The Steiner minimal tree (SMT) problem is an NP-hard problem, which is the best connection model for a multi-terminal net in global routing problem. This paper presents a unified algorithm for octagonal and rectilinear SMT construction based on hybrid transformation strategy (HTS) and self-adapting particle swarm optimization. Firstly, an effective HTS is proposed to enlarge the search space and improve the convergence speed. Secondly, the proposed HTS in the evolutionary process may produce an ineffective solution, and consequently the crossover and mutation operators of genetic algorithm (GA) based on union-find sets is proposed. Thirdly, a self-adapting strategy that can adjust the acceleration coefficients is proposed to further improve the convergence and the quality of the proposed algorithm. Finally, the hybrid transformation can be applied to GA and the proposed algorithm can be applied to rectilinear architecture. To our best knowledge, the proposed algorithm is the first unified algorithm to solve the SMT construction under both octagonal and rectilinear architecture. The experimental results show that the proposed algorithm can efficiently provide a better solution for SMT problem both in octagonal and rectilinear architectures than others. Moreover, the algorithm can obtain several topologies of SMT, which is beneficial for optimizing congestion in VLSI global routing stage.

A unified algorithm based on HTS and self-adapting PSO for the construction of octagonal and rectilinear SMT

For some $\epsilon > 10^{-36}$ we give a $3/2-\epsilon$ approximation algorithm for metric TSP.

A (Slightly) Improved Approximation Algorithm for Metric TSP

In advanced technology nodes, detailed routing becomes the most complicated and runtime consuming stage. To spur detailed routing research, ISPD 2018 initial detailed routing contest is hosted and it is the first ISPD contest on detailed routing problem. In this contest, the benchmarks synthesized by industrial tool and library are released, which consider the design rules like spacing table, cut spacing, end-of-line spacing, and min-area rules. In addition, the global routing guide is provided associated to each benchmark, and detailed routers are required to honor the routing guides as much as possible meanwhile minimize design-rule-checking (DRC) violations. The biggest benchmark released in this contest has near-millions of nets, so the runtime and memory scalability for detailed routers need to be well addressed. To reduce routers' runtime, the deterministic multithreading framework is encouraged but optional in this contest.

ISPD 2018 Initial Detailed Routing Contest and Benchmarks

In this paper, we study the approximability of the metric Traveling Salesman Problem (TSP) and prove new explicit inapproximability bounds for that problem. The best up to now known hardness of approximation bounds were 185/184 for the symmetric case (due to Lampis) and 117/116 for the asymmetric case (due to Papadimitriou and Vempala). We construct here two new bounded occurrence CSP reductions which improve these bounds to 123/122 and 75/74, respectively. The latter bound is the first improvement in more than a decade for the case of the asymmetric TSP. One of our main tools, which may be of independent interest, is a new construction of a bounded degree wheel amplifier used in the proof of our results.

New Inapproximability Bounds for TSP.

We show that there is a polynomial-time algorithm with approximation guarantee 3/2+e for the s-t-path TSP, for any fixed e > 0. It is well-known that Wolsey’s analysis of Christofide algorithm also works for the s-t-path TSP with its natural LP relaxation, except for the narrow cuts (in which the LP solution has a value less than two). A fixed optimum tour has either a single edge in a narrow cut (then call the edge and the cut lonely) or at least three (then call the cut busy). Our algorithm “guesses” (by dynamic programming) lonely cuts and edges. Then, we partition the instance into smaller instances and strengthen the LP, requiring a value of at least three for busy cuts. By setting up a k-stage recursive dynamic program, we can compute a spanning tree (V,S) and an LP solution y such that (½+O(2−k))y is in the T-join polyhedron, where T is the set of vertices whose degree in S has the wrong parity.

Approaching 3/2 for the s-t-path TSP

We show how to incorporate global static timing constraints into global routing. Our approach is based on the min–max resource sharing model that proved successful for global routing in theory and practice. Static timing constraints are modeled by a linear number of additional resources and customers. The algorithm dynamically adjusts delay budgets and can, thus, tradeoff wiring congestion for delay. As a subroutine, the algorithm routes a single net. If this subroutine is near-optimal, we will find near-optimal solutions for the overall problem very efficiently. The approach works for many delay models; here we discuss a linear delay model (before buffering) and the Elmore delay model (after buffering). We demonstrate the benefit of our timing-constrained global routing algorithm by experimental results on industrial chips.

Global Routing With Timing Constraints

We revisit the constant-factor approximation algorithm for the asymmetric traveling salesman problem by Svensson, Tarnawski, and Vegh. We improve on each part of this algorithm. We avoid the reduction to irreducible instances and thus obtain a simpler and much better reduction to vertebrate pairs. We also show that a slight variant of their algorithm for vertebrate pairs has a much smaller approximation ratio. Overall we improve the approximation ratio from $506$ to $22+\epsilon$ for any $\epsilon > 0$. This also improves the upper bound on the integrality ratio from $319$ to $22$.

An improved approximation algorithm for ATSP

We present a black-box reduction from the path version of the Traveling Salesman Problem (Path TSP) to the classical tour version (TSP). More precisely, we show that given an α-approximation algorithm for TSP, then, for any e >0, there is an (α+e)-approximation algorithm for the more general Path TSP. This reduction implies that the approximability of Path TSP is the same as for TSP, up to an arbitrarily small error. This avoids future discrepancies between the best known approximation factors achievable for these two problems, as they have existed until very recently. A well-studied special case of TSP, Graph TSP, asks for tours in unit-weight graphs. Our reduction shows that any α-approximation algorithm for Graph TSP implies an (α+e)-approximation algorithm for its path version. By applying our reduction to the 1.4-approximation algorithm for Graph TSP by Sebő and Vygen, we obtain a polynomial-time (1.4+e)-approximation algorithm for Graph Path TSP, improving on a recent 1.497-approximation algorithm of Traub and Vygen. We obtain our results through a variety of new techniques, including a novel way to set up a recursive dynamic program to guess significant parts of an optimal solution. At the core of our dynamic program we deal with instances of a new generalization of (Path) TSP which combines parity constraints with certain connectivity requirements. This problem, which we call Φ-TSP, has a constant-factor approximation algorithm and can be reduced to TSP in certain cases when the dynamic program would not make sufficient progress.

Reducing path TSP to TSP

We consider the Connectivity Augmentation Problem (CAP), a classical problem in the area of Survivable Network Design. It is about increasing the edge-connectivity of a graph by one unit in the cheapest possible way. More precisely, given a $k$-edge-connected graph $G=(V,E)$ and a set of extra edges, the task is to find a minimum cardinality subset of extra edges whose addition to $G$ makes the graph $(k+1)$-edge-connected. If $k$ is odd, the problem is known to reduce to the Tree Augmentation Problem (TAP) -- i.e., $G$ is a spanning tree -- for which significant progress has been achieved recently, leading to approximation factors below $1.5$ (the currently best factor is $1.458$). However, advances on TAP did not carry over to CAP so far. Indeed, only very recently, Byrka, Grandoni, and Ameli (STOC 2020) managed to obtain the first approximation factor below $2$ for CAP by presenting a $1.91$-approximation algorithm based on a method that is disjoint from recent advances for TAP. 
We first bridge the gap between TAP and CAP, by presenting techniques that allow for leveraging insights and methods from TAP to approach CAP. We then introduce a new way to get approximation factors below $1.5$, based on a new analysis technique. Through these ingredients, we obtain a $1.393$-approximation algorithm for CAP, and therefore also TAP. This leads to the currently best approximation result for both problems in a unified way, by significantly improving on the above-mentioned $1.91$-approximation for CAP and also the previously best approximation factor of $1.458$ for TAP by Grandoni, Kalaitzis, and Zenklusen (STOC 2018). Additionally, a feature we inherit from recent TAP advances is that our approach can deal with the weighted setting when the ratio between the largest to smallest cost on extra links is bounded, in which case we obtain approximation factors below $1.5$.

Vera Traub

Papers

Approaching 3/2 for the s-t-path TSP

Global Routing With Timing Constraints

An improved approximation algorithm for ATSP

Reducing path TSP to TSP

Bridging the Gap Between Tree and Connectivity Augmentation: Unified and Stronger Approaches.