Combining Approximation Algorithms for the Prize-Collecting TSP

Com bining Appro ximation Algorithms for the Prize-Collecting TSP Mic hel X. Go emans ∗ Abstract W e present a 1 . 9 1457-approximation algorithm for the prize-co llecting tra velling salesman problem. This is obtained b y combining a randomized v aria nt of a rounding algorithm of Biensto c k et al. [2] and a primal-dua l alg orithm of Go emans a nd Williamso n [5 ]. 1 In tro duction In the prize-collec ting tr a v el ling salesman pr oblem (PC-TS P), we are giv en a verte x set V (with | V | = n ), a metric c on V × V ( i.e. c satisfies (i) c ij = c j i ≥ 0 for all i, j ∈ V and (ii) triangle inequalit y: c ij + c j k ≥ c ik for all i, j, k ∈ V ), a sp ecial vertex r ∈ V (the dep ot), p enalties π : V → R + , and th e goal is to find a cycle T with r ∈ V ( T ) suc h that c ( T ) + π ( V \ V ( T )) is minimized, where c ( T ) = P ( i,j ) ∈ T c ij , π ( S ) = P i ∈ S π i , and V ( T ) denotes the vertice s spanned b y T . The fi rst constant app ro ximatio n algo rithm for PC-T S P was giv en b y Biensto c k et al. [2 ]. It is based on roud ing the optim um solution to a n at ural LP relaxation for the p r oblem, and p ro vides a p erformance guarantee of 2 . 5. G o emans and Williamson [5] hav e d esig ned a pr im al-du al algorithm based on t he same LP relaxa tion, an d this giv es a 2-approximat ion algo rithm for the problem. In 1998, Go emans [4] has sh own that a simple impro ve ment of the algorithm of Biensto c k et al. giv es a guarant ee of 2 . 055 · · · = 1 1 − e − 2 / 3 . Rece ntly , Arc her et al. [1] are the fi rst to break the barrier of 2 and p ro vide an impro v emen t of the prim al-du al alg orithm of Go ema ns and Williamson; their p erformance guarante e is 1 . 990283. I n th is note, we s ho w that by com bining the round ing algorithm of Biensto c k et al. and the prim al-du al algorithm of Goemans and Willi amson, we c an obtain a guaran tee of 1 . 9145 6 · · · = 1 1 − 2 3 e − 1 / 3 . The analysis uses the tec hnique in [4] together with an impro v ed analysis of the primal-dual algorithm as observ ed in [3] and used in Arc her et al. [1]. ∗ MIT Departmen t of Mathematics. goemans@m ath.mit.edu . Sup ported b y NS F contract CCF-0829878 and b y ONR gran t N 00 014-05-1-0148. 1 2 Com bining Appro ximation Algorithms W e start by b riefly reviewing the roun ding result of Bienstock et al. [2]. Consider a classical LP relaxation of P C-TSP: Min X e ∈ E c e x e + X v π ( v )(1 − y v ) sub ject to: x ( δ ( v )) = 2 y v v ∈ V \ { r } ( LP ) x ( δ ( S )) ≥ 2 y v S ⊂ V , r / ∈ S, v ∈ S 0 ≤ x e ≤ 1 e ∈ E 0 ≤ y v ≤ 1 v ∈ V y r = 1 , where E d enote s the edge se t of the c omplete g raph on E . F or concise ness, w e u se c ( x ) + π ( 1 − y ) to denote the ob jectiv e function of th is LP . Let x ∗ , y ∗ b e an optim um solution of th is LP r ela xation, and let LP = c ( x ∗ ) + π ( 1 − y ∗ ) denote its v alue. Biensto c k et a l. [2] sho w the follo wing (based on the analysis of Christofides’ algorithm due to W olsey [8] and S hmo ys and Williamson [7]). Prop osition 1 (Biensto c k et al.) . L et 0 < γ ≤ 1 and let S ( γ ) = { v : y ∗ v ≥ γ } . L et T γ denote the cycle on S ( γ ) output by Christofides’ algorithm when given S ( γ ) as vertex set. Then: c ( T γ ) ≤ 3 2 γ c ( x ∗ ) . The 2.5-appro ximation algorithm can th en b e deriv ed b y setting γ = 3 5 since we g et c ( T 3 / 5 ) ≤ 5 2 c ( x ∗ ) and π ( V \ S (3 / 5)) ≤ 5 2 π ( 1 − y ∗ ). I n [4], w e hav e shown that one can get a b etter p erformance guaran tee b y ta king the best cycle outpu t o v er all p ossible v a lues of γ ; notic e that this leads to at most n − 1 differen t cycles. The p rimal-dual algorithm in [5] constru ct s a cycle T and a du al solution to the linear p ro- gramming r ela xation ab o v e suc h that their v al ues are within a factor 2 of eac h other, sh o wing a p erformance guaran tee of 2 since the v alue of an y d ual s ol ution is a lo w er b ound on L P . C h udak, Roughgarden and Williamson [3] (see th eir Th eorem 2.1) observ e that the analysis of [5] actually sho ws a str onge r guaran tee on the p enalt y side of the ob jectiv e f unction, n amely that the cycle T returned satisfies: c ( T ) +  2 − 1 n − 1  π ( V \ V ( T )) ≤  2 − 1 n − 1  LP . (1) This increased factor on the p enalt y sid e is exploited in Ar c her et al. [1], and this m ot iv ate d the result in this n ote. Su pp ose no w that w e apply the pr imal- dual algorithm to an in s ta nce in wh ic h w e replace th e p enalties π ( · ) by π ′ ( · ) giv en by π ′ ( v ) = 1 2 − 1 / ( n − 1) π ( v ) . (2) Th us, (1) im p lies that the cycle T return ed satisfies: c ( T ) + π ( V \ V ( T )) ≤  2 − 1 n − 1  LP ′ , (3) 2 where LP ′ denotes the LP v alue for the p enalties π ′ ( · ). As th e optim um solution x ∗ , y ∗ of LP (with p enalties π ( · )) is feasible for the linear pr og ramming relaxation w ith p enalties π ′ ( · ), we derive that the cycle T pd output satisfies: c ( T pd ) + π ( V \ V ( T pd )) = c ( T pd ) +  2 − 1 n − 1  π ′ ( V \ V ( T pd )) ≤  2 − 1 n − 1  LP ′ ≤  2 − 1 n − 1   c ( x ∗ ) + π ′ ( 1 − y ∗ )  =  2 − 1 n − 1  c ( x ∗ ) + π ( 1 − y ∗ ) . Summarizing: Prop osition 2. The primal-dual algorithm applie d to an instanc e with p enalties π ′ ( · ) give n by (2 ) outputs a c yc le T pd such that c ( T pd ) + π ( V \ V ( T pd )) ≤ 2 c ( x ∗ ) + π ( 1 − y ∗ ) . W e claim that th e b est of the algorithms giv en in Prop ositions 1 and 2 give s a b etter than 2 appro ximation guarantee f or PC-TSP . Theorem 3. L et H = min(min γ ( c ( T γ ) + π ( V \ V ( γ ))) , c ( T pd ) + π ( V \ V ( T pd ))) . Then H ≤ α ( c ( x ∗ ) + π ( 1 − y ∗ )) = αLP , wher e α = 1 1 − 2 3 e − 1 / 3 < 1 . 9145 7 . As men tioned earlier, the min imum in the theorem inv olv es only n d ifferen t algorithms as we need only to consider v alues γ equal to some y ∗ v . Pr o of. W e constr u ct an app ropriate probabilit y distribution ov e r all the algorithms inv olv ed such that the exp ected cost of the solution pro duced is at most α ( c ( x ∗ ) + π ( 1 − y ∗ )). First, assume that w e select γ rand omly (according to a certain distribution to b e sp ecified). Then, by Prop osition 1, we hav e that E [ c ( T γ )] ≤ 3 2 E  1 γ  c ( x ∗ ) , while the exp ected p enalt y we hav e to pay is E [ π ( V \ V ( γ ))] = X v ∈ V P r [ γ > y ∗ ( v )] π ( v ) . 3 Th us, th e ov erall exp ected cost is: E [ c ( T γ ) + π ( V \ V ( γ ))] ≤ 3 2 E  1 γ  c ( x ∗ ) + X v ∈ V P r [ γ > y ∗ ( v )] π ( v ) . (4) Assume now that γ is c hosen uniformly b et w een a = e − 1 / 3 = 0 . 7165 3 · · · and 1. Then, E  1 γ  = Z 1 a 1 1 − a 1 x dx = − ln( a ) 1 − a = 1 3(1 − a ) = 1 3(1 − e − 1 / 3 ) , and P r [ γ > y ] = ( 1 − y 1 − a a ≤ y ≤ 1 1 ≤ 1 − y 1 − a 0 ≤ y ≤ a. Therefore, (4) b ecomes: E [ c ( T γ ) + π ( V \ V ( γ ))] ≤ 1 2(1 − e − 1 / 3 ) c ( x ∗ ) + 1 1 − e − 1 / 3 π ( 1 − y ∗ ) . (5) Supp ose we n o w s elect, with probabilit y p , the primal-dual algorithm as giv en in Prop osition 2 or, w ith probabilit y 1 − p , the rounding algorithm with γ chose n randomly ac ording to γ ∼ U [ e − 1 / 3 , 1]. F rom (5) and Prop osition 2, w e get that the exp ected cost E ∗ of the r esu lting alg orithm satisfies: E ∗ ≤  2 p + (1 − p ) 1 2(1 − e − 1 / 3 )  c ( x ∗ ) +  p + (1 − p ) 1 1 − e − 1 / 3  π ( 1 − y ∗ ) . Cho osing p = (1 − p ) 1 2(1 − e − 1 / 3 ) , i.e. p = 1 3 − 2 e − 1 / 3 , w e get E ∗ ≤ 3 p ( c ( x ∗ ) + π ( 1 − y ∗ )) = 3 pLP . Therefore, the b est of the algorithms in v olv ed outpu ts a solution of cost at most 3 pLP = αLP where α = 1 1 − 2 3 e − 1 / 3 < 1 . 9145 7 . One can sh o w that the pr obabilit y distribution giv en in the pr oof is optimal for the p urp ose of this pro of; this is left as an exercise f or th e reader. Theorem 3 shows that the linear programming r ela xation of PC-TSP has an inte gralit y ga p b ounded by 1 . 9145 7; in contrast, the result of Archer et al. [1 ] do es not imply a b etter than 2 b ound on the int egralit y gap. As a final remark, if we replace C hristofides’ algorithm with an algorithm f or the symmetric TSP that outpu ts a solution within a factor β of th e standard L P relaxation for the T S P then the approac h d escribed in this note giv es a guarante e of 1 1 − 1 β e 1 − 2 /β for PC-TSP . 4 References [1] A. Arc her, M. Bateni, M. Ha jiagha yi and H. 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