On Friedmann’s Subexponential Lower Bound for Zadeh’s Pivot Rule

Abstract

The question whether the Simplex method admits a polynomial time pivot rule remains one of the most important open questions in discrete optimization. Zadeh’s pivot rule had long been a promising candidate, before Friedmann (IPCO, 2011) presented a subexponential instance, based on a close relation to policy iteration algorithms for Markov decision processes (MDPs). We investigate Friedmann’s lower bound construction and exhibit three flaws in his analysis: We show that (a) the initial policy for the policy iteration does not produce the required occurrence records and improving switches, (b) the specification of occurrence records is not entirely accurate, and (c) the sequence of improving switches described by Friedmann does not consistently follow Zadeh’s pivot rule. In this paper, we resolve each of these issues. While the first two issues require only minor changes to the specifications of the initial policy and the occurrence records, the third issue requires a significantly more sophisticated ordering and associated tie-breaking rule that are in accordance with the Least-Entered pivot rule. Most importantly, our changes do not affect the macroscopic structure of Friedmann’s MDP, and thus we are able to retain his original result.

This work is supported by the ‘Excellence Initiative’ of the German Federal and State Governments and the Graduate School CE at TU Darmstadt.

Full version digitally published at the University and State Library Darmstadt, available at http://tuprints.ulb.tu-darmstadt.de/id/eprint/7557.

A Proofs of Selected Statements

This section contains the proofs of the main statements. The proofs use the following two statements whose proofs can be found in [3].

Lemma A.1

Let $i\in \{2,\dots ,n-2\}$ and $l<i$. Then, there is a number $b\in \mathbb {B}_n$ with $\ell (b+1)=l$ such that for all $j\in \{i+2,\dots ,n\}$, $\phi ^{\sigma _b}(k_i,k_{\ell '})<\phi ^{\sigma _b}(k_j,k_{\ell '})$ and $(k_{i},k_{\ell '}),(k_j,k_{\ell '})\in L_{\sigma _b}^3.$

Lemma A.2

Assume that for any transition, the switches that should be applied during phase 3 were applied in some order. Let $i\in \{2,\dots ,n-2\}$ and $l<i$. Then there is a $b\in \mathbb {B}_n$ with $\ell (b+1)=l$ such that $\phi ^{\sigma _b}(k_{i+1},k_{\ell '})<\phi ^{\sigma _b}(b_{i,r}^1,k_{\ell '})$, where $r \in \{0,1\}$ is arbitrary and $(k_{i+1},k_{\ell '}),(b_{i,r}^1,k_{\ell '})\in L_{\sigma _b}^3.$

We now prove the main statements of this paper.

Issue 4

Applying the improving switches as described in [4, Lemma 5] does not obey the Least-Entered pivot rule.

Proof

According to [4, Lemma 5], the switches of phase 3 should be applied as follows (See footnote 2): “[$\dots$] we need to perform all switches from higher indices to smaller indices, and $k_i$ to $k_{\ell '}$ before $b_{i,l}^j$ with $(b+1)_{i+1}\ne j$ or $(b+1)_i=0$ to $k_{\ell '}$”.

Let $i\in \{2,\dots ,n-2\}, l<i$ and $j\in \{i+2,\dots ,n-2\}$. By Lemma A.1, there is a number $b\in \mathbb {B}_n$ such that $l=\ell (b+1)$ and $\phi ^{\sigma _b}(k_i,k_{\ell '})<\phi ^{\sigma _b}(k_j,k_{\ell '})$. In addition, $(k_i,k_{\ell '}),(k_j,k_{\ell '})\in L_{\sigma _b}^3$. Therefore, the switch $(k_j,k_{\ell '})$ should be applied before the switch $(k_i,k_{\ell '})$ during the transition from $\sigma _b$ to $\sigma _{b+1}$ when following the description of [4].

Consider the phase 3 policy $\sigma$ of this transition in which the switch $(k_j,k_{\ell '})$ should be applied. Then, since $j>i$ and we “perform all switches from higher indices to smaller indices”, the switch $(k_i,k_{\ell '})$ was not applied yet. However, it still is an improving switch for the policy $\sigma$. This implies $\phi ^{\sigma _b}(k_j,k_{\ell '})=\phi ^{\sigma }(k_j,k_{\ell '})$ and $\phi ^{\sigma _b}(k_i,k_{\ell '})=\phi ^{\sigma }(k_i,k_{\ell '}).$ Consequently, $\phi ^{\sigma _b}(k_i,k_{\ell '})<\phi ^{\sigma _b}(k_j,k_{\ell '})$ implies that $\phi ^{\sigma }(k_i,k_{\ell '})<\phi ^{\sigma }(k_j,k_{\ell '})$. Thus, since the edge $(k_i,k_{\ell '})$ is an improving switch for $\sigma$ having a lower occurrence record than $(k_j,k_{\ell '})$ and $\sigma$ was chosen as the policy in which $(k_j,k_{\ell '})$ should be applied, the Least-Entered rule is violated.    $\square$

Issue 5

Consider some $b\in \mathbb {B}_n$ and the transition from $\sigma _b$ to $\sigma _{b+1}$. Suppose that the switches of phase 3 are applied “level by level” according to any fixed ordering of the levels as described above. Further suppose that this ordering only depends on $\ell (b+1)$. Then, the Least-Entered pivot rule is violated.

Proof

To prove Issue 5, we show that applying the improving switches as discussed before violates Zadeh’s Least-Entered rule several times by showing the following statement: Let $S^i$ be an ordering of $\{1,\dots ,n\}$ for $i\in \{1,\dots ,n\}$. Suppose that the improving switches of phase 3 of the transition from $\sigma _b$ to $\sigma _{b+1}$ are applied in the order defined by $S^{\ell (b+1)}$ for all $b\in \mathbb {B}_n$. Then, for every possible least significant bit $l\in \{1,\dots ,n-4\}$, assuming that the ordering $S^l$ obeys the Least-Entered rule results in a contradiction.

Fix some $l\in \{1,\dots ,n-4\}$. Consider the ordering $S^l=(s_1,\dots ,s_n)$. For any $k\in \{1,\dots ,n\}$, we denote the position of k within $S^l$ by $k^{\star }$. Towards a contradiction, assume that applying the improving switches level by level according to the ordering $S^{l}$ obeys the Least-Entered rule. We show that this assumption yields $(l+1)^{\star }<(n-1)^{\star }$ and $(n-1)^{\star }<(l+1)^{\star }$.

Let $i\in \{l+1,\dots ,n-2\}.$ Then, $i>l$ and therefore, by Lemma A.2, there is a number $b\in \mathbb {B}_n$ with $\ell (b+1)=\ell '=l$ and $\smash {\phi ^{\sigma _b}(k_{i+1},k_{\ell '})<\phi ^{\sigma _b}(b_{i,r}^1,k_{\ell '})}$ such that $\smash {(k_{i+1},k_{\ell '}),(b_{i,r}^0,k_{\ell '})\in L_{\sigma _b}^3}$. Thus, both switches need to be applied during the transition from $\sigma _b$ to $\sigma _{b+1}$. Because of $\smash {\phi ^{\sigma _b}(k_{i+1},k_{\ell '})<\phi ^{\sigma _b}(b_{i,r}^1,k_{\ell '})}$, level $i+1$ needs to appear before level i within the ordering $S^l$. Since this argument can be applied for all $i\in \{l+1,\dots ,n-2\}$, the sequence

$$\begin{aligned} (n-1,n-2,\dots ,l+1) \end{aligned}$$

needs to be a (not necessarily consecutive) subsequence of $S^l$. In particular, $(n-1)^{\star }<(l+1)^{\star }$ since $l+1\ne n-1$ by assumption.

Let $i=l+1$ and $j\in \{i+2,\dots ,n\}$. Then, by Lemma A.1, there is a number $b\in \mathbb {B}_n$ with $\ell (b+1)=l$ such that $\phi ^{\sigma _b}(k_i,k_{\ell '})<\phi ^{\sigma _b}(k_{i+2},k_{\ell '})$ and $(k_i,k_{\ell '}),(k_{i+2},k_{\ell '})\in L_{\sigma _b}^3$. Now, both switches need to be applied during the transition from $\sigma _b$ to $\sigma _{b+1}$. Therefore, for all $i\in \{l+1,\dots ,n-2\}$, level i needs to appear before any of the levels level $j\in \{i+2,\dots ,n\}$ within $S^l$. But this implies that the sequence

$$\begin{aligned} (l+1,l+3,l+4,\dots ,n-1,n) \end{aligned}$$

needs to be a (not necessarily consecutive) subsequence of $S^l$. In particular, $(l+1)^{\star }<(n-1)^{\star }$ since $n-1\ge l+3$ as we have $l\le n-4$ by assumption. This however contradicts $(n-1)^{\star }<(l+1)^{\star }$.    $\square$

Theorem 3

There is an ordering of the improving switches and an associated tie-breaking rule compatible with the Least-Entered pivot rule such that all improving switches contained in $L_{\sigma _b}^3$ are applied and the Least-Entered pivot rule is obeyed during phase 3.

Proof

Let $\sigma$ denote the first phase 3 policy of the transition from $\sigma _b$ to $\sigma _{b+1}$. Then, $L_{\sigma }^3=L_{\sigma _b}^3$. By Lemma 5, there is an edge $e_1\in L_{\sigma }^3$ minimizing the occurrence record $I_\sigma$. Applying this switch results in a new phase 3 policy $\sigma [e_1]$ such that $L_{\sigma [e_1]}^3=L_{\sigma }^3\setminus \{e_1\}$. Now, again by Lemma 5, there is an edge $e_2\in L_{\sigma [e_1]}^3$ minimizing the occurrence record $I_{\sigma [e_1]}$.

We can now apply the same argument iteratively until we reach a phase 3 policy $\hat{\sigma }$ such that $\left| L_{\hat{\sigma }}^3\right| =1$ while only applying switches contained in $L_{\sigma _b}^3$. Then, by construction and by Lemma 5, $(e_1,e_2,\dots )$ defines an ordering of the edges of $L_{\sigma _b}^3$ and an associated tie-breaking rule that always follow the Least-Entered rule. When the policy $\hat{\sigma }$ with $\left| L_{\hat{\sigma }}^3\right| =1$ is reached, applying the remaining improving switch results in a phase 4 policy. Then, all improving switches contained in $L_{\sigma _b}^3$ were applied and the Least-Entered pivot rule was obeyed.    $\square$