Parameterized Complexity of Finding Dissimilar Shortest Paths

TL;DR

This work studies the Dissimilar Shortest Paths problem, seeking $k$ pairwise dissimilar $s$-$t$ shortest paths by enforcing a minimum symmetric-difference distance $d$ between paths. It delivers a deterministic fixed-parameter algorithm with running time $2^{O(3^k d k^2)}n^{O(1)}$, using a hybrid approach that combines preprocessing on shortest-path DAGs, greedy path construction, ball-based partitioning, color-coding, and dynamic programming. The paper also proves strong hardness results: W[1]-hardness parameterized by $k$ (even on DAGs with pathwidth at most $4$) and paraNP-hardness parameterized by $d$, highlighting the necessity of jointly parameterizing both $k$ and $d$ for tractability. The results contrast with recent randomized approaches and illuminate the combinatorial structure of diverse shortest-path families via structured colorings and bypass decompositions. Together, these findings advance our understanding of when diverse shortest-path solutions can be efficiently computed in directed graphs.

Abstract

We consider the problem of finding ``dissimilar'' $k$ shortest paths from $s$ to $t$ in an edge-weighted directed graph $D$, where the dissimilarity is measured by the minimum pairwise Hamming distances between these paths. More formally, given an edge-weighted directed graph $D = (V, A)$, two specified vertices $s, t \in V$, and integers $d, k$, the goal of Dissimilar Shortest Paths is to decide whether $D$ has $k$ shortest paths $P_1, \dots, P_k$ from $s$ to $t$ such that $|A(P_i) \mathbin{\triangle} A(P_j)| \ge d$ for distinct $P_i$ and $P_j$. We design a deterministic algorithm to solve Dissimilar Shortest Paths with running time $2^{O(3^kdk^2)}n^{O(1)}$, that is, Dissimilar Shortest Paths is fixed-parameter tractable parameterized by $k + d$. To complement this positive result, we show that Dissimilar Shortest Paths is W[1]-hard when parameterized by only $k$ and paraNP-hard parameterized by $d$.

Figures (10)

• Figure 1: The figure illustrates a bypass $B$ consisting of red and blue paths from $v_1$ to $v_\ell$. The arc set of $B$ can be decomposed into three minimal bypasses $B_1$, $B_2$, and $B_3$.
• Figure 2: The figure illustrates the graph $D$. The square boxes represent directed paths of some length. All arcs are directed from left to right.
• Figure 3: The left figure depicts the edge gadget ${\rm gad}(e)$. There are eight special vertices, called input and output gates, which are depicted as triangles. For aesthetic purposes, we rather use the right figure, which is obtained by rearranging auxiliary gates.
• Figure 4: The three paths $P_a$, $P_v$, and $P$ are illustrated as red, blue, and green lines.
• Figure 5: A partial construction of the graph $D$. There are four layers (separated by dotted lines). If $E_c$ contains an edge $e = \{v_i, v_j\}$, the layer $L_c$ contains the edge gadget ${\rm gad}(e)$ and the paths corresponding to $v_i$ and $v_j$ enter ${\rm gad}(e)$ at its vertex-input gates and leave at its vertex-output gates. All arcs are directed from left to right.

Theorems & Definitions (36)

• theorem 1
• theorem 2
• theorem 3
• proof
• theorem 4: NaorSS95:FOCS:Splitters
• lemma 1
• lemma 2
• lemma 3
• proof
• lemma 4