Computer Networks 54 (2010) 208–217

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Computer Networks

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Complexity results on labeled shortest path problems from wirelessrouting metrics

Charles B. Ward a,*, Nathan M. Wiegand b

a Department of Computer Science, Stony Brook University, Stony Brook, NY 11794-4400, United Statesb Department of Computer Science, The University of Alabama, Box 870290, Tuscaloosa, AL 35487-0290, United States

a r t i c l e i n f o a b s t r a c t

Article history:Available online 30 June 2009

Keywords:Wireless routing metricsLabeled pathsShortest pathsApproximation algorithms

1389-1286/$ - see front matter � 2009 Elsevier B.Vdoi:10.1016/j.comnet.2009.04.012

* Corresponding author. Tel.: +1 205 246 6686.E-mail addresses: charles@cs.sunysb.edu (C.B. W

ua.edu (N.M. Wiegand).

Metrics to assess the cost of paths through networks are critical to ensuring the efficiencyof network routing. This is particularly true in multi-radio multi-hop wireless networks.Effective metrics for these networks must measure the cost of a wireless path based notonly on traditional measures such as throughput, but also on the distribution of wirelesschannels used. In this paper, we argue that routing metrics over such networks may beviewed as a class of existing shortest path problems, the formal language constrained pathproblems.

On this basis, we describe labeled path problems corresponding to two multi-radio wire-less routing metrics: Weighted Cumulative Expected Transmission Time (WCETT), devel-oped by Draves et al., and Metric for Interference and Channel-switch (MIC), developedby Yang et al. For the first, we give a concise proof that calculating shortest WCETT pathsis strongly NP-Complete for a variety of graph classes. We also show that the existing heu-ristic given by Draves et al. is an approximator. For the second, we show that calculatingloop-free (simple) shortest MIC paths is NP-Complete, and additionally show that the opti-mization version of the problem is NPO PB-Complete. This result implies that shortest sim-ple MIC paths are only poorly approximable in the worst case.

Furthermore, we demonstrate how the polynomial-time algorithm for shortest MICpaths is derivable from an existing language constrained shortest path algorithm. Weuse this as a basis to exhibit the general utility of viewing multi-channel wireless routingmetrics as labeled graph problems, and discuss how a class of related polynomial-timecomputable metrics are derivable from this algorithm.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

Creating appropriate metrics to assess the cost of differ-ing paths through a network is crucial for developing effi-cient routing algorithms. Such metrics should account for avariety of differences in the quality of paths, ranging fromsimple hop-count, to issues such as throughput and packetdelay. Shortest path metrics which fail to do so risk creat-ing routing paths which are inferior in practice.

. All rights reserved.

ard), nwiegand@cs.

Although these factors are important in traditionalwired networks, these differences are particularly acutein wireless networks. In situations where some or all linksmay be using wireless mediums, new difficulties arise,such as link interference (cross-talk between wirelesschannels). These difficulties are exacerbated by the well-known problems in constructing wireless MAC protocols,such as with the implementation of collision detection. Be-cause of such difficulties, the multi-radio multi-channelwireless network model has been introduced. This modelallows individual wireless nodes to transmit over multipleradios using non-interfering channels. In this way, perfor-mance degrading interference may be reduced.

C.B. Ward, N.M. Wiegand / Computer Networks 54 (2010) 208–217 209

However, this model introduces new complexities tothe problem of adequately assessing the cost of a routingpath. Paths through a multi-radio, multi-hop wireless net-work consist not only of the intermediate wireless nodesbut also the wireless channel assignments for transmissionbetween them. Thus, in creating routing algorithms forsuch networks we have two major problems to consider.

First, metrics must be developed which give realisticassessments of the effect of channel assignment on net-work transmissions. Second, algorithms which computethe most effective paths according to these metrics mustbe constructed. In this paper, we argue that calculatingshortest paths over any metric which exploits channelassignment can be viewed as a shortest path problem overa multigraph with labeled edges. Specifically, we can relatethese problems to work done on formal language con-strained path problems.

Typically, formal language constrained shortest pathproblems take as input a (weighted or unweighted) multi-graph in which the edges have been labeled with terminal(alphabet) symbols from some formal language [1]. A paththrough the graph can then be associated with a string ofalphabet symbols corresponding to the edges constitutingthe path. Thus, the shortest language constrained path isthe shortest path through the graph with a correspondingstring which is in the given formal language.

Formally, we consider an all-pairs version of thisproblem:

Definition 1. The all-pairs language constrained shortestpath problem.

Input: Finite set of symbols R, a Labeled Directed Mult-igraph Graph G ¼ ðV ; EÞ where E # V � V � Rwith weight function w : E! R, and formal lan-guage K over the alphabet R.

Output: jV j � jV j matrix d such that dðu; vÞ ¼ minimumdistance from u to v over any path such thatthe concatenation of the edge labels over thepath is a valid string in K.

This is a generalization of regular language and context-free language constrained problems investigated by Men-delzon and Wood [2]; Yannakakis [3,4]; Barrett et al. [1];and others [5,6].

While we will retain the above definition of a labeledmultigraph throughout this paper, for notational conve-nience we will treat the labeling of edges as the outputof a labeling function L : E! R, rather than as the thirdterm of the ordered triple.

1.1. Contributions

In this paper, we discuss how routing metrics whichtake into account wireless channel assignment relate tothe field of labeled graph problems. We argue that viewingwireless routing metrics in this light allows us to elegantlydescribe the calculation of many such metrics, and pro-vides an existing base of knowledge which can be appliedto such metrics. Specifically, we relate the polynomialalgorithm given by Yang et al. [7] for shortest paths over

Metric of Interference and Channel-switch to existing algo-rithms for labeled path problems studied by Barrett et al.[1], Yannakakis [3], and others.

Furthermore, we demonstrate that the decision prob-lem corresponding to calculating shortest paths over theWeighted Cumulative Expected Transmission Time [8]metric is Strongly NP-Complete on general directed andundirected graphs, directed acyclic graphs, and series-par-allel graphs. We show the Strong NP-Completeness ofcomputing shortest WCETT paths, which implies thatthere exists no pseudo-polynomial time algorithm forthe problem. We also briefly discuss the provable approx-imability of this metric using the method given by Draveset al. [8].

We also show that calculating shortest simple pathsover the Metric of Interference and Channel-switch [7] isNP-Complete over general directed graphs. We show thata slight generalization of the metric is NP-Complete forgeneral directed and undirected graphs. Moreover, weshow that the optimization version of both problems be-longs to the NPO PB-Complete complexity class; that is,the most difficult NP optimization problems with objectivefunctions polynomially bounded by the input size. This im-plies very poor approximability results for these problems,and implies that the MIC metric can theoretically havearbitrarily bad worst case behaviour when compared tothe simple path if weighting constants are not chosen tomitigate this issue.

1.2. Prior work

1.2.1. Routing metrics in wireless networksThere is widespread interest in creating routing metrics

which appropriately account for the complexities of multi-radio multi-hop wireless networks. Proposed by De Coutoet al. [9], Expected Transmission Count (ETX) is one earlymetric which has served as a basis for more recent metrics.In essence, ETX is the expected number of packet transmis-sions required to successfully deliver a packet across apath. It is argued that in practice, networks which mini-mize this metric will have high throughput. Performingsignificantly better than simple hop-count, ETX is also usedas a component of several other metrics, including thosewe specifically examine in this paper.

Draves et al. [8] extended the ETX metric, describingtwo metrics: Expected Transmission Time (ETT) andWeighted Cumulative Expected Transmission Time(WCETT). ETT extends ETX to incorporate the bandwidthof a link, while WCETT extends ETT by incorporating anadditional constraint which works to spread a path overmultiple wireless channels. We discuss these metrics inmore detail in Section 2.1.

Yang et al. [7] suggest the Metric for Interference andChannel-switch (MIC). Like WCETT, MIC uses ETT as a com-ponent of the metric, but the channel-specific componentdiffers. MIC is concerned with the interference a flowmay cause to itself when it is retransmitted through thesame wireless channel as it arrived. Thus, MIC assigns high-er costs to paths in which two consecutive hops of a pathare transmitted over the same channel. We discuss MICin more detail in Section 2.2.

210 C.B. Ward, N.M. Wiegand / Computer Networks 54 (2010) 208–217

Other metrics such as Exclusive Expected TransmissionTime (EETT) and Adjusted Expected Transfer Delay (AETD)have been suggested [10,11]. Proposed by Jiang et al.[10], EETT also extends the concept of Expected Transmis-sion Time, taking into account not only the rapidity oftransfer over a link, but also the potential interference ofneighboring links. Proposed by Zhou et al. [11], AETD usesan estimate of the jitter of links, and is designed to select aroute in which links using the same wireless channel areseparated ‘‘as far as possible.”

1.2.2. Formal language constrained path problemsFormal language constrained path problems have

found applications in a wide variety of subject areasthroughout the field of computer science, from databasesto constraint programming [2,3,6]. A number of generalalgorithms exist to calculate shortest paths over varioustypes of languages, while NP-Completeness results havebeen shown for others. Below, we describe the applica-tions to which language constrained path problems havebeen applied. We will discuss theoretical results fromthe area in Section 3.

In one of the first papers to examine the problem offinding paths in labeled graphs, Mendelzon and Wood[2] examined simple labeled paths. They noted thatmany recursive database queries that arose in practicewere easily and naturally expressible as graph traversals.Yannakakis also studied labeled path problems as theyapplied to database theory [3,4]. Specifically, because‘‘relational databases can be viewed as labeled directedhypergraphs,” he examined labeled path problems asthey applied to computing transitive closures in dat-abases. In a similar vein, Abiteboul and Vianu examinednavigational queries expressed in regular expressions[12]. These queries are useful for any number of con-texts, ‘‘ranging from hypertext data to object-orienteddatabases” [12].

Barrett et al. have applied language constrained short-est path algorithms to transportation planning [13]. Barrettet al. have used these algorithms and greatly extended thetheoretical literature [1] in their work on the TRANSIMSproject, a transportation planning system supported byLos Alamos.

Choppella and Haynes investigated the problem ofsource-tracking unification [6]. Unification has applica-tions in a variety of areas such as automated theoremproving, artificial intelligence, programming languages,and logic programming. Non-unifiability has direct appli-cations to type errors in other programming languages,as well [6,14,15]. Choppella and Haynes investigatedthe problem of deriving the shortest unification pathfor a given problem and determined that it is simply ashortest paths problem with a context-free languageconstraint.

Bradford [16] considered a slightly different version ofthe context-free constrained shortest path problem, exam-ining quickest labeled paths. Quickest paths were origi-nally discussed by Moore [17] in reference to flowrate-constrained networks. Bradford extends the notion ofquickest paths to include context-free language constraintsand applies this notion to cryptographic routing.

1.3. Paper structure

In Section 2 we consider the WCETT and MIC metrics inmore detail, and in Section 3 we discuss the correspon-dence between shortest paths over these metrics and for-mal language-constrained path problems. In Sections 4and 5 we describe labeled graph decision problems corre-sponding to calculating paths over the WCETT and MICmetrics and give our complexity results. Finally, we con-clude the paper in Section 6.

2. Two routing metrics for wireless mesh networks

In all networks, effectively evaluating the cost of usingcommunications links is the basis for determining routingpaths. In wireless networks particularly, however, accu-rately assessing the cost of a routing path is problematic.The inherent qualities of individual links must be ac-counted for, as should the interference effects that linkusage causes on other links. These are particularly interest-ing problems in the case where individual routing nodeshave multiple wireless radios which can be tuned to sepa-rate, non-interfering channels. We now describe two met-rics which attempt to account for both of these factors tocreate a unified cost estimate for routing paths.

Fig. 1 gives an example of a simple multi-radio multi-hop wireless network which we will use to elucidate ourdiscussion of the WCETT and MIC metrics. In this figure,W represents the edge weight, while C represents thechannels available to each node. Each node has two radios,tuned to two different channels. Edges simply representradios within transmission range on the same channel,and the channels available between nodes are expressedas edge labels.

2.1. Weighted cumulative expected transmission time

The first metric we will examine is WCETT, from thework of Draves et al. [8]. WCETT is designed to combineestimates of transmission time across links with channelinformation in wireless networks. WCETT consists of twocomponents. The first is simply the sum of the ETT mea-sures for each link along a given path. This measure gener-alizes the ETX metric created by De Couto, et al. [9].

ETT adjusts the ETX metric to take into account thebandwidth of the link being considered. Thus, for a givenpath, the sum of the ETT measures gives an estimate ofthe total transmission time of packets across the path.

The second component of the WCETT metric is a mea-sure designed to take wireless channel usage into account.This measure is the total ETT cost of the most used (by ETT)channel in the path. Thus, this term is generally minimizedover channel diverse paths, though particularly high band-width and low loss channels may still take precedence overchannel diversity.

Formally, WCETT is defined as below, see [8].

Definition 2. Let p be a path in the network graphG ¼ ðV ; EÞ, let C be a set of valid edge labels (wirelesschannels), let b be a tunable parameter in the range [0,1],fixed for any instance of the problem, and let ETT : E! Rþ

Fig. 1. Multi-radio multi-hop wireless network example.

C.B. Ward, N.M. Wiegand / Computer Networks 54 (2010) 208–217 211

be the weight function on the edges. Then the cost functionWCETT is computed over p as follows:

WCETTðpÞ ¼ ð1� bÞXe2p

ETTðeÞ þ b maxj2CfXjðpÞg;

XjðpÞ ¼Xe2p

labelðeÞ¼j

ETTðeÞ:

Taking Fig. 1 as an example, allow the W values to bethe ETT values for each edge. Suppose that b ¼ 0:5. Thenthe shortest WCETT path from Node 1 to Node 5 is:1!1 2!6 4!6 5. This path has a cost of 6 � 0:5 from the firstterm, and 3 � 0:5 from the second, for a total cost of 4.5.The direct path from node 1 to node 5 through node 3,has a cost of 6 from each term, for a total cost of 6. How-ever, if b ¼ 0, then these paths have equal cost of 6, andthe shortest path has a cost of 5: 1!6 2!6 4!6 5.

We will demonstrate the NP-Completeness of calculat-ing shortest paths over this metric in Section 4. Draves,et al. compute an approximation of WCETT by using Dijk-stra’s Algorithm [8]. We note that this is a jCj-approxima-tor; that is, the solution will be within a multiplicativefactor of jCj of the optimal solution. However, it is also truethat the unlabeled shortest path (i.e., the shortest path by atraditional shortest path metric) is also a jCj-approximator.

Lemma 3. The shortest unlabeled path is within a factor ofjCj of the shortest WCETT path.

Proof. Let pw be the shortest WCETT path and po be theshortest ETT path: ETTpo 6 ETTpw. Then the smallest possi-ble value for WCETTðpwÞ is ð1� bÞETTðpwÞ þ b ETTðpwÞ

jCj , occur-ring in the case where the labels split the edge weightsperfectly evenly. The worst possible value for WCETTðpoÞis ETTðpoÞ, occurring in the case where the edges along po

have only one label. Thus, the worst case is that b ¼ 1and thus WCETTðpoÞ 6 jCjWCETTðpwÞ. h

Lemma 4. The shortest path as computed by Dijkstra’s algo-rithm using WCETT distance as the heuristic function is withina factor of jCj of the shortest WCETT path.

Proof. In order for the modified Dijkstra algorithm to failto compute the shortest WCETT path, it must be that atsome intermediate node x between the source s and thedestination t there are two paths from s to x: (1) a pathwhich is the shortest WCETT path from s to x but whichdoes not fall on the shortest WCETT path from s to t, (2) apath which is not the shortest WCETT path from s to xbut which does fall on the shortest WCETT path from s tot. Then, the worst possible case for the algorithm is thatpath (1) uses all channels equally to a level just under thatof path (2), and that path (2) uses only one channel. Thus,path (2) essentially has a factor of jCj more potential toabsorb channel cost in the remaining portion of the pathfrom x to t. By repeating this on jCj intermediate vertices,the worst possible case of cost arbitrarily close to a factorof jCj times optimal can be achieved. It is never possibleto do worse than this as, at each intermediate vertex, path(2) cannot beat path (1) in terms of WCETT cost. h

When the number of channels is a small constant, thisindicates that modified Dijkstra is a reasonable approxima-tor. In the case of the 802.11b/g wireless standards, thereare only 11 channels, only three of which can be non-over-lapping simultaneously. However, this constant can also befairly high. For example, the 802.11a wireless standard al-lows for up to 23 non-overlapping channels [18]. Further-more, although the number of interfaces per node shouldbe relatively small, due to antenna interference even on

212 C.B. Ward, N.M. Wiegand / Computer Networks 54 (2010) 208–217

non-overlapping channels, this does not affect the approx-imability of either algorithm.

2.2. Metric of interference and channel-switch

The MIC metric for shortest paths in wireless mesh net-works was created by Yang et al. [7], and was designed toimprove upon WCETT. MIC consists of two principalcomponents.

The first component is the sum of individual weights onall links in the path. These weights are the IRU values: esti-mates of the total amount of wasted transmission time.That is, it is the expected transmission time multipliedby the number of links which interfere with the wirelesschannel being used. Thus, this component captures the po-tential for the path to interfere with other paths.

The second component consists of a sum of weights forintermediate nodes in the path, where the weight is deter-mined by the incoming and outgoing channels of the pathat that node. That is, if two consecutive links on a path aretransmitted on the same channel, then a flow along thispath will interfere with its own transmission, reducingthe viability of the path. Thus, the second term capturesthe potential for the path to interfere with itself.

Formally, MIC is defined as below, see [7].

Definition 5. Let p;G;C; ETT , be defined as for WCETT,while minðETTÞ is the smallest edge weight in the graph.Then if v 2 V , let prevðv; pÞ be the node preceding v in pathp or ; if there is no such node, and define Lð;Þ ¼ �. Let Ne bethe set of neighboring edges which link e interferes with(has the same channel label). Finally, w1;w2 2 Rþ;0 6w1 < w2, are channel-switching costs. Then define the costfunction MIC over p as follows:

MICðpÞ ¼ 1jV j �minðETTÞ

Xe2pe2E

IRUe þXe2pe2E

CSCe;p;

IRUe ¼ ETTe � Ne;

CSCv;p ¼w1 if Lðprevðv;pÞÞ– LðvÞ;w2 if Lðprevðv;pÞÞ ¼ LðvÞ:

�

Let us take Fig. 1 again as an example; for simplicity al-low the W values to be the scaled ETT values for eachedges. First, let us take w1 ¼ 5 and w2 ¼ 13. In this case,the shortest path from Nodes 1 to 5 is: 1!1 3!1 5. This pathhas a cost of 6 from the weights, and 13 from the channel-switching cost, for a total cost of 19. If we reduce w1 to 1,however, the path 1!6 2!1 3!11

4!6 5 becomes the shortest,having a total weight of only 18, 15 from weights and 3from channel-switching.

Yang et al. give an algorithm that computes shortestpaths according to this metric. This algorithm does thisby augmenting the graph, such that each vertex in the ori-ginal graph is replaced with a number of vertices equal totwice the number of edges at that vertex. Thus by connect-ing these vertices with directed edges with appropriateweights, the CSC cost can be enforced.

It is important to note that a shortest path according tothis metric may have a loop, as it may be possible to cir-cumvent the increased cost of utilizing two edges in a

row with the same channel by leaving a vertex and return-ing across a different channel. However, such a loop createsonly a faux savings, as the disruption to the transmissionwill, in practice, be the same, but with the addition ofwasted bandwidth in the loop. Unfortunately, a shortestpath according to the metric may not be the shortestloop-free path according to the metric, so although wemay remove the loop in practice, this does not necessarilyresult in a shortest path according to the metric. It is argu-able that this is actually a practical problem, of course, asthis problem can only occur when the w2 �w1 cost ex-ceeds the IRU cost of the loop in question. This does, how-ever, reinforce the importance of choosing good CSC values.

Moreover, from a theoretical standpoint, we will showin this paper that for directed graphs there is very littleto be done to correct this problem. As we will demonstrate,for directed graphs it is not possible to calculate shortestsimple paths over MIC in polynomial time (assumingP – NP). Additionally, we show that a slight generalization,in which we allow arbitrary pairings of conflicting chan-nels, is NP-Complete for directed and undirected graphs.Thus, our results imply that for this metric mitigation ofthe problem through choice of appropriate CSC values isthe only practical solution.

3. Routing metrics and formal language constrainedpath problems

The formal-language-constrained path problems havebeen studied by a number of authors, most recently andin the most depth by Barrett et al. [1]. These problemsare concerned with finding shortest paths over graphswherein the set of valid paths are limited to those whichare labeled with strings from some formal language (giveneither as an input to the problem, or fixed as a constant).Edges in the graph are labeled with symbols from somealphabet R and the string corresponding to a path is simplytaken as the concatenation of the labels of the edges in thepath. It has been shown that these shortest path problemsare computable in polynomial time for context-free lan-guages (and hence all languages falling below in the Chom-sky hierarchy), with a variety of algorithms specialized tothe type of input language. These problems also becomeNP-Complete when restricted to simple paths, even for ex-tremely simple classes of input languages.

As an example, let us again consider Fig. 1. We may con-sider wireless channels to be alphabet symbols, and thechannels of the links themselves as edge labels. Suppose,then, we wish to calculate a shortest path from Nodes 1 to5, with the constraint that no label (channel) may be usedon two consecutive links. Under this constraint, there areonly three potential paths, all of which are forced to crossover the expensive edge between nodes 3 and 4. The short-est of these is 1!1 3!11

4!6 5, with a total cost of 13.We note that this is simply the calculation of shortest

paths according to MIC in the special case where w2 ¼ 1.Furthermore, note that this constraint can be expressedas a regular expression, and is thus solvable using an algo-rithm for the regular language constrained shortest pathproblem. It is easiest to express this constraint in terms

C.B. Ward, N.M. Wiegand / Computer Networks 54 (2010) 208–217 213

of the equivalent finite state machine. Simply, each state inthe finite state machine represents the last wireless chan-nel used in the path. From this state, it may transition toevery other state with the next edge, but may not remainin this state (reuse the wireless channel on the next hop).

Now, although this particular example only considersthe equivalence of shortest MIC paths when w2 ¼ 1, wewill now demonstrate that the algorithm which solvesthe regular-language constrained path problem (developedindependently by Barrett et al. [1] and Yannakakis [3]) canbe easily modified to work with arbitrary w2 values. Thiscanonical algorithm works by taking the cross product ofthe input graph and the graph representing the finite statemachine of the regular expression given as input. In thecross product graph, a vertex represents the combinationof both a vertex and an FSM state. Edges in the cross prod-uct graph correspond to crossing an edge in the originalgraph which results in a valid transition in the FSM.

Consider a graph G with two vertices v1 and v2 and a fi-nite state machine M with two states s1 and s2. Then, thecross product G�M of G and M will be a graph with fourstates: ðv1; s1Þ; ðv1; s2Þ; ðv2; s1Þ, and ðv2; s2Þ. G�M will havean edge ððvi; sjÞ; ðvk; smÞÞ if and only if there exists an edgeðvi; vkÞ in G with some label ‘ and there exists a transitionðsj; smÞ on label ‘ in M. In the specific case of this problem,the states of the machine would correspond to the possiblevalues for the last wireless channel used in the currentpath. Thus, an edge ððvi; sjÞ; ðvk; smÞÞ would exist only ifthere is an edge ðvi; vkÞ labeled with the wireless channelcorresponding to state sm and sj – sm.

Once the cross-product graph is obtained, the shortestpath can be determined by running a traditional shortestpath algorithm on the cross-product graph. That is, sup-pose that we wish to find the shortest regular languageconstrained shortest path from vertex u to vertex v. Then,this is simply the shortest path in the cross product graphfrom ðu; sÞ to ðv; f Þ where s is the start state and f is any fi-nal state. Hence, we can directly reduce the computation ofshortest MIC paths to the regular language shortest pathproblem in the case of w2 ¼ 1.

In order to handle the case when w2 –1, consider thatthe lack of edges corresponding to violations of the finitestate machine correspond to infinite w2 weights. By addingedges in the cross product which correspond to a violationof the state machine’s input language, we can ensure thatthese edges have the appropriate (non-infinite) w2 weight.That is, instead of disallowing edges which correspond tovalid edges in the original graph but violate the regularlanguage, we instead allow but raise the cost of the edgeby w2. In this case, the resulting graph is very similar tothe virtual node construction given for the polynomialtime algorithm given by Yang, et al. Furthermore, we ob-serve that any set of forbidden adjacent labels can be ex-pressed in a finite state machine of size at most OðjRj2Þ(jRj states and OðjRj2Þ transitions), and so the slight gener-alization of the problem which will be described in Section5 can be solved in a similar fashion when consideringshortest (not simple) paths. Moreover, it can be solvedwithout introducing additional computational complexity.

This analogy to an existing set of problems is obviouslyhelpful, as it gives us a set of existing polynomial time

algorithms which can be easily modified to compute short-est paths over routing metrics with channel assignmentconstraints. From this, we can also see that a number ofpotentially useful modifications are computable by slightmodification of this algorithm. For example, we can trivi-ally extend the ‘‘memory” of channel history, at the ex-pense of creating a larger finite state machine. We canalso express an arbitrary set of conflicting wireless chan-nels, each pair with its own penalty value. This could allowa metric capable of expressing interference caused by par-tially overlapping channels, such as could occur in a802.11b/g network.

Additionally, some of the results of formal languageconstrained path problems can be applied to computingrouting metrics of this type. For a simple example, as notedoriginally by Mendelzon and Wood [2], the regular-lan-guage-constrained simple path problem is solvable inpolynomial time on acyclic graphs, and hence this is alsotrue for both MIC and our generalization. As another, Bar-rett et al. [13] indicates that for certain subsets of regularlanguages (linear regular languages), the efficiency of theregular-language-constrained shortest path algorithmmay be improved by not calculating the entire cross-prod-uct graph in advance, and merely computing pieces of thegraph as required [13]. Although not identical, the regular-language corresponding to MIC is similar in construction toa linear regular language, and it is likely that this same in-crease in efficiency could be achieved.

Finally, our discussion in Sections 4 and 5 is heavilyinfluenced by the terminology and results of labeled pathproblems. From this basis, we will now discuss our com-plexity results.

4. Most used label shortest paths

In examining the WCETT metric, we note that the firstterm can be simply computed, as it is nothing more thana simple weighted shortest path problem. It is the secondcomponent of the metric, which seeks to minimize themost used channel, which provides difficulties. It can beverified that the addition of the first component does notfundamentally alter the problem of computing a shortestpath over the metric, and so we consider the following la-beled shortest path decision problem corresponding to thesecond term of the WCETT metric:

Definition 6. The most used label shortest paths (MUL)decision problem:

Input: Directed Multigraph G ¼ ðV ; EÞ, a set of labels R,weight function U : E! R�, labeling L : E! R,source s 2 V , destination t 2 V , maximum weightW.

Output: True if 9 path psuch that mulðpÞ 6W where

mulðpÞ ¼maxj2CfXjðpÞg;

XjðpÞ ¼Xe2p

labelðeÞ¼j

:UðeÞ

False, otherwise.

214 C.B. Ward, N.M. Wiegand / Computer Networks 54 (2010) 208–217

Theorem 7. The MUL Problem is NP-Complete.

Proof.

(: We first note that we can verify in polynomial timewhether for a given path p it is the case thatmulðpÞ < W . Thus the problem is in NP.

): To show that the MUL problem is NP-Hard wereduce from K-Bin Packing [19]. Suppose we havean instance of the K-Bin Packing Problem in whichwe have k unit bins fB1; . . . ;Bkg and n itemsf1; . . . ;ngwith weights fI1; . . . ; Ing. The K-Bin PackingProblem asks whether it is possible to assign itemsto bins such that total weight in each bin is 6 1.

From this input we will construct the input to aninstance of the MUL problem. Let R ¼ f1; . . . ; kg, and createthe graph G ¼ ðV ; EÞ. Here V ¼ fv1; . . . ; vnþ1g. For eachvertex við1 6 i 6 nÞ we create edges ei;j ¼ ðvi; viþ1Þð1 6 j 6 kÞ in E. Then let Lðvi;jÞ ¼ j and Uðvi;jÞ ¼ Ii. Finally,set s ¼ v1; t ¼ vnþ1, and W ¼ 1.

As Fig. 2 shows, the reduction graph is essentially alinear chain from source to sink where the choice of label(wireless channel) in each hop corresponds to choosing abin for each element in the original bin packing problem.The minimization of the most used wireless channel thusyields the best bin packing.

To formally complete the reduction, we run MUL onðG;R;U; L; s; t;WÞ. If MUL returns true, then the shortestpath through G represents a bin-packing, as each elementhas a label representing the bin it is in, and mulðpÞ is thetotal weight of all elements put in the most filled bin. h

Fact 8. Antenna-interference even on non-overlappingchannels implies that practical mesh networks should havea restricted number of interfaces per node. However, binpacking remains NP-Complete even with only two bins[19]. This implies that computing WCETT is an NP-Com-plete optimization problem even with only 2 wirelessradios per node and only 2 wireless channels. By Lemma3, of course, modified Dijkstra is still a jCj-approximatorwhen jCj ¼ 2.

Corollary 9. The MUL Problem is Strong NP-Complete overgeneral undirected graphs, directed acyclic graphs, and seriesparallel graphs. The MUL Problem is Strong NP-Complete forany fixed number of labels k P 2.

Fig. 2. Reduction from the K

Proof. Note that if any shortest mul-path contains a loop,we can remove the loop and it remains a shortest mul-path. Thus, the reduction in Theorem 7 holds if the edgesare undirected. Furthermore, the graph given in Theorem7 is both a directed acyclic graph and a series parallelgraph, and thus the problem is NP-Hard over these setsof input graphs as well. K-Bin packing is Strong NP-Hardfor k P 2, and our reduction is a pseudo-polynomial trans-formation according to the four necessary propertiesdefined by Garey and Johnson [19]. These four conditionsare that:

� The reduction is a mapping reduction.� The reduction takes time polynomial in terms of the size

of the original problem.� The size of the resulting problem is within a polynomial

bound of the original problem.� All weights in the resulting problem are within a poly-

nomial bound of the original problem.

As all these conditions hold, MUL is Strong NP-Hard forany number of labels P 2 over all the aforementionedgraph classes. h

Fact 10. Recall that NP-Completeness in the strong senseimplies that there exists no pseudo-polynomial time algo-rithm for the problem [19]. Thus, there exists no pseudo-polynomial time algorithm for computing the shortestWCETT path on general directed, general undirected, direc-ted acyclic, or series parallel graphs for any number ofwireless channels k P 2.

5. Restricted adjacent-labels simple paths

The MIC metric attempts to minimize intra-flow inter-ference by increasing the cost of using the same wirelesschannel on two consecutive network links in a path. How-ever, this interference is not necessarily restricted to trans-missions on the same channel. Take, for example, 802.11gwireless networks where each of the 14 channels (11 inthe US, due to legal restrictions) causes interference to be-tween 4 and 8 other channels [18].

Thus, we generalize this to the case in which eachchannel may interfere with some number of otherchannels, and thus increase the cost of a transmission whichuses consecutive edges labeled with conflicting channels.

-bin packing problem.

C.B. Ward, N.M. Wiegand / Computer Networks 54 (2010) 208–217 215

The following formally expresses this generalization:

Definition 11. The restricted adjacent-labels simple pathdecision problem (RASP):

Input: Directed or Undirected Multigraph G ¼ ðV ; EÞ, aset of labels R, weight function U : E! R�,weights w1;w2 2 Rþ, labeling L : E! R, k unor-dered pairs of labels F ¼ fða1; b1Þ; . . . ; ðak; bkÞgwhere ai 2 R and bi 2 R for all 1 6 i 6 k, sources 2 V , destination t 2 V ,maximum weight W. Alsodefine CSC as in Section 2.2. ; if e begins p.

Output: True if 9 path p such that ralðpÞ 6W where

Fig. 3. Conflicting vertices in reduction from the DSPFP.

ralðpÞ ¼Xe2p

ðUðeÞ þ CSCe;pÞ

False, otherwise.

Note that in the case where the set of restricted pairs isfðl1; l1Þ; . . . ; ðljRj; ljRjÞg, each li 2 R, we have exactly the MICmetric given in Yang et al. [7]. Corollary 14 shows that thisis NP-Complete for directed graphs. Also note that, as dis-cussed earlier, in the case of shortest (rather than simple)paths, this problem is solvable in polynomial time, as givenin [7]. However, this result implies the impossibility ofimproving upon this flaw in the MIC metric and thusemphasizes the importance of avoiding the problem inpractice by proper choice of w1 and w2 values.

In order to show that RASP is NP-Complete, we reducefrom the Shortest Paths with Forbidden Pairs Problem(SPFP). SPFP was shown to be NP-Complete by Gabowet al. [20]. The input to the SPFP Problem consists of agraph and a set of pairs of vertices. The output of the prob-lem is the shortest path in an input graph subject to theconstraint that at most one vertex from each pair may ap-pear in the path.

Formally, we define the following decision version ofSPFP below:

Definition 12. Shortest paths with forbidden pairs:

Input: Directed Graph G ¼ ðV ; EÞ, weight functionU : E! Rþ, pairs of vertices fða1; b1Þ; . . . ; ðak;

bkÞg, maximum weight W.Output: True if 9 path p such that

Pe2pUðeÞ < W and 8

vertices u; v in path p there is no k such thatu � ak and v � bk.

Theorem 13. The RASP Problem is NP-Complete for generaldirected and undirected graphs.

Proof.

(: We first note that we can verify whether for a givenpath p it is the case that ralðpÞ < W . Thus, the prob-lem is in NP.

): We reduce from the Directed Shortest Paths withForbidden Pairs Problem [20] to the RASP Problemwith an undirected input graph (as the undirectedcase reduces directly to the directed case by the sub-

stitution of two directed edges for each undirectededge). The DSPFP Problem takes as input a graph Gwith weight function U, start vertex s, destinationvertex t; k pairs of vertices fða1; b1Þ; ða2; b2Þ; . . . ;

ðak; bkÞg, and maximum weight W.

The intuition of the reduction is as follows. We wish tomodify the input graph such that the adjacent labelingconstraints will enforce the forbidden pairs constraint forus. In order to do this, we will replace each vertex with aninput and an output vertex connected through a chain ofconflict vertices. Each of these conflict vertices correspondsto a forbidden pair in the input problem which this vertexis a member of.

That is, if the vertex is part of three conflicts, it will havethree intermediate conflict vertices between its input andoutput. These conflict vertices will be shared with theother vertices in the forbidden pairs. Fig. 3 shows twovertices, x and y, which are each part of a single forbiddenpair with each other. The labels given to these edges, andthe set of forbidden adjacent labels, will be set such that itis impossible for a path to ‘‘jump” to another part of thegraph by way of the conflict vertices.

The result of these shared conflict vertices is that, as theproblem requires the solution to be a simple path, eachconflict vertex can be used only once in the solution.Hence, the simple solution path, when translated back tothe original graph, will not violate the input set offorbidden pairs. A more formal description of the reductionfollows.

We will construct graph H ¼ ðV 0; E0Þ for input to theRASP Problem. Begin by setting V 0 ¼ ;; E0 ¼ ;; F ¼ ;. Also,let us denote the vertices of V as f1; . . . ; jV jg, and letR ¼ f1; . . . ;2jV j;x;wg. Here, x and w are additional labelswe will need in the graph construction. For each vertexv 2 V add three vertices vS; vI , and vO to V 0. For each conflictpair ðaj; bjÞ add vertex Cj to V 0. For each vertex v 2 V whichis in c conflict pairs, add c � 1 vertices vCj

ð1 6 j < c).For each edge ðu; vÞ 2 E add edge ðuO; vIÞ to E0; let

LðuO; vIÞ ¼ x. For each vertex v 2 V , add edge ðvS; vIÞ to E0;

216 C.B. Ward, N.M. Wiegand / Computer Networks 54 (2010) 208–217

let LðvS; vIÞ ¼ x. If for each vertex v 2 V , v is not in anyconflict pair, add edge ðvI; vOÞ to E0; let LðvI; vOÞ ¼ w. For allvertices v 2 V , let m be the number of conflict pairs for v,and add edges fðvI;C1Þ; ðC1; vC1 Þ; ðvC1 ;C2Þ; . . . ; ðCm; vOÞg toE0. Let LðvI;C1Þ ¼ LðvC1 ;C2Þ ¼ � � � ¼ LðvCm�1 Þ ¼ v and in eachconflict pair i, let v be ak, then set LðCi; vCi

Þ ¼ jV j þ bk, jV jplus the label corresponding to the other vertex in thisconflict pair. Let the weight function U0 on E0 be such thatfor u; v 2 V edges ðuO; vIÞ have weight from the instance ofthe DSPFP problem, and all other edges have weight 0.

Finally, for all vertices v 2 V add ðv; jV j þ vÞ to F, and foreach u which conflicts with v add ðv;uÞ. Also add ðx;xÞ toF, and set w1 ¼ 0 and w2 ¼ 1. Now, return the solutionfrom calling RASP with input ðG0;U0;w2; L; F; sS; tO;WÞ.

Fig. 3 shows the construction of a pair of vertices x and ywhich conflict only with each other.

There will exist a simple path with weight 6W from sI

to tO in G0 which does not pass over any consecutive labelslisted in F if and only if it is possible to find a path from s tot in G with weight 6W . This follows from the fact that iftwo vertices u and v conflict, any path which uses both uand v must pass through the conflict vertex correspondingto their conflict twice, violating simplicity of the resultantpath. This follows from the fact that the x labelingprevents the solution from creating a path which has noanalogue in G. This is because all edges with analogue in Ghave label x, forcing a path to pass through all conflictvertices associated with a vertex.

Thus, the shortest simple path from sS to tO in G0 willcorrespond exactly to the shortest path from s to t in G inwhich no forbidden pairs of vertices are used, and thesepaths will have the same weight. h

Corollary 14. The RASP Problem is NP-Complete for generaldirected graphs and the given set of forbidden adjacent labelsfðl1; l1Þ; . . . ; ðljRj; ljRjÞg (li 2 R).

Proof. In this case, we note that our previous reductionholds but that the four edges surrounding conflict verticescan no longer be appropriately constrained using labels.This is because there are two undirected edges which couldallow a ‘‘leak” of the path through the conflict vertices, andthe set of forbidden adjacent labels allows us to plug one ofthese at a time. In this case, the directedness supplies thenecessary constraint, demonstrating that the problemremains NP-Complete in this subcase on general directedgraphs. h

5.1. Approximation of RASP

In this section, we demonstrate that RASP belongs to aclass of NP-Complete optimization problems (NPO PB-Complete) which are only poorly approximable. This im-plies that MIC can have arbitrarily bad worst case behav-iour when compared to the optimal simple path ifweights are not chosen appropriately to mitigate theproblem.

The complexity class NPO PB consists of NP optimiza-tion problems in which the objective function being opti-mized is bounded by some polynomial in terms of the

size of the problem input [21]. The complete problems ofthe NPO PB class are those which every NPO PB problemcan be reduced to in polynomial time using an approxima-tion preserving reduction. An approximation preservingreduction is simply a reduction wherein an approximationof the problem reduced to yields an approximation for theoriginal problem. Thus, any hardness of approximation re-sults from the original problem can be applied. The com-plete problems of this class were formerly divided intoMin PB-Complete and Max PB-Complete, correspondingto the set of complete minimization and complete maximi-zation problems, respectively. However, it has been shownthat these are equivalent, and so the term NPO PB-Com-plete is used [22].

The NPO PB-Complete problems are difficult to approx-imate. This can be observed by a result of Halldórsson,which showed that the calculating the Minimum MaximalIndependence Number is not approximable within a factorof n1�� for any � > 0 [23]. As this problem belongs to theNPO PB-Complete class, and all NPO PB-Complete prob-lems are approximation preserving polynomially reducibleto each other, this result implies that NPO PB-Completeproblems are difficult to approximate [21].

Theorem 15. The optimization version of the RASP Problemis NPO PB-Complete for general directed graphs even with thegiven set of restricted labels fðl1; l1Þ; . . . ; ðljRj; ljRjÞgðli 2 RÞ,provided that the weights w1;w2, and the weight function ware restricted to some polynomial of the input size.

Proof. The Shortest Paths with Forbidden Pairs Problem isNPO PB-Complete by the work of Kann [21]. Thus, in orderto show that the RASP Problem is NPO PB-Complete it suf-fices to show that the reduction given in Theorem 13 andmodified in Corollary 14 is approximation preserving,and that the objective function of the RASP optimizationproblem is bounded by some polynomial function of theinput size.

(: First note that the original reduction preserves thecost of a path, as the edges added to enforce the pairrestriction do not contribute additional cost to thepath. Any valid path computed between start andend vertices in the RASP graph will correspondexactly to a valid path in the original SPFP graph,and have the same cost. Hence, any approximatepath in the RASP graph will correspond to anapproximate path in the SPFP problem, and haveexactly the same approximation ratio.

): Since we consider only simple paths, it is straightfor-ward to observe that the objective function isbounded by at most n� 1 times the largest edgeweight in the graph. Provided that the input weightsare restricted to some polynomial function of theinput size, the objective function is thus, alsorestricted to a polynomial function of the inputsize. h

Since NPO PB-Complete problems cannot be approxi-mated within n1�� for any � > 0, then RASP is difficult toapproximate. As a result, we can observe that on directed

C.B. Ward, N.M. Wiegand / Computer Networks 54 (2010) 208–217 217

graphs, simple MIC paths are, likewise, difficult to approx-imate. This implies that the work of Yang et al., in whichshortest MIC paths are computed, is the theoretical limitof this particular approach to metric design, within thebounds of polynomial-time computability.

6. Conclusions

In this paper we have demonstrated the NP-Complete-ness of calculating shortest loop-free routing paths overWCETT and MIC, which yield good routing solutions in mul-tichannel wireless networks. We have shown that comput-ing shortest WCETT paths is strongly NP-Complete, andthus there exists no pseudo-polynomial time algorithm,but we have also discussed the provable approximabilityof the modified Dijkstra’s algorithm, as given by Draveset al. [8]. Additionally, we have demonstrated that comput-ing simple MIC paths belongs to a class of optimizationproblems which are difficult to approximate. This impliesthat good choice of weights is important to avoid this the-oretical bad behaviour.

Furthermore, we have also related the calculation ofthese metrics to a class of existing problems, the formallanguage constrained path problems, and noted that an en-tire class of similar wireless routing metrics have polyno-mial-time algorithms simply derivable from algorithmsfrom this class. We believe that there is much to be gainedby viewing multi-radio multi-hop wireless metricsthrough the lens of labeled graph theory.

Acknowledgement

Special thanks to Phillip G. Bradford, for his numeroussuggestions which have improved this paper enormously.

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Charles Ward received his B.S. in Mathemat-ics and Ph.D. in Computer Science from theUniversity of Alabama in 2004 and 2008,respectively. He is currently a postdoctoralassociate in Dr. Steven Skienas lab at StonyBrook University. His research areas includealgorithms, graph theory, networking, formallanguages, computer security, and distributedalgorithms.

Nathan Wiegand received his B.S. in Mathe-matics and M.S. in Computer Science from theUniversity of Alabama in 2004 and 2007,respectively. He is currently a Ph.D. candidatein the Department of Computer Science at theUniversity of Alabama. His research areasinclude algorithms, graph theory, program-ming languages, networking, formal lan-guages, distributed algorithms, andapproximability.