Approximation algorithms for the arc orienteering problem

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<ul><li><p>Accepted Manuscript</p><p>Approximation algorithms for the arc orienteering problem</p><p>D. Gavalas, C. Konstantopoulos, K. Mastakas, G. Pantziou,N. Vathis</p><p>PII: S0020-0190(14)00218-XDOI: 10.1016/j.ipl.2014.10.003Reference: IPL 5189</p><p>To appear in: Information Processing Letters</p><p>Received date: 25 May 2014Revised date: 30 August 2014Accepted date: 3 October 2014</p><p>Please cite this article in press as: D. Gavalas et al., Approximation algorithms for the arcorienteering problem, Inf. Process. Lett. (2014), http://dx.doi.org/10.1016/j.ipl.2014.10.003</p><p>This is a PDF file of an unedited manuscript that has been accepted for publication. As aservice to our customers we are providing this early version of the manuscript. The manuscriptwill undergo copyediting, typesetting, and review of the resulting proof before it is publishedin its final form. Please note that during the production process errors may be discovered whichcould affect the content, and all legal disclaimers that apply to the journal pertain.</p><p>http://dx.doi.org/10.1016/j.ipl.2014.10.003</p></li><li><p>Highlights</p><p> We propose a polylogarithmic approximation algorithm for the AOP in directed graphs. A constant factor approximation algorithm for the AOP in undirected graphs is presented. An inapproximability result for the AOP is obtained. Approximation algorithms for the Mixed Orienteering Problem are given.</p></li><li><p>Approximation Algorithms for the Arc Orienteering</p><p>Problem</p><p>D. Gavalasa,f, C. Konstantopoulosb,f, K. Mastakasc,f,, G. Pantzioud,f, N.Vathise,f</p><p>aDepartment of Cultural Technology and Communication, University of Aegean, GreecebDepartment of Informatics, University of Piraeus, Greece</p><p>cSchool of Applied Mathematical and Physical Sciences, National Technical University ofAthens, Greece</p><p>dDepartment of Informatics, Technological Educational Institution of Athens, GreeceeSchool of Electrical and Computer Engineering, National Technical University of</p><p>Athens, GreecefComputer Technology Institute &amp; Press Diophantus, Patras, Greece</p><p>Abstract</p><p>In this article we present approximation algorithms for the Arc Orienteering</p><p>Problem (AOP). We propose a polylogarithmic approximation algorithm in</p><p>directed graphs, while in undirected graphs we give a (6 + + o(1)) and a</p><p>(4+)approximation algorithm for arbitrary instances and instances of unitprofit, respectively. Also, an inapproximability result for the AOP is obtained</p><p>as well as approximation algorithms for the Mixed Orienteering Problem.</p><p>Keywords: Arc Orienteering Problem, Orienteering Problem, Mixed</p><p>Orienteering Problem, Approximation Algorithms, Inapproximability</p><p>Corresponding authorEmail addresses: dgavalas@aegean.gr (D. Gavalas), konstant@unipi.gr (C.</p><p>Konstantopoulos), kmast@math.ntua.gr (K. Mastakas), pantziou@teiath.gr (G.Pantziou), nvathis@softlab.ntua.gr (N. Vathis)</p><p>Preprint submitted to Information Processing Letters October 14, 2014</p></li><li><p>1. Introduction1</p><p>The Arc Orienteering Problem (AOP) is a single route arc routing prob-2</p><p>lem with profits defined as follows [10]: Given a quadruple (G = (V,A), t, p, B)3</p><p>where G = (V,A) is a directed graph with V = {s = u1, u2, . . . , un = l} the4set of nodes and A the set of arcs, t : A R+ and p : A R+ i.e., each5arc a A is associated with a nonnegative travel time ta and a nonnegative6profit pa, and a nonnegative time budget B, the goal is to find an s l walk7with length at most B with the maximum collected profit from the traversed8</p><p>arcs. An arc may be traversed more than once, however its profit is collected9</p><p>only once. In a similar way, we may define the AOP in undirected graphs.10</p><p>The AOP is the arc routing version of the Orienteering Problem (OP) [7],11</p><p>where the nodes (instead of the arcs) are associated with profit. The algorith-12</p><p>mic research relevant to the OP and its extensions is broad [6, 11]. The OP13</p><p>in directed and undirected graphs is NP-hard [7] and hard to approximate14</p><p>within a factor of 14811480</p><p>[3]. The best approximation algorithm for the OP in15</p><p>undirected graphs is the (2 + )-approximation algorithm given in [4] while16</p><p>in directed graphs is the O( log2 n</p><p>log logn)-approximation algorithm given in [9].17</p><p>Contrary to the OP, limited body of literature deals with the AOP and18</p><p>its extensions. A metaheuristic algorithm for the AOP was proposed by19</p><p>Souffriau et al. [10] while a matheuristic approach and a branch-and-cut20</p><p>technique for the extension of the AOP to multiple tours were proposed21</p><p>in [1] and [2], respectively. In this article we study the AOP in directed22</p><p>and undirected graphs. In Section 2 we give an inapproximability result23</p><p>for the AOP and propose a polylogarithmic approximation algorithm for24</p><p>the problem using the polylogarithmic approximation algorithm for the OP25</p><p>2</p></li><li><p>in [9]. In Section 3 we present a (6 + + o(1))approximation algorithm26for the AOP in undirected graphs and a (4 + )approximation algorithm27for the unweighted version of the problem, using the (2 + )approximation28algorithm for the unweighted OP in [4]. Finally, we obtain approximation29</p><p>algorithms for the Mixed Orienteering Problem (MOP), the extension of both30</p><p>the OP and the AOP, where both nodes and arcs are associated with profit.31</p><p>2. The Arc Orienteering Problem32</p><p>In this section we first obtain an inapproximability result for the AOP.33</p><p>Then, we propose an approximation algorithm for the problem.34</p><p>The OP in directed graphs is reduced to the AOP. Given an OP instance35</p><p>an AOP instance is created as follows: Assign zero profit to each arc and for36</p><p>each node u add a node u, the arcs (u, u), (u, u) of zero travel cost and pu2</p><p>37</p><p>profit and remove us profit. Then, an OP solution yields an equal length38</p><p>and at least the same profit AOP solution and vice versa. Using the results39</p><p>in [7] and [3], we get the following theorem.40</p><p>Theorem 1. The AOP is NP-hard and hard to approximate within 14811480</p><p>.41</p><p>Theorem 2. An f(n)approximation algorithm for the OP in directed graphs,42where n is the number of nodes, yields an f(m+2)approximation algorithm43for the AOP, where m is the number of arcs.44</p><p>Proof. Given an instance of the AOP (G = (V,A), t, p, B), |A| = m, we con-45struct an instance of the OP in the directed network N = (V , A) with V =46</p><p>{s, l}{(u, v) : (u, v) A}, |V | = n = m+2, and A = {(s, (s, u)), ((v, l), l) :47(s, u), (v, l) A} {((u, v), (v, w)) : (u, v), (v, w) A}. The travel times of48</p><p>3</p></li><li><p>the arcs in N are defined as follows: t((u,v),(v,w)) =t(u,v)+t(v,w)</p><p>2, t(s,(s,u)) =</p><p>t(s,u)2</p><p>49</p><p>and t((v,l),l) =t(v,l)2</p><p>. The profit of each node (u, v) equals to p(u,v) = p(u,v) and50</p><p>B = B. Then, a solution of the AOP instance yields a solution of the OP51</p><p>instance of equal total profit and length and vice versa.52</p><p>The theorem yields a O( log2 m</p><p>log logm)approximation algorithm for the AOP,53</p><p>applying Nagarajan and Ravis algorithm [9] to the metric closure of the54</p><p>constructed OP instance and transforming the solution to an AOP solution.55</p><p>3. Approximation Algorithms for the AOP in Undirected Graphs56</p><p>In this section we study the AOP in undirected graphs. Similarly to the57</p><p>previous section the problem is NP-hard and hard to approximate within58</p><p>14811480</p><p>via a reduction from the OP in undirected graphs. We obtain a con-59</p><p>stant factor approximation algorithm for the problem by reducing it to the60</p><p>Unweighted OP (UOP) in undirected graphs, the restriction of the OP with61</p><p>nodes of unit profit. First, we reduce the AOP to the special case with poly-62</p><p>nomially bounded positive integer profits using a similar technique with [4, 8].63</p><p>Lemma 3. A approximation algorithm for the AOP in undirected graphs64with polynomially bounded positive integer profits yields a ( + o(1)) ap-65proximation algorithm for the AOP in undirected graphs.66</p><p>Proof. Given an AOP instance I = (G = (V,E), t, p, B), we construct an67</p><p>instance I = (G = (V , E ), t, p, B) with polynomially bounded positive in-68</p><p>teger profits. First, we guess the edge of highest profit (pmax) in the optimal69</p><p>walk and remove all higher profit edges. Then, we set pe = n3pe</p><p>pmax + 1 for70</p><p>each edge e. A feasible walk W , consisting of the distinct edges e1, e2, . . . , ek71</p><p>4</p></li><li><p>has profit profit(W ) =k</p><p>j=1</p><p>pej in I and profit(W ) =</p><p>k</p><p>j=1</p><p>pej &gt;n3profit(W)</p><p>pmax72</p><p>in I . Hence, OPT &gt; n3</p><p>pmaxOPT, where OPT(OPT) is the optimum in73</p><p>I(I ). On the other hand, profit(W ) =k</p><p>j=1</p><p>pej k</p><p>j=1</p><p>(n3pejpmax</p><p>+ 1) =74n3</p><p>pmaxprofit(W ) profit(W ) m profit(W ) mOPT</p><p>pmax, so profit(W ) 75</p><p>pmaxn3</p><p>profit(W ) mn3OPT. Then, a approximation solution W of I has76</p><p>profit(W ) OPT</p><p>, so profit(W ) 1pmaxn3</p><p>OPT mn3OPT &gt; (1</p><p> m</p><p>n3)OPT.77</p><p>Theorem 4. A approximation algorithm for the UOP in undirected graphs78yields a 3approximation algorithm for the AOP in undirected graphs with79polynomially bounded positive integer profits.80</p><p>Proof. Given an AOP instance, each edge e such that the shortest path81</p><p>from s to l passing through e exceeds the time budget is removed from the82</p><p>graph. Then, we construct an UOP instance by splitting each edge {u, v}83into puv+1 edges as follows: Each node of the AOP instance is a node of the84</p><p>UOP instance (basic node) and for each edge {u, v} of the AOP instance,85the UOP instance includes the auxiliary nodes {u, v}1, {u, v}2, , {u, v}puv86and the edges {u, {u, v}1}, {{u, v}1, {u, v}2}, , {{u, v}puv1, {u, v}puv},87{{u, v}puv , v}. The travel times of {u, {u, v}1} and {{u, v}puv , v} are set to88t{u,v}</p><p>2, while the travel time of the remaining edges is zero. The time budget89</p><p>of the UOP instance is set equal to the time budget of the AOP instance.90</p><p>An AOP solution yields an equal length UOP solution of at least the same91</p><p>profit by replacing each edge {u, v} by the segment (u, {u, v}1, , {u, v}puv , v).92On the other hand, we will show that any UOP solution yields an AOP so-93</p><p>lution of at least a third of the formers profit (pAOP pUOP3 ). A sequence of94nodes (u, {u, v}1, {u, v}2, , {u, v}puv , v) is called an appropriate segment,95</p><p>5</p></li><li><p>i.e. a segment representing the traversal of the edge {u, v} in the AOP in-96stance, while (u, {u, v}1, , {u, v}i1, {u, v}i, {u, v}i1 , {u, v}1, u) is called97an inappropriate segment, i.e. a segment representing the partial traversal98</p><p>of the edge {u, v} of the AOP instance. For each inappropriate segment we99consider that i = puv, otherwise we may extend it to the equal length and100</p><p>higher profit segment with i = puv.101</p><p>In an UOP solution, if pAS(pIS) is the profit gained by the appropriate102</p><p>(resp., inappropriate) segments, then pUOP = pAS+pIS. Note that pAS equals103</p><p>to the number of visited basic nodes plus the number of auxiliary nodes104</p><p>visited in the appropriate segments while pIS equals to the number of auxiliary105</p><p>nodes visited in the inappropriate segments, since any basic node visited in106</p><p>an inappropriate segment has already been counted in pAS. If all segments107</p><p>are appropriate, an AOP solution is obtained replacing the segments by their108</p><p>representing edges. This yields an AOP solution of pAOP &gt;pUOP</p><p>2, since pUOP109</p><p>equals to pAOP plus the number of visited basic nodes, apart from s, l. Hence,110</p><p>pUOP is less than pAOP plus the number of traversed edges, i.e. pUOP &lt; 2pAOP.111</p><p>If however inappropriate segments exist, let IS = {s1, s2, . . . , sk} be the112set of inappropriate segments of the UOP solution. Let also p1, p2, . . . , pk be113</p><p>the profits collected by traversing them (k</p><p>i=1</p><p>pi = pIS), assuming w.l.o.g that114</p><p>p1 p2 pk and t1, t2, . . . , tk be the travel times associated with them.115If p1 pUOP3 , then an AOP solution of pAOP pUOP3 is obtained returning the116shortest path from s to l passing through the edge represented by s1.117</p><p>Otherwise, we shall replace some of the inappropriate segments by their118</p><p>representing edge traversed in both directions, i.e. the inappropriate segment119</p><p>(u, {u, v}1, , {u, v}puv1, {u, v}puv , {u, v}puv1, , {u, v}1, u) shall be re-120</p><p>6</p></li><li><p>placed by the sequence of nodes (u, v, u) in the AOP instance. A subset121</p><p>RS of IS, with</p><p>sjRS2tj </p><p>k</p><p>i=1</p><p>ti =</p><p>sjIStj will be called a replaceable subset,122</p><p>i.e. replacing the appropriate segments by their representing edges and the123</p><p>segments in the replaceable set by their representing edges traversed in both124</p><p>directions, we obtain a feasible AOP solution. RS is a maximal replaceable125</p><p>subset, if inserting a segment (sm / RS) into the set would violate the time126constraint, i.e.</p><p>sjRS2tj+2tm &gt;</p><p>k</p><p>i=1</p><p>ti. Consider a maximal replaceable subset127</p><p>MRS of segments. We distinguish between the following cases: (i) pMRS pIS3128or pAS + pMRS 2pUOP3 , where pMRS is the total profit of segments in MRS.129Then an AOP solution is obtained consisting of the edges represented by130</p><p>the appropriate segments (contributing at least pAS2</p><p>profit) and the sequences131</p><p>that replace the segments in MRS (contributing pMRS profit), with profit at132</p><p>least pAS+pIS3</p><p>= pUOP3</p><p>or greater than pAS+pMRS2</p><p> pUOP3</p><p>, hence pAOP pUOP3 . (ii)133The set of inappropriate segments MRSc = IS\MRS has pMRSc &gt; pUOP3 . Note134that MRSc contains at least two segments, otherwise p1 pMRSc &gt; pUOP3 , a135contradiction. Furthermore, pMRSc &gt;</p><p>2pIS3, hence removing the lowest profit136</p><p>segment sm MRSc, MRSc\{sm} has profit pMRSc\{sm} pMRSc2 &gt; pIS3 . Since137</p><p>sjMRS2tj + 2tm &gt;</p><p>k</p><p>i=1</p><p>ti then</p><p>sjMRSc\{sm}2tj </p><p>pUOP3</p><p>.140</p><p>Using Lemma 3, Theorem 4 and the (2 + )approximation algorithm141for the UOP by Chekuri et al. [4] we obtain a (6 + + o(1))approximation142algorithm for the AOP in undirected graphs with execution time nO(</p><p>12</p><p>).143</p><p>The unweighted AOP (UAOP) is the special case of the AOP with edges144</p><p>7</p></li><li><p>of unit profit. Similarly to Theorem 4, a approximation algorithm for145the UOP in undirected graphs yields a 2approximation algorithm for the146UAOP in undirected graphs. If the optimal UAOP solution contains less than147</p><p>+2 nodes, it is found by exhaustive search since is a constant. Otherwise,148</p><p>OPTUOP OPTUAOP + . Then, an UOP solution of pUOP OPTUOP yields149an UAOP solution of pUAOP pUOP12 OPTUAOP2 by replacing its appropriate150and its half shortest inappropriate segments. Using Chekuri et al.s algorithm151</p><p>[4] we obtain a (4 + )approximation algorithm.152The Mixed Orienteering Problem (MOP) [5, 11] extends both the OP153</p><p>and the AOP, assigning profit to both nodes and arcs. The MOP is reduced154</p><p>to the AOP in a similar way with the reduction from the OP to the AOP,155</p><p>retaining though the arcs profit. As a result, any approximation algorithm156</p><p>for the AOP yields an approximation algorithm for the MOP.157</p><p>Acknowledgment. We thank the anonymous reviewers for their construc-158</p><p>tive comments and for suggesting the inapproximability result for the AOP.159</p><p>This work has been supported by the EU FP7/2007-2013 (DG CONNECT.H5-160</p><p>Smart Cities and Sustainability), under Grant Agreement no. 288094 (project161</p><p>eCOMPASS) and the EU FP7 Collaborative Project MOVESMART (Grant162</p><p>Agreement No 609026).163</p><p>[1] C. Archetti, A. Corberan, I. Plana, J. M. Sanchis, M. G. Speranza, A164</p><p>matheuristic for the team orienteering arc routing problem, Working165</p><p>p...</p></li></ul>

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