connec-
2
tivity is impossible outside [t ,t +1] and [t ,t +1]. For this, s+ is followed
1 1 2 2
by a path P from s+ to a vertex s′ composed of three edges that disconnect
s+ from s− in the time intervals [0,t ], [t +1,t ], and [t +1,T]. These edges
1 1 2 2
e have p = d −r . Furthermore, we construct the network such that the to-
e e e
tal connectivity time is greater than one if and only if the 3SAT-instance is a
YES-instance. And we show that if the total connectivity time is greater than
one,then there is a schedule with maximumtotal connectivity time oftwo.The
high-level structure of the graph we will be creating can be found in Figure 2,
with an expanded version following later in Figure 3.
throughvariable
c c ... c
1 gadgetsaccording 2 m
s+ toclauseliterals s−
x Gadget x Gadget ... x Gadget
1 2 n
Fig.2. High-levelview of theconstruction for Theorem 3.
Let Y(x ) be the set of clauses containing the literal x and Z(x ) be the set
i i i
ofclausescontainingthe literal¬x ,andsetsetk =2|Y(x )| andℓ =2|Z(x )|.
i i i i i
We define the following node sets
– V :={y1,...,yki |i=1,...,n},
1 i i
– V :={z1,...,zℓi |i=1,...,n},
2 i i
– V :={c |r =1,...,m+1},
3 r
– V :={v |i=1,...,n+1}
4 i
– and set V = 4 V ∪{v :v ∈P}∪{s−}.
j=1 j
We introduce thrSee edge types
– E :={e∈E :r =t ,d =t +1,p =t −t },
1 e 1 e 2 e 2 1
– E :={e∈E :r =t ,d =t +1,p =1},
2 e 1 e 1 e
– and E :={e∈E :r =t ,d =t +1,p =1}.
3 e 2 e 2 e
The graph G = (V,E) consists of variable gadgets, shown in Figure 3, to
which we connect the clause nodes c , r =1,...,m+1. We define the following
r
edge sets for the variable gadgets, namely,
– E :={{s′,v },{v ,s−}} of type E ,
1 1 n+1 2
– E :={{v ,y1},{v ,z1},{yki,v },{zℓi,v }:i=1,...,n} of type E ,
2 i i i i i i+1 i i+1 2
– E :={{yq,yq+1}:i=1,...,n;q =1,3,...,k −3,k −1} of type E ,
3 i i i i 1
– E :={{zq,zq+1}:i=1,...,n;q =1,3,...,ℓ −3,ℓ −1} of type E ,
4 i i i i 1
– E :={{yq,yq+1}:i=1,...,n;q =2,4,...,k −4,k −2} of type E ,
5 i i i i 2
– and E :={{zq,zq+1}:i=1,...,n;q =2,4,...,ℓ −4,ℓ −2} of type E .
6 i i i i 2
Notice that a variable x may only appear positive (ℓ = 0) or only negative
i i
(k = 0) in our set of clauses. In this case, we also have an edge of type E
i 2
connecting v and v besides the construction for the negative (z nodes) or
i i+1
positive part (y nodes). Finally, we add edges to connect the clause nodes to
the graph. If some positive literal x appears in clause C and C is the q-th
i r r
clause with positive x , we add the edges {c ,y2q−1} and {y2q,c } both of
i r i i r+1
type E . Conversely,if some x appears negated in C and C is the q-th clause
3 i r r
with ¬x , we addthe edges {c ,z2q−1} and{z2q,c } both oftype E . We also
i r i i r+1 3
connect c and c to the graphby adding {s′,c } and {c ,s−} of type E .
1 m+1 1 m+1 3
We define E to be the union of all introduced edges. Observe that the network
G has O(n+m) nodes and edges.
Wecallan(s+,s−)-paththatcontainsnonodefromV avariablepath andan
3
(s+,s−)-path with no node from V a clause path. An (s+,s−)-path containing
4
edgesoftypeE andE doesnotconnects+ withs− in[t ,t +1]orin[t ,t +1].
2 3 1 1 2 2
Therefore, all paths other than variable paths and relevant clause paths are
irrelevant for the connectivity of s+ with s−.
When maintaining all edges of type E in [t ,t ], we have connectivity in
1 1 2
[t ,t + 1] exactly on all variable paths. Conversely, maintaining all edges of
2 2
type E in [t +1,t +1] yields connectivity in [t ,t +1] exactly on all relevant
1 1 2 1 1
clause paths. On the other hand, any clause path can connect s+ with s− only
yi1 yi2 yi3 yiki
E E
1 2
vi vi+1
zi1 zi2 zi3 ziℓi
E
3
... cr cr+1 cr+2 cr+3 ...
Fig.3. Schematic representation of the gadget for variable xi, which appears negated
in clause Cr and positive in clause Cr+2 among others.
in [t ,t +1] and any variable path only in [t ,t +1]. We now claim that there
1 1 2 2
is a schedule with total connectivity time greater than one if and only if the
3SAT-instance is a YES-instance.
LetS beaschedulewithtotalconnectivitytimegreaterthanone.Thenthere
is a variable path Pv with positive connectivity time in [t ,t +1] and a clause
2 2
path Pc with positive connectivity time in [t ,t +1]. As the total connectivity
1 1
time is greater than one, Pc cannot walk through both the positive part (y
nodes) and the negative part (z nodes) of the gadget for any variable x . This
i
allowstoassumew.l.o.g.thatPv andPc aredisjointbetweens′ ands−.SayPv
and Pc share an edge on the negative part (z nodes) of the gadget for variable
x . Then we can redirect the variable path Pv to the positive part (y nodes)
i
withoutdecreasingthe totalconnectivity time.The sameworksif they sharean
edge on the positive part.
Now set x to FALSE if Pv uses the nodes y1,...,yki, that is the upper part
i i i
of the variable gadget, and to TRUE otherwise. With this setting, whenever Pc
uses edges of a variable gadget, e.g. the sequence c ,z2q−1,z2q,c for some
r i i r+1
r,q, disjointness of Pv and Pc implies that clause C is satisfied with the truth
r
assignment of variable x . Since every node pair c ,c is only connected with
i r r+1
paths passing through variable gadgets, and at least one of them belongs to Pc
we conclude that every clause C is satisfied.
r
Consider a satisfying truth assignment. We define a schedule that admits a
variable path Pv with connectivity in [t ,t +1]. This path Pv uses the upper
2 2
part (y -part) if x is set to FALSE and the lower part (z -part) if x is set to
i i i i
TRUE. That is, we maintain all edges of type E on the upper path (y -path) of
1 i
the variable gadget for x in [t ,t ] if x is FALSE and in [t +1,t +1] if x is
i 1 2 i 1 2 i
TRUE. Conversely, edges of type E on the lower path (z -path) of the variable
1 i
gadget for x are maintained in [t ,t ] if x is TRUE and in [t +1,t +1] if
i 1 2 i 1 2