A Constant-Factor Approximation for Directed Latency

TL;DR

The paper resolves the open question of whether Directed Latency admits a constant-factor approximation in polynomial time. It introduces a novel bucketing mechanism and a strengthened time-indexed LP with tour-intervals and roots, enabling tight control over how per-bucket subsolutions are assembled. A rounding framework leveraging ATSPP relaxations and directed splitting off yields a global, constant-factor approximation, significantly advancing beyond prior O(log n) results and matching the quasi-polynomial approach in spirit but achieving polynomial-time performance. The methods have potential implications for related directed routing and latency problems where asymmetric metrics complicate decompositions and rounding.

Abstract

In the Directed Latency problem, we are given an asymmetric metric on a set of vertices (or clients), and a given depot $s$. We seek a path $P$ starting at $s$ and visiting all the clients so as to minimize the sum of client waiting times (also known as latency) before being visited on the path. In contrast to the symmetric version of this problem (also known as the Deliveryperson problem and the Repairperson problem in the literature), there are significant gaps in our understanding of Directed Latency. The best approximation factor has remained at $O(\log n)$, where $n$ is the number of clients, for more than a decade [Friggstad, Salavatipour, and Svitkina, '13]. Only recently, [Friggstad and Swamy, '22] presented a constant-factor approximation but in quasi-polynomial time. Both results follow similar ideas: they consider buckets with geometrically-increasing distances, build paths in each bucket, and then stitch together all these paths to get a feasible solution. [Friggstad and Swamy, '22] showed if we guess a vertex from each bucket and augment a standard LP relaxation with these guesses, then one can reduce the stitching cost. Unfortunately, there are logarithmically many buckets so the running time of their algorithm is quasi-polynomial. In this paper, we present the first constant-factor approximation for Directed Latency in polynomial time by introducing a completely new way of bucketing which helps us strengthen a standard LP relaxation with less aggressive guessing. Although the resulting LP is no longer a relaxation of Directed Latency, it still admits a good solution. We present a rounding algorithm for fractional solutions of our LP, crucially exploiting the way we restricted the feasibility region of the LP formulation.

Figures (3)

• Figure 1: (a) We have a pair of vertices $u,v$ with $c_{\mathop{\rm OPT}\nolimits}(u)\in I_i$, $c_{\mathop{\rm OPT}\nolimits}(v)\in I_{j+1}$, $i<j$, and $c(v,u)\leq c(u,v)$. Let us call such pair a bad pair. Then $(v,u)$ with the subpath of $\mathop{\rm OPT}\nolimits$ shown in thick black color create a cycle of cost at most $2 c_{\mathop{\rm OPT}\nolimits}(v)\leq 2 t_{j+1}$ that spans all vertices visited in the time interval $I_j$. (b) We relax threshold of $I_j$ to be $t_j +t_{j+1}$ and we add $j$ to $A_{\textnormal{tour}}$. Then, we can visit $v$ in the new time interval $I_j$, and if we select a bad pair $u,v$ such that $c_{\mathop{\rm OPT}\nolimits}(v)$ is the maximum possible value, then \ref{['item:ordering']} is satisfied. Moreover, we choose $((v,t),(v',t')) \in E[G_T]$ joining $v$ and $v'$ (perhaps with $t'-t > c(v,v')$) that delays the visit of vertices coming after $v$ in $\mathop{\rm OPT}\nolimits$ so they are visited in the next time interval. (c) By adding the dashed edges, we can enforce to enter and leave $I_j$ via $v^*_j$. It is easy to see that the cost of all the added edges can be bounded by $O(1) t_{j+1}$. That is why we need to further relax the thresholds.
• Figure 2: In this example, we have $n=2^9-1$. The $s,s'$-path is $\mathop{\rm OPT}\nolimits$. The buckets are indicated with their first and last vertices. The number besides the semi-circles show the distances from $s$. In this example, we then have $\mathcal{G}^*_1:=\{1,2\}, \mathcal{G}^*_2:=\{3\}, \mathcal{G}^*_3:=\{4,5,6\}, \mathcal{G}^*_4:=\{7\}, \mathcal{G}^*_5:=\{8\}, \mathcal{G}^*_6:=\{9\}$. Also, we have $\bar{\ell}_{\max}(\mathcal{G}^*_1)=\bar{\ell}_{\max}(\mathcal{G}^*_2)=2^{i_1}$, $\bar{\ell}_{\max}(\mathcal{G}^*_3)=\bar{\ell}_{\max}(\mathcal{G}^*_4)=2^{i_2}$, $\bar{\ell}_{\max}(\mathcal{G}^*_5)=\bar{\ell}_{\max}(\mathcal{G}^*_6)=2^{i_3}$.
• Figure 3: (a) shows the construction of walk $P$ in the proof of \ref{['thm:existence']} up to $(i-1)$'th time interval with red color. Here we have both $i-1$ and $i$ are in $A^*_{\textnormal{tour}}$. The part of $\mathop{\rm OPT}\nolimits$ not used so far is in black. The dashed edges are the added edges as in the proof. The edges of $\mathop{\rm OPT}\nolimits$ are shown with a directed line segment, meanwhile the undirected line segments are a subpath of $\mathop{\rm OPT}\nolimits$ (e.g., the line segment between $u^*_i$ and $w^*_i$). (b) shows the final walk $P$ constructed up to $i$'th time interval. Note that $I^*_i$ is changed, i.e., the threshold $t^*_i$ increased so that the embedding of $P$ in $G_T$ satisfy the constraints of \ref{['eq:refined-LP']}.

Theorems & Definitions (36)

• Theorem 1
• Corollary 2
• Definition 3
• Theorem 4
• Theorem 5
• Definition 6
• Theorem 7
• Theorem 8
• Definition 9
• Lemma 10